STUDIES IN SOLITON BEHAVIOUR

I I M I S - m f — 1 0 4 7 1

PETER CORNELIS SCHUUR

HOW, IN FRAMES AT REST

THE TAU GOES WEST

WHILE THE EAST IS WON

BV THE SÖLITOM

STUDIES IN SOLITON BEHAVIOUROMTRENT SOLITONGEDRAG

(MET EEN SAMENVATTING IN HET NEDERLANDS)

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR INDE WISKUNDE EN NATUURWETENSCHAPPEN AAN DERIJKSUNIVERSITEIT TE UTRECHT, OP GEZAG VANDE RECTOR MAGNIFICUS PROF. DR. O.J. DE JONG,VOLGENS BESLUITVAN HET COLLEGE VAN DECANENIN HET OPENBAAR TE VERDEDIGEN OP MAANDAG

9 SEPTEMBER 1985 DES NAMIDDAGS TE 4.15 UUR

DOOR

PETER CORNELIS SCHUUR

GEBOREN OP 31 JANUARI 1950 TE ZWOLLE

DRUKKERIJ ELINKWIJK BV - UTRECHT

PROMOTOR : PROF. DR. IR. W. ECKHAUSCOPROMOTOR: DR. A. VAN HARTEN

TABLE OF CONTENTS

PREFACE

1. Historical remarks 12. 1ST for KdV: the gist of the method 53. Asymptotics for nonzero reflection coefficient:

main purpose of the thesis 74. Brief description of the contents 8References 1 1

ACKNOWLEDGEMENTS 12

CHAPTER 1 : THE EMERGENCE OF SOLITONS OF THE KORTEWEG-DE VRIES

EQUATION FROM ARBITRARY INITIAL CONDITIONS

1 . Introduction 132. Formulation of the problem 153. Analysis of flc and T(- 174. Solution of the Gel'fand-Levitan equation 205. Decomposition of the solution and estimates 216. Analysis of S and gj 24Appendix A: Case C 24Appendix B: Case P 26Appendix C: Generalization to higher KdV equations 29References 33

CHAPTER 2 : ASYMPTOTIC ESTIMATES OF SOLUTIONS OF THE KORTEWEC-DE VRIES

EQUATION ON RIGHT HALF LINES SLOWLY MOVING TO THE LEFT

1. Introduction 342. Preliminaries and statement of the problem

2.1 Direct scattering at t = 0 382.2 Inverse scattering for t > 0 432.3 Statement of the problem 45

3. Analysis of fic and Tc 464. Solution of the KdV initial value problem

in the absence of solitons 605. The operator (I+Td)~i 636. Solution of the Gel'fand-Levitan equation

in the presence of bound states 687. Decomposition of the solution of the KdV problem

when the initial data generate solitons 69References 76

CHAPTER 3 : MULTISOLITON PHASE SHIFTS FOR THE KORTEWEG-DE VRIES

EQUATION IN THE CASE OF A NONZERO REFLECTION COEFFICIENT

1. Introduction 782. Scattering data and their properties 803. Forward and backward asymptotics 824. An explicit phase shift formula 845. An example: the continuous phase shifts

arising from a sech initial function 86References 88

CHAPTER 4 : ON THE APPROXIMATION OF A REAL POTENTIAL IN THE

ZAKHAROV-SHABAT SYSTEM BY ITS REFLECTIONLESS PART

1 . Introduction 892. Construction and properties of the scattering data 903. Simplification of the inverse scattering algorithm 964. Statement of the main result 985. Auxiliary results ]006. Proof of theorem 4.1 105References 107

CHAPTER 5 : DECOMPOSITION AND ESTIMATES OF SOLUTIONS OF THE MODIFIED

KORTEWEG-DE VRIES EQUATION ON RIGHT HALF LINES SLOWLY

MOVING LEFTWARD 109

1. Introduction 1102. Some comments on the asymptotic structure of the

reflectionless part ]|33. Two useful results obtained previously lib4. Estimates of fic(x+y;t) 1175. Estimates of q(x,t)-qd(x,t) 119References 122

CHAPTER 6 : MULTISOLITOW PHASE SHIFTS FOR THE MODIFIED KORTEWEG-DE VRIES

EQUATION IN THE CASE OF A NONZERO REFLECTION COEFFICIENT

1. Introduction 1242. Left and right scattering data and their relationship 1263. Forward and backward asymptotics 1284. An explicit phase shift formula 1305. An example: the continuous phase shifts arising

from a sech initial function 132References 135

CHAPTER 7 : ASYMPTOTIC ESTIMATES OF SOLUTIONS OF THE SINE-GORDON

EQUATION ON RIGHT HALF LTNES ALMOST LINEARLY MOVING

LEFTWARD

1. Introduction 1362. The asymptotic structure of the reflectionless part 1403. A useful result obtained earlier 143

4. Estimates of £2c(x+y;t) 1445. Estimates of q(x,t)-qd(x,t) and q(x,t) 147References 149

CHAPTER 8 : ON THE APPROXIMATION OF A COMPLEX POTENTIAL IN THE

ZAKHAROV-SHABAT SYSTEM BY ITS REFLECTIONLESS PART

1. Introduction 1512. Direct scattering 1533. Inverse scattering 1564. Statement of the main result 1575. First steps to the proof 1586. Proof of theorem 4.] 1647. An application: cmKdV asymptotics 167References 171

CHAPTER 9 : INVERSE SCATTERING FOR THE MATRIX SCHRÓÜINGER EQUATION

WITH NON-HERMITIAN POTENTIAL

1. Introduction 1732. The inverse scattering problem for the matrix

Schrödinger equation 1752.1 Jost functions 1762.2 Scattering coefficients 1782.3 Bound states 1802.4 The Gel'fand-Levitan equation 184

References 189

CHAPTER 10 : UNIFICATION OF THE UNDERLYING INVERSE SCATTERING PROBLEMS

1. Introduction 1902. Jost functions 1913. Scattering coefficients and bound states 1934. Inverse scattering 196References 199

CONCLUDING REMARKS 201

APPENDIX : AN OPEN PROBLEM 2031

References 205 !

iSAMENVATTING 207 j

CURRICULUM VITAE 209 i

PREFACE

For centuries nonlinearity formed a dark mystery.

Nowadays, though things still look rather black, there are a few bright

spots where we may confidently expect steady progress. This thesis deals

with one of these sparkles of hope: the. i-nverse scattering tranafornation.

1. Historical remarks.

Many physical phenomena are nonlinear in nature. More often than not

they can be modelled by nonlinear partial differential equations offering

a wide range of complexity. Until the late sixties of this century the

analyst had, roughly speaking, the choice: approximate or apologize. In f

the past two decades this situation changed, since various powerful j}

nonperturbative mathematical techniques made their entrance. One of these '

is the inverse scattering technique (1ST), also called inverse scattering

transformation or spectral transform.

Its discovery is due to Gardner, Greene, Kruskal and Miura (GGKM for

short) and was first reported in 1967 in their famous two-pagei signal

paper [9]. In this paper GGK>! showed how to obtain the solution u(x,t)

of the Korteweg-de Vries (KdV) initial value problem

(1.1a) u - 6uu + u = O, -t» < x < +°°, t > 0

(1.1b) u(x,0) = uQ(x).

Here and in the sequel a subscript variable indicates partial differen-

tiation, e.g. u = -r— . Equation (1.1a) was first derived by Korteweg andX 0X

de Vries [13] in 1395 in the context of free-surface gravity waves

propagating in shallow water (see [4] for its historical background).

Below we shall discuss the GGKM method in some detail. Here we only

mention its amazing starting point, namely the introduction of the

solution u(x,t) of (1.1) as a potential in the Schro'dinger scattering

problem.

In 1968 Lax [16] put the GGKM method into a framework that clearly

indicated its generality and had a substantial influence on future

developments. In particular, Lax showed that (1.1a) is a member of an

infinite family of nonlinear partial differential equations that can all

be analysed in a similar fashion.

Guided by Lax' generalization of the pioneering work of GGKM, Zakharov

and Shabat [25] were able to solve the initial value problem for another

nonlinear equation of physical importance, the nonlinear Schrödinger

equation (NLS)

(1.2)

To this end they associated (1.2) with a spectral problem based on a

system of two coupled first order ordinary differential equations.

Incidentally, the NLS shows up in the description of plasma waves and

models plane self-focusing and one-dimensional self-modulation.

Subsequently, Tanaka [20], [22] extended and rigorized the direct

and inverse scattering theory for the Zakharov-Shabat system and showed

furthermore that another interesting nonlinear evolution equation could

be solved by this system, namely the modified Korteweg-de Vries equation

(mKdV)

(1.3) ut + 6u>ux + u x x x - 0

which appears in the continuum limit of a one-dimensional lattice with

quartic anharmonicity [5].

Ablowitz, Kaup, Newell and Segur [1], [2] then showed that NLS and

mKdV belong to a large class of nonlinear partial differential equations

that can be solved via a generalized version of the Zakharov-Shabat

scattering problem. Among these newly found integrable equations were

several of physical importance, such as the sine-Gordon equation

r rx 1(1.4) uc = è sin 2 u(x',t)dx'

which arises as an equation for the electric field in quantum optics [15],

though the related forms

(1.A) ' o = sin a andxt

(1.4)" o - o = sin oxx tt

appear more frequently in the literature (cf. [12]).

Herewith the triumphal march of the inverse scattering technique began.

We shall not follow it further but refer to fie survey articles [5], [10],

[15], [17], L18J as well as the many textbooks on solitons ['i\, [b], [7],

[3], [14], [24J currently available. We only mention that several other

classes of physically relevant equations were found to be solvable by

inverse scattering methods. In fact the process of finding new integrable

nonlinear evolution equations has continued until this very day and has

grown out into a major industry. Moreover, 1ST had its spin-off's to other

areas of mathematics, like algebraic and differential geometry, functional

and numerical analysis, etc. Nowadays - as stated in [6] - its applications

range from nonlinear optics to hydrodynamics, from plasma to elementary

particle physics, from lattice dynamics to electrical networks, from

superconductivity to cosmology and geophysics. Moreover, 1ST is <II'VL- lopirn;

into an interdisciplinary subject, since it has recently penetrated in

epidemiology and neurodynamics.

An essential reason for this wide applicability lias not been mentioned

so tar: a dominant feature of nonlinear evolution equations of physical

importance solvable via 1ST is that they admit exact solutions that

describe the propagation and interaction of .' .'' '..'.

At the moment there is no generally accepted mathematical definition

of a soliton. As a working definition of a soliton we mi;;ht take (cf. [5])

that it is a "localized" wave (in the sense of sufficiently rapidly

decaying) which asymptotically preserves its shape and speed upon

interaction with any other such localized wave. However, the concept of

a soliton has 'a great intuitive appeal and is a good illustration of the

fact that a happily chosen terminology is half of the success of a theory.

The soliton was discovered in 1965 by Zabusky and Kruskal [23] while

performing a numerical study of the KdV. Actually, the name "soliton"

was suggested by Zabusky, who originally used the term "solitron" instead

(see [6], pp. 176, 177).

Let us discuss their discovery in some detail. Already Korteweg and

de Vries theirselves knew [13] that the KdV had a special travelling

wave solution, the solitary wave

(1.5) u(x,t) = -2k2sech2[k (x - xQ - 4k^t)], (sech z = -~——)

e + e

where k„ and x„ are constants. Observe, that the velocity of this wave,

4ki, is proportional to its amplitude, 2ki. Now, in [23] Zabusky and

Kruskal considered two waves such as (1.5), with the smallest to the right,

as initial condition to the KdV. They discovered that after a certain

time the waves overlap (the bigger one catches up), but that next the

bigger one separates from the smaller and gradually the waves regain their

initial shape and speed. The only permanent effect of the interaction is

a phase shift, i.e. the center of each wave is at a different position

than where it would have been if it had been travelling alone.

Specifically, the bigger one is shifted to the right, the smaller to the

left. The name soliton was chosen so as to stress this remarkable

particle-like behaviour.

To conclude these introductory remarks, let us not forget to mention

that, although in the past few years soliton interaction has been

observed in various physical systems (see [3]), the first physical

observation of what is now known as the single soliton solution (1.5)

of the KdV already took place in the month of August 1334 by John Scott

Russell, during his celebrated chase on horseback of a huge wave in the

Union Canal, which from Edinburgh, joins with the Forth-Clyde canal and

thence to the two coasts of Scotland. His own report of this experience,

though classical by now, cannot be missed in any true soliton story.

It reads as follows [19]:

"I was observing the motion of a boat which was rapidly drawn

along a narrow channel by a pair of horses, when the boat

suddenly stopped - not so the mass of water in the channel

which it had put in motion; it accumulated round the prow

of the vessel in a state of violent agitation, then suddenly

leaving it behind, rolled forward with great velocity,

assuming the form of a large solitary elevation, a rounded,

smooth and well defined heap of water, which continued its

course along the channel apparently without change of form

or diminution of speed. I followed it on horseback, and over-

took it still rolling on at a rate of some eight or nine miles

an hour, preserving its original figure some thirty feet long

and a foot to a foot and a half in height. Its height

gradually diminished, and after a chase of one or two miles

I lost it in the windings of the channel. Such, in the month

of August 1334, was my first chance interview with that

singular and beautiful phenomenon ...".

2. 1ST for KdV: the gist of the method. |

iTo comfort the reader who is completely new to the subject, let us j

at least give a rough sketch of how 1ST works, referring to [8] for the •

many intricate mathematical details. To this end we indicate here very j

briefly the basic features of the GGKM method, which is the first and

undoubtedly the most fundamental example of an inverse scattering method.

Let us consider the KdV initial value problem (1.1) with u_(x) an

arbitrary real function, sufficiently smooth and rapidly decaying for

x * ±". The surprising discovery of GGKM is now, that the nonlinear

problem (1.1) can be solved in a series of linear steps, schematically

representable in the following diagram

initial functionu(x,0) = uff(x)

solution u(x,t)at t • 0

direct

lution ination space

inverse

Schrödinger

Schrödinger

scatterin'scattering data

at t = 0

time evolution in >spectral space

scattering

f

scattering dataat t • 0

The manipulations suggested by this diagram are the following:

For each t "" 0, introduce the real function u(x,t) as a potential in the

Schrödinger scattering problem

(2.1) + (k2 - u(x,t)h> = 0,

_ 2• 9

— i 2For t = 0, compute the associated bound states —•

K. v 0, right normalization coefficients c. and right reflection

coefficient b (k) (see Chapter 2 for their definition and properties),

in other words, compute the right scattering data 1b (k),> . ,i'. 1 associated

with u_(x). Then, as u(x,t) evolves according to the KdV, its right

scattering date evolve in the following simple way:

(2.2a)

(2.2b) cT(t) =ryexpU»U}, j = 1,2 ÏI

(2.2c) b (k,t) = b (k) exp{8ik 3t}, -™ • k

To recover u(x,t) from these data, one applies the inverse scattering

procedure for the Schrödinger equation found by Gel'f and and Levitan

[11], and defines

N -2i..r

(2.3) .;(•'. ;t) = 2 .Ij [cf(t)]2e J f br(k,t)e2lk'dk.

Next, one solves the Gel'fand-Levitan equation

(2.4) ;-:(y;x,t) + !-'(x+y;t) + >:(x+y+z; t ):•' (z ;x, t )d7. = 00J

with y • O, x £ K, t • 0 . The s o l u t i o n i-;(y;x,t) has the important

p r o p e r t y

(2.5) 3 ( 0 + ; x , t ) = u ( x ' , t ) d x ' , x £ K, t • 0 ,x-'

and so we find that the solution of the KdV problem (1.1) is given by

(2.6) u(x,t) = - ~ B(0+;x,t), x £ JR, t • 0.

Notice that the original problem for the nonlinear partial differential

equation (1.1) is essentially reduced in this way to the problem of

solving a one-dimensional linear integral equation.

Explicit solutions of (2.4) have only been obtained for b = 0. The

solution u,(x,t) of the KdV with scattering data '0,< . ,c . (t) } is called

the pure N-soliton solution associated with u„(x), on account of its

asymptotic behaviour displayed in the following remarkable result due to

Tanaka L21]

(2.7 a) lim supN

a' ' ' p=lx£E '= 0

+ _ 1 J p •• f (XP - ir

lon^~^i V^ p

Thus for large positive time u,(x,t) arranges itself into a parade of N

solitons with the largest one in front and this happens uniformly with

respect to x on E.

3. Asymptotics for nonzero reflection coefficient: main purpose of the

thesis.

As illustrated by the previous section, the inverse scattering method

enables us to obtain rather explicit exact solutions to nonlinear wave

equations and to determine their asymptotic behaviour, which generally

corresponds to a decomposition into solitons. Evidently, the problem of

the asymptotic behaviour evolving from an arbitrary initial condition is

in this way far from exhausted. It is still necessary to determine the

asymptotic properties of the "nonsoliton part" of the solution whose

presence is connected with the reflection coefficient being nonzero. In

this thesis we concern ourselves with this problem.

Rather than to give an elaborate general discussion, let us

illustrate the ideas involved by considering again the KdV problem (1.1).

Suppose un(x) is not a reflectionless potential. Then, in view of the

fact that the linearized version of (1.1a) is a dispersive equation with

associated group velocity v = -3k2 = 0, one expects that for lf.rge time

the soliton part and the dispersive component will separate out, the

dispersive wavetrain moving leftward and the solitons nicely arranging

theirselves into a parade moving to the right similar to that described

by (2.7). However, this is only heuristic reasoning. In fact it is

dangerous reasoning too, since for nonlinear equations there is no such

thing as a superposition principle.

The circumstance that at the time the question of validity of the

above "plausible" conjecture had not been answered in a mathematically

satisfactory way, formed the impetus for the research laid down in the

present thesis.

The main purpose of this thesis is therefore to give a complete and

rigorous description of the emergence of solitons from various (classes

of) nonlinear partial differential equations solvable by the inverse

scattering technique.

Throughout the thesis we focus our attention on coordinate regions

where the dispersive component is sufficiently small, e.g. x • -t for

the (m)KdV problem. The behaviour of the solution in other regions, where

the dispersive waves interact, is not discussed, since entirely different

techniques are needed. For recent results in those regions we refer to [3].

4. Brief description of the contents.

The chapters in this thesis are largely self-explanatory. Only

Chapter 1 forms an exception. We therefore advise the reader new to the

field to start with Chapter 2. In fact, both chapters deal with the KdV.

However, in Chapter t the central ideas of our asymptotic method are

exposed in the simplest nontrivial setting, wheroas Chapter 2 serves to

extend the results of Chapter 1, as well as to supply the details of the

inverse scattering machinery. Also, the discussion of existence and

uniqueness for the KdV initial value problem is postponed to Chapter 2.

In Chapter 1 we present a rigorous demonstration of the emergence of

solitons from the KdV initial value problem with arbitrary initial

function. We show that for any choice of the constants v > 0 and M 5 0

there exists a function a(t) tending to zero as t -*•<*>, such that

(4.1) sup |u(x,t) - u (x,t)| = 0(o(t)), as t •» ••xS-M+vt

The asymptotic analysis given in Chapter 1 is extended in Chapter 2. It

is shown that in the absence of solitons the solution of (1.1) satisfies

(4.2) sup |u(x,t)| =0(t~ 2 / 3), ast-.-»,xa-t1'3

whereas in the general case

(4.3) sup |u(x,t) - u (x,t)| = 0(t~' / 3), ast-»-».xè-t1'3

The emergence of solitons is clearly displayed by the remarkable

convergence result

(4.4) lim suiIim s upt-«» xS-t'/>3

N ,

£, (-2u(x,t) - Z, -2K2sech2U' (x-x -4K 2 t)] = 0P=1 \ P P P P

with x as in (2.7b).P

In Chapter 3, while studying multisoliton solutions of the KdV in the

general case of a nonzero reflection coefficient we derive a new phase

shift formula which shows that each soliton experiences in addition to

the ordinary N-soliton phase shift an extra phase shift to the left

caused by the interaction with the dispersive wavetrain.

In Chapter 4 we consider the question how well a solution of a nonlinear

wave equation is approximated by its soliton part in a more general

setting: we derive an estimate which indicates how well a real potential

in the Zakharov-Shabat system is approximated by its reflectionless part.

Moreover, the associated inverse scattering formalism is simplified

considerably.

In Chapter 5 we use the estimate derived in Chapter 4 to obtain

asymptotic bounds of solutions of the mKdV of the same type as those

found in Chapter 2 for the KdV.

Jsing the results from Chapter 5 we derive in Chapter 6 a general

phase shift formula for the mKdV remarkably similar in form to that

found in the KdV case in Chapter 3; the only difference is, however, that

now the extra phase shift is to the right, i.e. mKdV solitons are

advanced by their interaction with the dispersive wavetrain.

Chapter 7 is devoted to the asymptotic analysis of the sine-Gordon

equation on right half lines almost linearly moving leftward. Again the

estimate found in Chapter 4 is shown to be of vital importance.

In Chapter 8 we study the Zakharov-Shabat system with complex potential

and show that the results obtained in Chapter 4 can be generalized to

this case. As an illustration we investigate the long-time behaviour of

the solution of the complex modified Korteweg-de Vries initial value

problem.

In Chapter 9 we develop an inverse scattering formalism for a higher

order scattering problem than those considered in the rest of this thesis,

namely the matrix Schrödinger equation with non-Hermitian potential.

Finally, in Chapter 10 we show how this matrix Schrödinger problem can be

seen as a synthesis of all the scattering problems discussed in the

present thesis.

In an appendix we illustrate that our methods fail to give any result if

the associated group velocity is not of constant sign. As an example we

discuss the NLS and pose an interesting open problem.

10

References

[ 1) M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Method forsolving the sine-Gordon equation, Phys. Rev. Lett. 30 (1973),1262-1264.

[ 2] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inversescattering transform - Fourier analysis for nonlinear problems,Stud. Appl. Math. 53 (1974), 249-315.

[ 3] M.J. Ablowitz and H. Segur, Solitons and the Inverse ScatteringTransform, Philadelphia, SIAM, 1981.

[ 4] F. van der Blij, Some details of the history of the Korteweg—de Vriesequation, Nieuw Archief voor Wiskunde 26 (1978), 54-64.

[ 5] R.K. Bullough and P.J. Caudrey, The soliton and its history, in:Solitons, Topics in Current Physics 17, Springer, New York, 1980(edited by the same).

[ 6] F. Calogero and A. Degasperis, Spectral Transform and Solitons,Amsterdam, North-Holland, 1982.

[ 7] R.K. Dodd, J.C. Eilbeck, J.D. Gibbon and H.C. Morris, Solitons andWonlinear Wave Equations, Academic Press, 1982.

[ 3] W. Eckhaus and A. van Harten, The Inverse Scattering Transformationand the Theory of Solitons, North-Holland Mathematics Studies 50,1931 (2nd ed. 1983).

[ 9] C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Method forsolving the Korteweg-de Vries equation, Phys. Rev. Lett. 19 (1967),1095-1097.

[10] C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Korteweg-deVries equation and generalizations VI, Cornm, Pure Appl. Math. 27(1974), 97-133.

[11] I.M. Gel'fand and B.M. Levitan, On the determination of a differentialequation from its spectral function, Izvest. Akad. Nauk 15 (1951),309-360, AMST 1 (1955), 253-309.

[12] D.J. Kaup and A.C. Newell, The. Goursat and Cauchy problems for thesine-Gordon equation, SIAM J. Appl. Math. 34 (1978), 37-54.

[13] D.J. Korteweg and G. de Vries, On the change of form of long wavesadvancing in a rectangular canal, and on a new type of longstationary waves, Phil. Mag. 39 (1895), 422-443.

[14] G.L. Lamb, Jr. , Elements of Soliton Theory, Wiley-Interscience, 1980.

[15] G.L. Lamb, Jr. and D.W. McLaughlin, Aspects of soliton physics, in:Solitons (Ed. R.K. Bullough and P.J. Caudrey) Topics in CurrentPhysics 17, Springer, New York, 1980.

[16] P.D. Lax, Integrals of nonlinear equations of evolution andsolitary waves, Comm. Pure Appl. Math.21 (1968), 467-490.

[17] R.M. Miura, The Korteweg-de Vries equation: A survey of results,SIAM Review 18 (1976), 412-459.

1 1

[18] A.C. Scott, F.Y.F. Chu and D. McLaughlin, The soliton: a new conceptin applied science, Proc. IEEE 61 (1973), 1443-1483.

[19] J. Scott Russell, Report on waves in: Report of the fourteenthmeeting of the British association for the advancement of science,John Murray, London, 1844, 311-390.

[20] S. Tanaka, Modified Korteweg-de Vries equation and scattering theory,Proc. Japan Acad. 48 (1972), 466-439.

[21] S. Tanaka, On the N-tuple wave solutions of the Korteweg-de Vriesequation, Publ. R.I.M.S. Kyoto Univ. 8 (1972), 419-427.

[22] S. Tanaka, Non-linear Schrödinger equation and modified Korteweg-deVries equation; construction of solutions in terras of scatteringdata, Publ. R.I.M.S. Kyoto Univ. 10 (1975), 329-357.

[23] N.J. Zabusky and M.D. Kruskal, Interactions of "solitons" in acollisionless plasma and the recurrence of initial states, Phys.Rev. Lett. 15 (1965), 240-243.

[24] V.E. Zakharov, S.V. Manakov, S.P. tlovikov and L.P. Pitaievski, Theoryof Solitons. The Inverse Problem Method, Nauka, Moscow, 1980 (inRussian).

[25] V.E.. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linearmedia, Soviet Phys. JETP (1972), 62-69.

IACKNOWLEDGEMENTS i

J wish to focus my thanks on Viktor Eakhaus and Aart van Harten,my thesis super- and advisors, for their stimulating guidanoeand continuous interest in my work and for their pleasant wayof combining moral support with valuable criticism.

To Wilma van Nieuwamerongen I am indebted for skilfully typingthe manuscript.

12

CHAPTER ÖME

THE EMERGEIICE OF SOLITONS OF THE KORTEWEG-DE VRIES EQUATION FROM

ARBITRARY INITIAL CONDITIONS

We study the solution u(x,t) of the Korteweg-de Vries equation

u - &uu + u = 0 evolving from arbitrary real initial conditionst x xxx

u(x,0) = u-Cx), uo(x) decaying sufficiently rapidly as | x] -+ <». Using the

method of the inverse scattering transformation we analyse the Gel'fand-

Levitan equation in all coordinate systems moving to the right and give

a complete and rigorous description of the emergence of solitons.

*)1. Introduction.

The discovery by Gardner, Greene, Kruskal and Miura [8], [9] of a

method of solution for the Korteweg-de Vries equation by the inverse

scattering transformation has led to a rapid and impressive development,

which one can find described for example in [1], [7]. The rapid progress

has produced a wealth of results, however, it has also left certain

questions unanswered.

Let us recall that, by the inverse scattering transformation, the

initial value problem for the (nonlinear) Korteweg—de Vries equation, is

13

reduced to the problem of solving the (linear) Gel'fand-Levitan integral

equation. The initial values u..(x), prescribed for the solution of the

KdV equation, when introduced as a potential in the Schrödinger equation,

provide the scattering data that are needed to define the kernel of the

Gel'fand-Levitan equation. However, explicit solution of that equation

has been obtained only in the case that the reflection coefficient

corresponding to u„(x) is zero. One then has the famous "pure" M-soliton

solution, with N being the number of discrete eigenvalues in the

Schrödinger scattering problem.

If ur,(x) is not a reflectionless potential, then by a heuristic

reasoning one arrives at a conjecture about the behaviour of the solutions,

as follows: one knows that solitons, if present, move to the right, while

the dispersive waves that are expected to be present when the reflection

coefficient is not zero, move to the left. This leads to the expectation

that for large time the two ingredients of the solution will separate

out, and that observers moving with suitable speeds to the right will

eventually see a parade of solitons, each one followed by a decaying

train of dispersive waves, as described in [10]. In spite of attempts

such as [11], the question of validity of this "plausible" conjecture has

not been answered in a mathematically satisfactory way.

In this paper we study the solution u(x,t) of the Korteweg-de Vries

equation u - 6uu + u = 0 with arbitrary real initial conditionst x xxx

u(x,0) = u_(x), u^Cx) decaying sufficiently rapidly as |x| -> « for the

whole of the inverse scattering transformation to hold. We analyse the

Gel 'fand-Levitan equation in all coordinate systems moving to the right

and give a complete and rigorous description of the emergence of solitons.

It will probably not be a surprise to most readers that we find " solitons

emerging if un(x) is a potential that produces N discrete eigenvalues in

the Schrödinger equation, but it may be a surprise to some that this fact

has never been demonstrated mathematically, except for the reflectionless

potentials. We further show that the nature of decay of the dispersive

wave trains behind the solitons is essentially related to properties of

the reflection coefficient b (k), at t = 0, such as differentiability and

See also the more detailed introduction to Chapter 2, especially i'ordetails about existence and uniqueness.

14

behaviour for |k[ -+ »>, or the possibility of extension of b (k) to an

analytic function. These properties are in turn related to properties of

the initial function u~(x).

The problem of relations between properties of potentials uQ(x) and

corresponding reflection coefficients b (k) belongs to the scattering

theory and is not discussed in detail in this paper (see Chapter 2,

subsection 2.1). From the literature it is known that, if u„(x) decays

exponentially for x -+ +•», then b (k) can be extended to an analytic

function on a strip in the upper half plane, as can be seen from [6].

Furthermore, if un(x) and its derivatives up to fourth one decay

algebraically for |x| -+ °> sufficiently fast, then b (k) belongs to

C ( q )(K) for some q and b£m)(k) = 0<|k|~5) as |k| > », m = 0,1 ,... ,q

(see [5]).

We also no not discuss the behaviour of solutions of the KdV equation

for large time viewed in coordin te systems moving to the left, where

the dispersive waves interact. Recent results on that problem (in the

case of no discrete eigenvalues) have been given in [3],

The analysis given in this paper consists of a rather simple

reasoning in an abstract setting, supplemented by hard labour that is

needed to obtain the necessary estimates. The reasoning is developed in

sections 2 to 5, the labour is performed in section 6 and in two

appendices. In the last appendix we show that our method also works in a

more general setting by considering the so-called higher KdV equations.

2. Formulation of the problem.

We consider the Gel'fand-Levitan equation

(2.1) 3(y;x,t) + ft(x+y;t) + S2(x+y+z; t)iUz;x,t)dz = 0,0J

w i t h y > 0 , x £ K, t > 0 ,

( 2 . 2 ) s i ( f . ; t ) = a d ( c ; t ) + ? 2 c ( £ ; t ) ,

N -2K.t',(2.3) f!d(J;;t) = 2 ^ [ c j ( t ) p e J , 0 < KN < . . . - <2 <

15

(2

(2

(2

.4)

.5)

.6)

^c

b

rc.

( 5 ; t )

(k.t)

(t) =

= |

- b

cTeJ

r(k)e

4KUJ

b ( k ,

' t

t )2ik£„e dk,

Here -<2., c , j = 1,2,...,N are the bound states and (right)

normalization coefficients and b (k) is the (right) reflection coefficient

associated with the potential ur/x) in the Schrödinger scattering problem

(see Chapter 2, section 2, for their definition and properties). In the

integral equation (2.1) the unknown g(y;x,t) is a function of the variable

y, whereas x and t are parameters. The solution of the KdV equation is

given by

(2.7) u(x,t) = - Y- (3(0+;x,t).crX

We shall study the solution of (2.1) in moving coordinates in the parameter

space x,t, defined by

(2.8) "x = x - vt, v = 4c2 , c > 0.

In particular we shall examine the behaviour for large positive times,

with x confined to arbitrary half lines x S -M, where M £ 0 is independent

of t. For each c = K. we expect to see a soliton emerging.

We now give the problem an abstract formulation.

Let V be the Banach space of all real continuous and bounded functions %

on (0,»), equipped with the supremum norm

IIgil = sup |g(y) | .

For g £ V we write

(2.9) (Tdg)(y) = j {?d(x+y+z;t)g(z)dz,

(2.10) (T g)(y) = [ oc(x+y+z;t)g(z)dz.

C 0J

T clearly is a mapping of V into V; T will be investigated in the next

section.

Our problem is thus to find an element P € V such that

16

(2.11) (I + Td)3 + Tc@ = -fi,

(2.12) fi = nd + n c,

where I is the identity mapping.

We know the solution (3, ofa

(2.13) (I + T,)6, = -S?,,da a

which yields the pure N-soliton solution of the KdV equation. We intend to

study the full problem as a perturbation of the pure N-soliton case.

3. Analysis of Ü and T .

We consider

/•> i\ „ /•"./ ?- .- 1 f , „ , 2ik(x+y) 8itk(c2+k2) ,.(3.1) Slc(x+4c2 t+y;t, = — b (k)e } e dk.

It should be clear that Q is an oscillatory integral for large t, x ï -M,

tending to zero as t tends to infinity. The precise behaviour depends on

the behaviour of b (k), which in turn is determined by the initial

condition for the KdV equation.

Imposing suitable conditions on b (k) we shall establish that ft is

strongly differentiable in V with respect to x and obeys an estimate of

type

(3.2a) |fic(jc+vt+y;t)[ + |si^(x+vt+y;t) | :- H(y,t), t S tQ, x > -M,

such that

(3.2b) H(y,t) is a monotonically decreasing function of y for fixed t,

(3.2c) o(t) 5 H(z,t)dz + sup H(y,t) < +»,(r O'y-+<»

(3.2d) o(t) ->• 0 as t ->• ».

In (3.2a) we introduced the prime as a quick notation for the derivative

with respect to x.

17

We shall work out in detail two cases of conditions on t>r(k) that are

more or less typical (for a priori knowledge about b (k) as well as the

motivation for these conditions see Chapter 2, subsection 2.1).

Case C There exists an E > 0 such that b (k) is analytic on— — — ^ — f

0 < Im k < € and continuous on 0 s lm k S c, while in that

strip br(k) = o(|k|2) for |k| •+ «.

Case P ( i) b (k) is n S 2 times differentiable on the real axis;b

z . . * , (m) /, 2m+2 \ i, i „ - <( n ) b = o(k ) , |k •+ o°, m = 0,1,...,n-1;

r b b 0 )

(iü) ^ , — ^ , ••., b^" 1 ), (H|kl)b^n) belong1 | | n 2

to L 1(R).

In Case C - treated in Appendix A - one finds by means of contour

integration

(3.3) H(y,t) = Ye~2 f ye" a t, o(t) = ü(e" a t),

where y and a are positive constants.

In Case P - the subject of Appendix B - integration by parts pro 'uces

(3.4) H(y,t) = , o(t) = 0(— l- T),

(-M+y+vt)n tn~'

where again p is a positive constant.

With the result (3.2), examplified by (3.3) and (3.A), we proceed to

investigate the mapping T .

In moving coordinates we have

(3.5) (T g)(y) = j <; fx"+vt+y+z;t)g(z)dz,

which is a continuous function of y, since the integrand is dominated by

H(z,t)|g(z)|. Furthermore,

CO

(3.6) IIT gll i llg» H(z,t)dz S llglla(t).

We have thus established that T is a continuous mapping of V into V with

norm tending to zero for t -* •» uniformly on x * —M.

18

We next claim that T is strongly x-differentiable in V with derivative

f -(3.7) (T';;)(y) = fl'(x+vt+y+z;t)g(z)dz.

c 0J

The proof consists in showing that

(3.3) A, = sup (A, (y+z)-Q'(x+vt+y+z;t))g(z)dzn 0<y<+» 0Jy 0

tends to zero as h + 0, where

(3.9) Ah(y+z)=-£1» (x+vt+y+z+h;t)-U (x+vt+y+z;t)

Clearly \ S J B ( z ) | g ( z ) | d z , with

(3.10) B j / Z ) = SUP \

Since Si is strongly x-dif f erentiable, B (z) tends to zero as h •+ 0. Thus

in virtue of the dominated convergence theorem it suffices to show that

(3.11) B (z) < G(z) with G £ L1(0,»).

Let x S -M+6, 6 > 0. For |h| t' ó one has

(3.12) |Ah(y+z)| = |ic''(x-ö+vt+y+z+Oh+iS;t)| for some •? € (0,1)

S H(y+z+0h+6,t)

i H(z,t).

Consequently B (z) •.-' 2H(z,t) which is in L (0,«).

Finally, we deduce from (3.2) and the above

( 3 . 1 3 ) n i a x i II I I . I l T I I , il ' H . l l r M I j • r ( t ) ,L c c c c J

w h e r e j ( t ) -• 0 a s t • -".

19

4. Solution of the Gel 'fand-Levitan equation.

We consider

(4.1) (I + Td)fc! = -(£3 + T cS).

Since T is an integral operator with degenerate kernel, solutions of

(1 + Td)g = f, f,g £ V

can be studied explicitly. In lemma 6.1 we shall show that the inverse

(I+T,) indeed exists as a mapping of V into V and that furthermore

Il(l+T )~1II is bounded for t •> 0, it 5 -M.

To simplify the notation we shall write

(4.2) (I + T d ) ~1 = S

and we have

(4.3) ItSlI £ a for t > 0, 1 è -M.

We can thus "invert" (4.1) and obtain the equation

(4.4) B = - Sn - ST 3.

It can be easily shown that (4.4) possesses a unique solution p € V.

Indeed, consider the mapping T defined by

(4.5) Tg = f - STcg, f,g € V.

By the results (3.13) and (4.3) one has

(4.6) IIST II s IISIIHT If ï ao(t).

Hence, for sufficiently large t, we find IIST II - 1 and T is a contractive

mapping in the Banach space V. It follows that a unique solution g of

(4.7) g = f - STcg, f,g £ V

exists. Furthermore, one easily obtains an estimate for the solution. In

fact, since

'23

( 4 . 3 ) II gil < IIfII + IIST gil S IIfII + IIST I l l lg l l ,

w e o b v i o u s l y h a v e

(4.9) «g« S , _ 1 „ S T „ llfll.c

5. Decomposition of the solution and estimates.

We write

(5.1) d = Sd + 3c,

with

(5.2) sd = -snd.

In lemnia 6.2 i t will be shown that fi (y;x+vt,t) is uniformly bounded for

t •• 0, x £ -M, y "•• 0. Ue recall that p, produces the pure H-soliton

solution of the KdV equation through the formula

(5.3) u,(x+vt,t) = = B (0+;x'+vt,t).a 3x a

Introducing the decomposition (5.1) into (4.4) we have the equation

(5.4) p + ST p = -Ss> - ST B,.c c c c c d

From the analysis of the preceding section we know that a unique

solution B € V exists. To estimate the solution we proceed as folLows: i

( 5 . 5 ) l i p II < II S T I I I I H II + I I S l l l l ' ! II + II S T I I I I B . I I . 'c c c c c d •

Using (3.13), (4.3) and (4.6) one gets

(5.6) up ii < , ao(t), . (1 + I I B . H ) .c 1 - ao(t) d

Our final result at this stage is that in all moving coordinates

x = x-vt, v ~' 0, in any half line x 't. —M, for large t

21

(5.7a) llBcll S ba(t),

where b is some constant and o(t) + 0 as t •» ».

Furthermore, in the first approximation we have

(5.7b) @c = -S(«c + Tc(5d) + 0(a2(t)).

Unfortunately, the labour is not finished yet. The solution of the

KdV equation is given by

(") .(5.3) u(x+vt,t) = ud(x+vt,t) = a (0 ;x+vt,t).

We thus need estimates of the derivative of (-! with respect to x. To obtain

these estimates we return to equation (5.4). One verifies without

difficulty that both S and 3, are strongly x-differentiable. From

equation (5.4) we then see that g , too, is strongly x-differentiable.

Oifferentiating both sides with respect to x we find

(5.9) 13 + STc3^ = -S{T<;(0c+ed) + 9.'c + T fj)

Jsing section 4 we again conclude that a unique solution 8' exists and

proceed to estimate the solution.

By lemma 6.2 Bj(y;x+vt,t) is uniformly bounded for t > 0, x "• -M, y • 0,

while lemma 6.1 tells us that

(5.10) IIS 'II < a for t > 0, ~x. > -M.

From ( 5 . 9 ) a n d t h e e s t i m a t e s ( 3 . 1 3 ) , ( 4 . 3 ) , ( 4 . 6 ) , ( 5 . 7 a ) and ( 5 . 1 0 ) o n e

f i n d s

( 5 . 1 1 ) II6 Ml S 1 ! ° a ^ ( t ) ( 2 + 2«SdH + ll!?dll + 2 b o ( t ) ) .

T h u s f o r l a r g e t

( 5 . 1 2 ) II6 Ml S B o ( t ) ,

c

where B is some constant. Evidently

22

(5.13) sup •-= 0 (y;x+vt,t

= IIBMI

é Ba(t).

We thus arrive at our final result, which can be stated as follows:

Theorem 5.1. Let u(xst) be the solution of the Korteweg-de Vries -problem

( 5 . 1 4 )r U - 6UU + U = 0, -oo < X < +oo, t > 0j t X XXX ' '

*• u(x,0) = u Q ( x ) ,

where the real initial function un(x) is sufficiently smooth and decays

sufficiently rapidly for |x| •* •» for the whole of the inverse

scattering method to work and to guarantee an estimate of type (3.2).

Then for any choice of the constants v > 0 and M > 0 there is a function

o(t) such that

(5.15) sup |u(x,t) - ud(x,t) | = 0(a(t)) as t •» «.xa-M+vt

Here u,(x,t) is the pure H-soliton solution, N being the number of

discrete eigenvalues corresponding to the potential un(x). The function

o(t) tends to zero as t + « and the exact behaviour of o(t) depends on

properties of the reflection coefficient b (k).

If there exists an c > 0 such thai b (k) is analytic on 0 < lm k * r

and continuous on 0 ï lm k :-v z, while in thai strip b (k) = o(|k|2) for

ik| -»• », then o(t) = 0(e ) , a - 0. •

If b (k) is n ; 2 i f me s differentiab'V /«; '»r r<?i7.' axis, with

b<m)r= o(k 2 m + 2), |k| - », m = 0,),...,n-1 Ji.i

(1 + lk|)1-nbr> (1 + |k|)

2"nbJl),...,b^n-1),(1 + |k|)b^n) belong to

;fe« o(t) = 0(t'"n).

6. Analysis of S and B,.

In this se.ction we present estimates concerning S, 8, and their

x-derivatives that were essential in the previous investigation of the

Gel'fand-Levitan equation. The original proofs as given in Chapter 2,

reference [d], have been improved considerably and are therefore omitted.

Instead we refer to the corresponding proofs occurring in Chapter 2.

Lemma 6.1. I + T, is an invertible operator1 on the Banaoh space V. Writing

(6.1) S = (I + T d)~'

we have

(6.2) llsll, US Ml i a for t > 0, 1 ^ -M,

where a is some constant and S' denotes the strong x-de.rivative of S.

Proof: See Chapter 2, lemma 5.1.

Lemma 6.2. (5, (y ;x+vt,t) and S!(y;x+vt,t) arc uniformly bounded for t > 0,

JC a - M , y •> 0.

i

Proof; This follows from combining Chapter 2, (5.15) with Chapter 2, (5.7). j

Appendix A: Case C.

Assuming that

(A.1) there exists an c > 0 such that b (k) is analytic on

0 ' In t < E and continuous on 0 S lm k S c, while in that

strip b (k) = o(|k|2) for |k| •* »,

we shall derive the estimate

(A.2) |«c(x+4c2t+y;t)| + |i2 (x+4c* t+y;t) | f- y e " 2 ^ . ^ ^ ,

t .> t0, x > -M,

24

where a and y are positive constants.

Remark. The reason why we use no steepest descent method is that generically

b (k), if at all extendable to the upper half plane, has poles on the

imaginary axis, corresponding to i« IK ,..., itc . Working as in (A.1)

we avoid them.

Let us fix c > 0 and choose 0 < r. < c.

Putting w = x+4c2t+y we integrate b (k)e ' e around a rectangle in

the complex k-plane with vertices at -R, R, R+it;, -R+ir. By (A. 1) one has

for r = ±R

,-, I f r + 1 £ v r, \ 2ikw Sitk' | . 2Me f' ,, , . ., -24r 2ts,(A.3) b (k)e e dk S e |b (r+is)|e dslrJ I 0 J

f0J

2Mf= kr • -p- max lb

r(r+ i s )!

MtQ r Oisr:

= o(1) for R - ».Thus, using Cauchy's theorem

,. . , fR , ,, s 2ikw 3 i t k 3 , , fR , ,, . . 2i(k+ic)w 8 i t (k+ir.)3 ,.(A.4) b (k)e e dk = b (k+j.c)e e dk

-RJ r -RJ r

Now the integrand on the right clearly belongs to L (E).

Hence, if we let R -* •» in (A.4) we find that the integral in (3.1) is

Cauchy convergent and equals

(A.5) H (x+4c2t+y;t) =

where the integral on the right converges absolutely.

From (A.5) one easily deduces

(A.6) ]H c&4c z t+y;t) | < y^e~2 c ye~a t ,

—24c];2 twith Y- = - e2cA f°° jb (k+ie) (e °dk and a = 8r (c2-r2) - 0.

Obviously we can differentiate (A.5) with respect to x uniformly in y.

Estimating the derivative one finds

25

(A.7) |^ ^

r» -24ck21with Y 2 = \ e

2 £ M |k+ie||br(k+iE)|e dk.

Hence the proof of (A.2) is complete.

Appendix B: Case P.

Let us demonstrate that the conditions

(B.1) b (k) is n 'i 2 times differentiable on the real axis;

(B.2) b ( m ) = o(k 2 m + 2), |k| * -, m - 0,1,....n-1;

b b 0 )

(B.3) r-—,, T—^=I b<n"1), <i + |k|)b<n)

(1+|k|)n 1 ( H | k | ) n 2

belong to L (E);

guarantee the estimate

(E.4) |fi Cx+vt+y;t)| + | SI' (x+vt+y;t) | Sfi Cx+vt+y;t)| + | SI(x+vt+y;t) | S ,c c (-a+y+vt)n

-M, t 2 t - > —, v = 4c2 , c - 0.

For brevity of writing we put

(B-J' t 3x t' f W *' f

Furthermore we define

(B.6) w = jc + vt + y ,

(B.7) (j> = 2k(x+y)+atk(c2+k2) = 2kw + 8tk3 ,

(B.3) s = * °^j 2 s + 2 y + 2 v t + 24tk2 * °-

Now, since b ,b ,...,b are locally integrable we can integrate by

parts n times to find

26

R , R .(B.9) [ ei+b dk = -ise1* ï <iT)b +

_RJ r *• w r _ R _RJ

where the operator T is defined by

(B.10) Tf = (sf) ( 1 ) = s(1)f + sf(1).

Induction reveals that the t-th iterate of T has the structure

(ii. 11a) T f = s Z ctp f P » with ap = 1, whereas for p

U \ Ii - i . . . i\ s / \ s

^ =p

where a,, » » » are nonnegative integers, independent of s and i.

Applying Leibniz' formula to the identity (2w+24tk2)s = 1, we find

.12) (w+12tk2)s(j) + 24jtks(j"1) + 12 j (j-1 )ts ( j" 2 ) = 0,(£

from which it is easily seen that

05.13)(j)

3 1,2

where M. i s a c o n s t a n t .J

Thus, in view of (B.11) there are constants A„ such that^ P

Returning to (C.9) we extract from (B.2) and (B.14) that

fR • .(B.15) e l l | ) [b - ( i T ) " b ]dk = o ( 1 ) for R - » .

-R J r r

i lence, u s i n g ( B . 3 ) , (B.14) and ( 3 . 1 ) , we o b t a i n

e l(iT)b rdk,

where tlie integral is absolutely convergent. From (H.14) we find the bound

_(B.17) |:.'c(x+vt+y;t)| (-M+y+vt)

1 " f ibr Iwith u. = I_ A dk.

1 n p=0 n,p _ J , | h P

27

Next, we consider the x-derivative of the integrand in (B.16). Since

n , _ .we are led to examine (Tnb )' = £ n (s"a )'b^n~p , in which (s"a )' is

r p=0 n,p r n,pa linear combination of terms of the following form

where l} + 2^ + ... + pi = p, lQ + ^ + . . . + £ = n.

Since s' J = (-2s2) J we obtain from (B.13) and Leibniz' formula

(3.20) |s' J | S M. , where M. is a constant.J (l+|k|)J J

Estimating each term (B.19) by (B.13) and (B.20) one gets

(TnbrV p |bj

where the A are constants.n,P

Finally, combining (B.14), (B.18) and (B.21), we find constants B suchn,p

that

(B.22)1 7i r ' , .n p=ü ,, i I .p-1 ' r

(-M+y+vt) v (1+|k|)

where by (B.3) the right hand side belongs to L (K).

To prove that Ü is strongly x-differentiable we proceed as follows.

For h > M-vtQ let g(h) denote T br with x replaced by x + h. One has

I Si (x+h+vt+y;t) - Q (x+vt+y;t)(B.23) M r—£ — I (eM'Tnb )'dk!11 -J S br

(B.24) Gh = |e2lkh(g(h)-g(0)-hg'(0)) + hg'(0)(e2lkh-l) +

+ g(0)(e2ikh-1-2ikh)|

|S"(0h)|+|hg'(0)|.]e2ikh-1|+|g(0)|.|e2ikh-1-2ikh!

for some 6 £ (0,1).

Examining the x-derivative of (B.I 9) we obtain the estimate

n + 2 ^ n P )03.25)

where the A are constants.n,p

From (B.14), (B.21), (B.24) and (B.25) it is clear that

(B.26) lim -JT- f ( sup G )dk = 0.h+0 l

hl -J Vy<+» VHence, (B.23) yields that fi is strongly x-differentiable, the

derivative being given by

. n r0

(B.27) n'(x+vt+y;t) = — (e *T b )'dk,C TT ' _ ' »

satisfying the obvious bound

(B.28) |fi'(x+vt+y;t)| S

U2

(-M+y+vt)n

n p» |b n P |

=0 Bn I " =T d k '

This completes the proof of (B.4).

i iAppendix C: Generalization to higher KdV equations. [

We now show that the method described in this paper is still working

in a more general setting.

In [2], Appendix 3, the class of evolution equations

(C.1.a) qt + CQ(L*)qx = 0, q(x,0) = qQ(x)

+ _ _ i 32 _ i j

x-*

where C is a ratio of entire functions, is found to be solvable by

29

the inverse scattering transformation associated with the Schrödinger

equation.

Setting q = -ü, c„(z) = -a(-4z) we can rewrite (C.1.a) as

(C.1) u = a(L)u , u(x,0) = un(x)

L = ^ L _ 4 u + 2ux j dy.In this form the class has been investigated by Calogero (see [4] and

subsequent papers). Introducing the solution u(x,t) of (C.1) as a

potential in the Schrödinger equation one obtains the following simple

time evolution of the spectral parameters

(C.2) br(k,t) = br(k)exp{2ik«(-4k2)t},

(C.3) <.(t) = K-J(O) = ^ , j = 1,2,...,N,

(C.4) c'i(t) = crexp|-K.ti(4K2.)t}, j = 1,2... . ,N.

In particular, choosing n(z) = —z we rediscover the KdV equation and its

wellknown time evolution in spectral space (see (2.5) and (2.6)). For

simplicity we shall assume a(z) to be a polynomial. In this case the

equations (C.1) are generally called "higher KdV equations", though, of

course, this appellation is only relevant if •» has degree higher than one.

Let us first consider some special solutions of (C.I).

If b (k) = 0 and N = 1, we find

(C.5) u(x,t) = - 2 K 2 2

(C.7) v1 = - U ( 4 K 2 ) ,

which is immediately recognized as the celebrated single soliton

solution.

If b (k) = 0 and N • 1 one obtains by an exercise in linear algebra

the so-called pure H-soliton solution, which is such that a transformation

to moving coordinates with speed

(C.3) v. = -u(4>-2.), j = 1,2,...,N

30

makes the j-th soliton stationary as 11 •* «•, provided that all v.'s are

different.

In the case b (k) # 0 dispersive waves enter in the solution. The

linearized version of (C.I), reading

(C.9) ut = u ( ^ r ) v u(x,0) = uo(x),

is a dispersive equation with associated group velocity

(CIO) vn = - -LkaHc 2)], k £ R.

Now, if all v.'s have the same sign, while v has the opposite sign (as

J S

is the case for the KdV equation with v. = 4K2. and v = -3k 2), one

expects that for large time the soliton part and the dispersive component

will separate out. In order to convert this expectation into a

mathematical fact we shall impose the following conditions upon the

function a occurring in (C.1):

(C.11a) a is a polynomial,

(C.1 1b) v = - ^ [ku(-k2)] • 0, k 6 R, j

( d i e ) V J = - C X ( 4 K ? ) - 0 , j = 1,2 N» J

i

(C. lid) v. * v. for i * j. !

To reassure the reader let us mention a class of functions meeting these j

requirements j

(C.12) oc(z) = -z , m • 1 an odd integer. ;

For convenience we shall order the solitons so that

(C.13) 0 < vN < ... < v2 <• v,.

We can now generalize theorem 5.1 as follows:

31

Let u(x,t) be the solution of the higher Korteweg-de Vries equation

(C.I), with a as in (C.ll), evolving from an arbitrary real initial

function u.(x) I such that uQ(x) is sufficiently smooth and decay o

sufficiently rapidly for |x| •* °° for the whole of the inverse scattering

method to work and to guarantee an estimate of type (3.2)).

Then fop any choice of the constants v > 0 and M % 0 there is a function

o(t) such that

(C.14) sup |u(x,t) - u,(x,t)| = O(a(t)) as t + «.xê-M+vt

Here u,(x,t) is the pure ii~soliton solution, H being the number of

discrete eigenvalues corresponding to the potential u„(x). The function

a(t) tends to zero as t •+ « and the exact behaviour of o(t) depends

on properties of the reflection coefficient b (k).

If b (k) is n ï 2 times differentiable on th<: real axis, with

b^ m ) = o(k 2 m + 2), |k| - », tn = 0,1,....n-1 and

belona to

L'(IR), then o(t) = 0(t 1~ n).

To prove this we repeat the analysis given in sections 2 to 6 and

Appendix B with the following obvious adaptations:

In (2.3) the ordering is replaced by (C.13). Furthermore (2.5) and

(2.6) are replaced by (C.2) and (C.4). Throughout, the speed 4c2 is

identified with v and case C is left out.

(B.7) becomes :• = 2kw + 2s(k)t; g(k) = ka(-4k2).

(B.8) becomes s = —rvy = — TTK • 0.:• ' 2x+2y+2vt+2gU J(k)t

(2w+24tk-')s = 1 is replaced by (2w+2g^1 (k)t)s = 1.

(11.12) becomes (w+g( ' } (k) t)s (j } = -t ^ ( )g ( r + I ) (k)s ( j" r ) .

32

The reader is Invited to verify in detail that our analysis remains

valid once the alterations summarized above have been carried out.

References

[ 1] M.J. Ablowitz, Lectures on the inverse scattering transform,Stud. Appl. Hath. 53 (1978), 17-94.

[ 2] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inversescattering transform - Fourier analysis for nonlinear problems,Stud. Appl. Math. 53 (1974), 249-315.

[ 3] M.J. Ablowitz and H. Segur, Asymptotic solutions of the ICorteweg-de Vries equation, Stud. Appl. Math. 57 (1977), 13-44.

[ 4] F. Calogero, A method to generate solvable nonlinear evolutionequations, Lett. ÏJuovo Cimento 14 (1975), 443-447.

[ 5] A. Cohen, Existence and regularity for solutions of the Korteweg-de Vries equation, Arch, for Rat. Mech. and Anal. 71 (1979),143-175.

[ ó] P. üeift and E. Trubowitz, Inverse scattering on the line, Comm.Pure Appl. Math. 32 (1979), 121-251.

[ 7] W. Eckhaus and A. van Harten, The Inverse Scattering Transformationand the Theory of Solitons, North-Holland Mathematics Studies 50,1981.

[ 8] C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Method forsolving the Korteweg-de Vries equation, Phys. Rev. Lett. 19 (1967),1095-1097.

[ 9] C.S. Gardner, J.M. Greene, M.D. Kruskal and K.M. Miura, Kortewer;-deVries equation and generalizations VI, Comm. Pure Appl. Math. 27(1974), 97-133.

[10] R.M. Miura, The Korteweg-de Vries equation: A survey of results,SIAM Review 18 (1976), 412-459.

[11] H. Segur, The Korteweg-de Vries equation and water waves, J. FluidMech. 59 0973), 721-736.

33

CHAPTER TWO

ASYMPTOTIC ESTIMATES OF SOLUTIONS OF THE KORTEWEG-DE VRIES

EQUATION ON RIGHT HALF LINES SLOWLY MOVING TO THE LEFT

We consider the Korteweg—de Vries equation u - 6uu + u = 0 witht x xxx

arbitrary real initial conditions u(x,0) = u_(x), sufficiently smooth and

rapidly decaying as |x| •-> °». Using the method of the inverse scattering

transformation we analyse the behaviour of the solution u(x,t) in coordinate

regions of the form t -? tQ, x ~- - n - \>T, T = (3t) where u, v and t_ are

nonnegative constants. We derive explicit x and t dependent bounds for the

nonsoliton part of u(x,t). These bounds enable us to prove a convergence

result, which clearly displays the emergence of solitons. Furthermore, they

help us to establish some interesting momentum and energy decomposition

formulae.

1 . Introduction.

We study the Korteweg-de Vries (KdV) problem

(1.1a) ut - 6uux + u x x x = 0, ~» •- x •• +», t - 0

(1.1b) u(x,0) = uQ(x),

where the initial function u-(x) is an arbitrary real function on R, such

34

that

(1.2) un''x) ^s sufficiently smooth and (along with a number of its

derivatives) decays sufficiently rapidly for |x| -<• •» for the

whole of the inverse scattering method to work and to guarantee

certain regularity and decay properties of the right reflection

coefficient, to be stated further on.

To make the discussion less abstract let us quote a definite example from

[4] in which (1.2) is fulfilled:

(1.3a) u_ is of class C3 on E and has a piecewise continuous fourth

derivative,

(1.3b) U Q J ) ( * ) = 0(|x|~M) as |x| •> « for j •• 4,

where M > y. Here y is a constant which is 8 in the generic case (see

(2.13)) but which is 10 in the exceptional case (see (2.14)). In [4] it

is shown by an inverse scattering analysis that condition (1.3) guarantees

the existence of a real function u(x,t), continuous on H*[0,t»), which

satisfies (1.1) in the classical sense.

Let us recall that there is uniqueness of solutions of (1.1) within

the "Lax-class", i.e. the class of functions which, together with a

sufficient number of derivatives vanish for |x| -• °», as discussed in [12].

Whenever, in the sequel, we speak of "the solution" of (1.1) we shall

refer to the solution obtained by inverse scattering (unique within the

Lax-class).

The long-time behaviour of the solution u(x,t) of (1.1) has been

discussed by several authors, the general picture being, that as t •* <"

the solution decomposes into a finite number of solitons moving to the

right and a dispersive wavetrain moving to the left. The emergence of

solitons from initial conditions as arbitrary as (1.2) was demonstrated

rigorously in [8] corresponding to Chapter 1 of this thesis. It was

proven there, that for any choice of the constants v • 0 and M - 0 there

is a function o(t) such that

(1.4) sup |u(x,t) - u (x,t)| = 0(o(t)) as t ->• »>,xS-M+vt

35

where u,(x,t) denotes the pure N-soliton solution (see (5.16)), N being

the number of bound states corresponding to u (x), when introduced as a

potential in the Schrödinger scattering problem. The function a(t) tends

to zero as t + • and the exact behaviour of a(t) depends on properties

of u0.

If u„ decays exponentially for x -*•+•» then u(t) = 0(e ) for some

constant a > 0.

If u„ and its derivatives up to the fourth one decay algebraically

for ]x| •+ •» sufficiently fast, then a(t) = (Kc ) f°r some constant m 2 1.

Earlier results in this direction, though less detailed and not widely

known, were given in [16], where it was shown that

(1.5) lim sup |u(x,t) - u,(x,t)| = 0 for v > 0 arbitrarily fixed.t-H» x>Vt

In fact, in recent years, most of the asymptotic attention was devoted to

the solitonless KdV initial value problem. In this case, the analysis

given in [1] led to the recognition of four distinct asymptotic regions

i. x a 0(t) ii. |x| s 0(t1/3)

III. -x = 0{t1/3(ln t)2/3} IV. -x 'i 0(t),

where 0 denotes positive proportionality. Within each region, the

solution u(x,t) has an asymptotic expansion, characteristic for that

region. However, interesting as they may be, the results are far from

rigorous. Indeed, discussing the matter in their book [2], p. 68 the

authors remark

"Two warnings should be made before we begin the analysis. The

first is that almost none of the results to be described in this

section are known rigorously. These results are formal, and have

great practical value, but proofs of asymptoticity are yet to be

given. The second (related) warning is that some of the existing

literature on this question contains errors".

In this paper we extend the asymptotic analysis given in Chapter 1. We

use the method of the inverse scattering transformation to analyse the

behaviour of the solution u(x,t) of (1.1) - both in the absence of

solitons as well as in the general case - in coordinate regions of the

36

form

1 73(1.6) t È t0, x 2 -c, C = y + vT, T = (3t)

where u, v and t. are nonnegative constants. Here |j is arbitrary, but the

values of v are restricted to 0 S u < v where v is some generic number

connected with the Airy function, its numerical value being 1.39. Further-

more, t„ depends on u, v and the behaviour of u„ as well. Note that (1.6)

covers region I and almost all of region II.

It is shown that in the absence of solitons

(1.7) sup |u(x,t)| =0(t" 2 / 3) as t °°.x>-c

In the general case we improve (1,5) by

(1.8) sup |u(x,t) - ud(x,t)| =0(t"1 / 3) as t •* ».

XÈ-5

This leads to the convergence result (7.17), which clearly displays the

emergence of solitons. Moreover, we construct several remarkably explicit

x and t dependent bounds for the nonsoliton part of the solution valid in

regions (1.6). With the help of these bounds we establish the momentum and

energy decomposition formulae

(••» N( 1 . 9 a ) u ( x , t ) d x = - 4 I . K + 0 ( t "

_ J P = 1 P

( 1 . 9 b ) j u ( x , t ) d x = - | Qf°° l o g ( 1 - | b r ( k ) | 2 ) d k + 0 ( t ~ 1 / 3 ) a s t ^ «

f°° if, N - 1 / 1

(1.9c) u2(x,t)dx =-if ï t ' + 0 ( t " J ) a s t > »_qJ P P

(1.9d) u2(x,t)dx = - | 0 / k2 log(1- |b r(k) |2)dk+0(t" l / J) as t - »,

where -K 2 (K > 0) p = 1,2,...,N are the bound states and b (k) is the

right reflection coefficient associated with the potential un(x) in the

Schrödinger scattering problem.

Let us point out three major differences between the present paper and

[1]. Firstly, it is our main purpose to obtain explicit bounds for the

nonsoliton part of the solution of (1.1), valid in the region (1.6),

whereas in [i] the emphasis lies on the construction of asymptotic

37

expansions of the (solitonless) solution in the various regions. Secondly,

the analysis in [1] requires that u~ decays faster than exponentially as

x -+ +<=. For our results, however, except explicitly stated otherwise, an

algebraic decay rate of type (1.3) is sufficient. The third difference

lies in the fact that we venture to call our results rigorous.

The composition of this paper is as follows. In section 2 we briefly

discuss the direct and inverse scattering problem for the Schrödinger

equation and formulate our problem. In section 3 we isolate certain

properties of the function Q and the operator T , both occurring in the

Gel'fand-Levitan equation. These properties are used in section 4 to

investigate the solution of (1.1) in the absence of solitons. In the sub-

sequent sections it is supposed that solitons are present. In that case

the operator I + T appears in the Gel'fand-Levitan equation. Section 5

is devoted to showing that this operator is invertible. In section 6, the

operator S = (I + T.) is applied to both sides of the Gel'fand-Levitan

equation. It is shown that the resulting equation has a unique solution fi,

representable by a Neumann series. Finally, in section 7, we write R as

the N-soliton state plus a perturbation. This leads to a decomposition of

the solution u(x,t) ot (1.1) into an N-soliton part and a nonsoliton part.

For the nonsoliton part we derive explicit estimates and discuss their

consequences.

The notation used is similar to that in Chapter 1. Since the constants

appearing in this paper have a simple structure, we have traced them all.

In this way the interested reader can obtain numerical estimates in a

practical case.

2. Preliminaries and statement of the problem.

2.1. Direct scattering at t = 0.

Let us briefly review the direct scattering problem for the

Schrödinger equation

\ £• • i / 1 ~ v, is. r\ ''T — " > T ^ " j

38

where u„ is the initial function in (1.1b) and k a complex parameter. For

details we refer to [A], [6] and [7], Our notation closely resembles that

used in [7] .

The results of this subsection are valid for any real function u„(x),

continuous on K, vanishing at infinity and integrable with respect to

x2dx.

For In k i 0 we introduce the Jost functions ijj (x,k) and (Jr„(x,k), two

special solutions of (2.1) uniquely determined by

(2.2a) ij, (x,k) = e~lkxR(x,k), lira R(x,k) = 1, lira R (x,k) = 0r x

X-S--1» K-»-»

(2.2b) i^(x,k) - e l k x L ( x , k ) , lira L(x,k) = 1, lira Lx(x,k) = 0.x-v+°° x->-+w

The functions R, R , R , L, L , L are continuous in (x,k) on RxC and' x' xx' ' x' xx ' +

analytic in k on C + for each x 6 TR. Furthermore, for k real, k * 0, the

pairs i(.^,(x,k), i|) ,(x,-k) and J<r(x,k), V'r(x,-k) constitute fundamental

systems of solutions of equation (2.1).

In particular we have for x £ R, k £ E\{0}

(2

(2

(2

.3a)

.3b)

• 3c)

r+(k)

r (k)

r+(k)i|,£(x,k)

where W[I|J. ,!(/„] = 'i'A'n ~ ^1 ' 2 e n o t e s t n e Wronskian of ifi. and \p~. It is

easily verified that for k £ R\(0}, writing for complex conjugation:

(2.4) r*(k) = r.(-k) r*(k) = r (-k)+ +

(2.5) |r_(k) 2

The representation (2.3c) enables us to extend r_(k) to a function

analytic on Im k > 0 and continuous on lm k J 0, k * 0. One can prove,

that r_(k) has at most finitely many zeros, all simple and on the

imaginary axis. Let us denote them by itc ., j = 1,2,...,N and order

(2.6) K 1 > K 2 > ... •> < N > 0.

It turns out that the eigenvalues of (2.1) (the so-called bound states)

are given by -K? < -<% < ... < ~K*. The associated L2-eigenspaces are

one-dimensional and spanned by the real-valued exponentially decaying

functions iK(x,iK.) , j = 1,2,...,N. In terms of these one defines the*~ J

(right) normalization coefficients

* = [ [•* '-—00''

(2.7)

Note that

(2.8) c* = lira if). (x)e -* ,

where ty. (x) stands for the normalized eigenfunction lliK(. ,1K .) II 2i)i.(x,i<.).

In [6] it is proven that (2.1) has a bound state if and only if i(;-(x,0)

vanishes for some x.

Next, we introduce the following functions for k £ E\{0}

(2.9a) ar = r_ , the (right) transmission coefficient

(2.9b) b = r+r , the (right) reflection coefficient.

The appellation is motivated by the asymptotic behaviour of \j> for

|x| -* co with k £ E\{0} fixed:

(2.10) ar(k)i//r(x,k) « e"l k x + br(k)e

lkx as x •* +»

,. s -ikx*** ar(k)e as x -+ -•».

The growth of u_ permits us to extend a and b in a natural way to

continuous functions on all of R, where they satisfy

(2.11) a*(k) = ar(-k), b*(k) = br(-k)

(2.12) |ar(k)|2 + |br(k)|

2 = 1 and for k £ E\fO} |ar<k) | > 0.

It is shown in [7], that br is an element of L1 D L2(1R), which behaves

as o( |k| ) for k -+ ±».

We shall call the aggregate of quantities {b (k),K.,c.} the (right)

scattering data associated with the potential u_. It is a remarkable fact

that a potential is completely determined by its scattering data.

AO

In general, starting from a given potential u it is not possible

to obtain the scattering data in closed form. An exception constitutes

the potential U Q ( X ) = ~^U + 1)sech2x, X > 0, as can be seen from

Chapter 3, section 5.

Let us examine the behaviour of the reflection coefficient b (k) in

some detail. Following [4], we shall from now on distinguish two cases,

the "generic case" and the "exceptional case". In the generic case, the

Jost functions ty (X,0) and <]i„(x,0) are linearly independent

(2.13) lim 2ikr_(k) = W[if> (x,0),<f/„(x,0)] * 0,k->-0 r

whereas the exceptional case is characterized by

(2.14) lim 2ikr (k) = W[i|i (x,0) ,1>. (x,0) ] = 0,k->0 r Z

imkiOexpressing the linear dependence of <p (x,0) and iK,(x,0). The value of b

at k = 0 will frequently appear in our analysis. Therefore, let us give

it due attention.

In the generic case, we obtain from (2.3-13)

(2.15) br(k) = -1 + ark + o(|k|) as k -+ 0,

where a * 0 is some constant,r

In the exceptional case, there is a constant a„ € R\{0} such thato

(2.16) i r(x,0) = a Q^(x,0).

As a consequence of (2.3-14) both r+ and r_ have a finite limit as k + 0.

Taking k -+ 0 in (2.3a) we find

(2.17) r+(0) + r_(0) = aQ.

Hence, in view of (2.4-5)

(2.18) r+(0) = è(c0 - a"'), r_(0) = J(aQ + a^1),

(2.19) br(0) = — ^ 6 (-1,1).

Observe that b (0) = 0 if and only if we are in the "degenerate exceptional

.'.1

case" characterized by

(2.20) i> (x,0) =±(J(„(x,0).

As an example of an exceptional case consider the potential

u _ = v + v2, v € C (R), where v and v decay rapidly enough. Then

(2.21a) ijj (x,0) = exp v(y)dy , iK(x,0) = exp - v(y)dy

(2.21b) aQ = expf J v(y)dyj, br(0) = tanhf j v(y)dyj.

Since \p„(x,0) does not vanish, there are no bound states in this example.

On the other hand, any potential u„ S 0, u» 4 0, is generic with no bound

states.

In [13] it was supposed that the zeros of b (k) are associated with the

discontinuities of u~(x) and that b (k) would have no zeros for a

perfectly smooth initial condition. The previous example shows that this

supposition is incorrect: e.g., any rapidly decaying odd function v of

class C (K) produces a perfectly smooth u~ with b (0) = 0.

Next, let us mention some regularity and decay properties of b (k). If

u.(x) merely satisfies the conditions stated at the beginning of this

subsection, then we have b (k) = o(|k] ) for k •+ +<*> and no better.

However, if u„(x) has rapidly decaying derivatives, then b (k) has

rapidly decaying derivatives as well. The converse is also true.

Specifically, u~ is in the Schwartz class if and only if b is in the

Schwartz class.

To be less demanding, suppose u 0 satisfies the hypotheses (1.3). Then [4],

lemma 2.4 tells ua

(2.22a) In the generic case, b is of class C on R.

ÏÏ1Ü-2(2.22b) In the exceptional case, b is of class C on R\{0} and of

class C on all of R.

(2.22c) In either case, b^^(k) = 0CIUj~5> as k •+ ±°° for j < M-2.

Here IMI denotes the largest integer strictly less than M. As a final

remark, let us mention that it is straightforward to obtain the following

result from the representation formulae (4.2.4.2-4) in [7]:

42

If u,, satisfies (1.3) and there exists an E. > 0 such that UQ(X) =

= O(exp(~2£gx)) as x ->• +« then for any E^ with 0 < e^ < min(eo,KN) the

function b (k) is analytic on 0 < lm k < E. and continuous onr I — *

0 S Im k i e,, while in that strip b (k) = 0(|k| ) as |k| •+ °°. In

particular, if u- decays faster than exponentially as x + t» and has no

bound states, then b (k) is analytic on the open upper half plane.

In the derivation of the asymptotic expansions given in [1] the

extension of b to the upper half plane plays a crucial role. It is there-

fore surprising that we shall not need any supposition of this kind for

our asymptotic analysis, except for an example given at the end of section

7.

2.2. Inverse scattering for t > 0.

From now on we shall impose stronger conditions on the initial function

u„, than those stated at the beginning of the preceding subsection. In

fact, we shall assume that (1.2) is fulfilled.

As a specific example it is good to keep in mind that, except explicitly

stated otherwise, all the results of this paper are valid if (1.3) holds.

Consequently, there is a unique (within the Lax-class) real function

u(x,t), continuous on Rx[0,<=) and satisfying (1.1b), which is a

classical solution of the KdV equation (1.1a) for all t > 0.

Moreover, for fixed t S 0, the function u(x,t) decays rapidly enough for

|x| + • to fall in the class of potentials discussed in the previous sub-

section.

The important discovery by Gardner-Greene-Kruskal-Miura [9], [10] is the

following:

For each t a 0, introduce u(x,t) as a potential in the Schrödinger

scattering problem

(2.23) ijj + (kz - u(x,t))i|> = 0 , -» < x ••- +»;

then the bound states -K 2 •- -K* < ... < -K* do not change with time,

whereas the associated normalization coefficients and reflection

coefficient change in a simple way

(2.24) cT(t) = cTexpUKU}, j = 1,2,...,N

(2.25) b (k,t) = b (k)exp{8ik3t}, -•» < k < +~.

To determine the solution of the KdV problem for all t > 0, one

exploits the fact that the potential u(x,t) can be recovered from the

scattering data {b (k,t),K.,c.(t) } by solving the inverse scattering

problem. For that purpose we introduce the real functions

(2.26a) «(£;»•) = « dU;t) + a^-.t), f, e R, t > 0

N -2K.C

(2.26b) SJd(S;t) = 2 j|1 [cT(t)]2e j , 0

(2 Joo t

br(k,t)e lkCdk.

Since b (k,t) is in L1 fl C_ (-» < k < +•»), the integral in (2.26c)

converges absolutely and J2 (C;t) belongs to L2 D C- (-•» •' 5 < +»).

Consider now the Gel'fand-Levitan equation (see [7])

(2.27) g(y;x,t) + S2(x+y;t) + 0/™ fi(x+y+z;t)S(z;x,t)dz = 0,

with y > 0, x £ R, t > 0.

In this integral equation the unknown S(y;x,t) is a function of the

variable y, whereas x and t are parameters. For each x € E, t > 0 there

is a unique solution 8(y;x,t) to (2.27) in L2 (0 < y < +»). Furthermore,

g(y;x,t) is real and belongs to L1 D L2 fl C (0 < y < +»), such that both

limits at the boundary of 0 < y < +°° exist . In fact we have

(2.28a) lim S(y;x,t) = 0y-M-00

as well as the important property

(2.28b) B(0+;x,t) = f°° u(x,t)dx, x € E, t > 0.

Herewith the inverse scattering problem is solved, since the solution of

the KdV problem is given by

(2.29) u(x,t) = - ~ B(0+;x,t), x € K, t - 0.

Note that the original problem for the nonlinear partial differential

equation (1.1) is essentially reduced in this way to the problem of solving

a one-dimensional linear integral equation. Explicit solutions of (2.27)

44

have only been obtained for b = 0. On account of its asymptotic

behaviour (cf. (5.21)), the solution ud(x,t) of the KdV equation with

scattering data {0,<.,c.(t)} is called the pure N-soliton solution

associated with u„(x).

2.3. Statement of the problem.

We shall study the solution of (2.27) in parameter regions of the

form

(2.30) t £ t0, x > - u - vT, T = (3t) 1 / 3

where p, v and t„ are nonnegative constants. Here y is arbitrary, but the

values of v are restricted to 0 S v < v where v is some generic numberc c

to be specified later on. Furthermore t_ depends on u, v and u_.

It is essential to give our problem a convenient abstract formulation.

Remarkably enough the function space introduced in Chapter 1 also works in

the more general setting of this chapter.

So, once again, let V denote the Banach space of all real continuous

and bounded functions g on (0,°°), equipped with the supremum norm

and

(2.

write

31)

II g U

(Tdg)

sup0<y<+

(y) = d(x+y+z;t)g(z)dz

(2.32) (T g)(y) = f 'u (x+y+z;t)g(z)dz.c nJ c

0J

As in Chapter 1, it is readily verified that T, maps V into V. In the next

section it is shown that if the right reflection coefficient b (k) satisfies

certain, rather modest regularity and decay conditions, then T is indeed

a raapping of V into V. Nevertheless, there is a fundamental difference

with the situation of Chapter 1. There, IIT II tends to zero as t •* » (cf.

Chapter 1,(3.13)). In the region (2.30), however, T is only a small

operator. In fact, v must be chosen with due care so as to guarantee that

llT II < 1. More so, it is an amusing question whether T is a small

operator in the region (2.30) with >> arbitrary! Plainly, this difference

in the long-time behaviour of llT II has its impact on the technicalities

(see section 6).

In the above notation our problem amounts to analysing the solution of

(2.33a) (I + Td)S + 1 6 = -fi, 8 6 V

(2.33b) S. = «d + «c,

- where I is the identity mapping - in the parameter region (2.30).

As in Chapter 1, we know the solution B. of

(2.34) (I + Td)Pd = -nd,

yielding the pure N-soliton solution of the KdV equation. Despite the

differences in the long-time behaviour of IIT II, the basic thought of the

analysis in this chapter is rln same as in Chapter 1, namely to treat the

full problem (2.33) as a perturbation of the pure N-soliton case (2.34).

3. Analysis of a and T .

Let us examine

(3.1) I2(x+y;t)-l f b (k)e^"VXTy^oiK"Ldk

1/3in the parameter region T = (3t) > 0, x > - v - vT, where v and ~J are

nonnegative constants. Performing the change of variables s = 2kT,

T = (3t) 1 / 3 we find

(3.2) ^ J " br<^)exp

which shows that the asymptotic behaviour of L? for t •+ » is intimately

related to the behaviour of the Airy function

1 T s3

(3.3) Ai(n) = -=- lim exp[ins + i -]ds, ri e E.2" R-» -RJ 3

Of course we can rewrite (3.3) as

(3.4) Ai(n) = lira Re /R exp[ins + i -y]dsR-KO

46

1 . . [E

= — l i m I11 R ~ O-"

scos[ns + -^-]

a form which is also frequently encountered in the literature.

Let us list some wellknown [14] properties of (3.3) that will be used in

the sequel. The function Ai(n) has an analytic continuation to the whole

complex n plane, which satisfies

(3.5) ~ Ai(n) = ') Ai(n), i, e C

For n - 0 one has the estimates

(3.6a) 0 < Ai(n) S w 7

n2

(3.6b) x = fn2 3/2

Together, (3.5) and (3.6) imply that for any constant y • 0

(3.7) a o(v) = sup (|Ai(m)(n)|exp[|o(^ + n)372]} - +»

ffl' 1)5— v I LJ JJ

for 0 i 6 < 1, m = 0,1,2

Let us conclude with an innocent looking property of the Airy function that

plays a key role in this paper

(3.3) Qr Ai(n)dn = .

Starting point for our analysis of in tho parameter region1/3 • r

T = (3t) • 0, x • — p - i-T is the following transcription of (3.2)

f'" s 3

(3.9a) :Mx+y;t) = r b(, s ,,;)oxp [ i- s + i -j-]ds, with

(3.9b) b(k,u) = br(k)e"2ik!', • = 2T , ' = Y^ •

Thanks to this representation our task is n-duced to the examination of

the integral (3.9a) in the parameter region 0, • -..

Basically, this examination is performed in the next two lemmas. These

are then combined to give theorem 3.3 describing the structure of

Throughout the following notational convention is used. If f : IR>.[0, ) - C

is bounded, then we write for u •• 0

k€R

The first lemma isolates certain regularity and decay properties with

respect to n of integrals of type (3.9a).

Lemma 3.1. Let g be of class C2[0,=°). Assume that the derivatives g -1 (s),

j = 0,1,2 satisfy

( 3 . 1 0 ) g ( s ) = 0 ( s ) , g O ) ( s ) = 0 ( s ) , g ( 2 ) ( s ) = 0 ( O

for s -* +°°. Then

fR

(3.11) I(n) = lim g(s)exp[ins + i ^-]dsR-«° 0J

i s well-defined for all n E R and I G C2 (-» < n < +») uit^z

(3.12) l i m ( J - ) ^ I ( n ) = 0 , j = 0 , 1 , 2 .

Furthermore, if instead of (S. 10) we make the isii'onjur assurni'tion

(3.13) g(j)(s) = 0(D for s •* +«•

iten ite following representation holds

(3.14) T — I(n) = lim isg(s)exp[ins + i -^-]ds.

Proof: The idea of the proof is to rewrite I(n) as a nice integral over a

finite interval plus a remainder which can be treated using integration

by parts.

In fact, we claim the following. Let n_ ' 0 be an arbitrary constant. Set

s = / i + n . Then for ri 5 —n one has

(3.15) I(n) = IyM + I 2 (n ) + I 3 (n) , with

(3.16a) I ^ n ) = QSS° g(s)exp [ins + i ^ - ] d s ,

(3.16b) I2(r,) = ^

oo S3

(3.16c) 1'n) = - / G(n,s)exp[iiis + i -=-]ds , whoro sn - s and

^ , ( 2 ) ( S ) .

To prove this it obviously suffices to show that

43

(3.18) limR —

Let us write

(3.19a) * =

rR ig(s)exp[iris + i -y]ds = I2(n)

(3.19b)

Then for s £ s_, n 5 ~n„ one has

(3.20) 0 <-nQ + s

2 S 1.n + sz

Now, integrating by parts twice we find for R > s

R R

so so'gds,

1(3.21) e Tgds = -I>I>É J-

where the operator T is defined by

(3.22) Tg = -^-(^g)-

Clearly,

(3.23a) (Tg)(n,s) = ^ ' ^ ^ g t s )

(3.23b) (T2g)(n,s) = G(n,s)

with G as in (3.17).

Substituting (3.23) into (3.21), taking R ->• » and usini; (3.10) we arrive

at the desired identity (3.13), where the integral defining I3 is

absolutely convergent.

H e r e w i t h I ( ' i ) i s w e l l - d e f i n e d f o r n ' ~ r U . N e x t , l e t u s show t h a t

1 e C2 (-nQ <, n < +») with (3.12).

Note first that I. and I„ behave perfectly. Actually, both I. and I?

belong to C (~1Q - n < +ro) and all derivatives vanish for n -+ +<». As for

I., the first statement follows from dominated convergence, the second

from the Riemann-Lebesgue lemma. As for I_, both statements are evident

from its explicit form (3.16b).

Thus it remains to consider I.. From (3.17) we readily obtain that for

each fixed s 5 s_ the function n> ->• e G(n,s) belongs to C2 (-'U s " '; +m)

with

(3.24) lira (- ) = 0, j = 0,1,2.

Moreover, in view of (3.10-20) there are constants c.(n,>,g), depending

only on r\~ and g, such that for n Ï -rin' s ' s0

(3.25) S c.(no,g)s \ j = 0,1,2.

Since the right hand side of (3.25) clearly belongs to L (s S s < +=°),

we may apply the dominated convergence theorem and conclude that

I £ C2 (-nQ S n < +<*>) with

(3.26) lira (4-)JI3(n) = 0 , j = 0,1,2.

As a consequence I £ C2 (~nn S n < +00) with (3.12), as was to be proven.

Finally let us prove (3.14) for n £ -ru under the assumption (3.13).

By virtue of the dominated convergence theorem it is sufficient to show

that

(3.27) ^- lira [ e^gds = lim -i- f ei<J>gds.3n R— sQ

J R - 3n SQ-1

Now, insert (3.21) into (3.27). Then, by the above it is clear that (3.27)

holds provided that

(3.28) lim ^- [-i^e1* ^ Q (iT)£g] = 0.

S->-+oo

It is an easy matter to show that under the assumption (3.13) condition

(3.28) is indeed satisfied.

To conclude with note that since n was arbitrarily chosen we have in

fact shown that I £ C2 (-•» < n < +•») . Likewise (3.14) under the assumption

(3.13) holds for all n £ E. Herewith the proof of the lemma is completed.D

Let us observe at this point that as a consequence of (3.14) and the

above lemma the first derivative of the Airy function can be represented

as follows

(3.29) Ai(1)(n) = -$- lim f is exptins + i -]ds, n £ E./1T R+<= - R j i

We proceed with H rather general lemma, which provides a good insight in

the structure of integrals of type (3.9a) when considered in the

parameter region e > 0, n i -u. To prove it we use some fruitful ideas

already developed in the proof of lemma 3.1.

50

Lemma 3.2. Let b be any function satisfying

(3.30a) b is of class C 2(K), such that b*(k) = b(-k)

(3.30b) The derivatives b J (k), j = 0,1,2 are bounded on R.

Then

fR(3 .31) j ( c , n ) = -s— lira j b ( e s )exp[ i r i s + i -^-]ds

71 R-H» - R J

is well-defined for all n £ K, e > 0 and belongs to C2 (-<= < n < +t»)

(3.32) lim (J-)Jj(e,n) = 0 , j =0,1,2.on

Furthermore one has the representation

(3.33) J(e,n) = b(0)Ai(n) - ieb(1)(0)Ai(1)(n) + R2<c,n>,

where the remainder term can be estimated as follows.

Let e > 0 be arbitrarily fixed.

Then for n > 0

(3 .34a) | R , ( E , r , ) | £ c 2 l lb ( 2 ) l l 4 " 3 / 2 , Hb (2 )ll = sup | b ( 2 ) (k) |

and for n > -n wit/2 n„ S 0 ani/ constant

(3.34b) |R 2 (e , r i ) | S e Jllb (2 )llMC<n0),

where C(nn) ïö ff constant depending only on n , which can be given inexplicit form.

If, instead of (3.30b), we make the stronger asswrrption

( 3 . 3 5 ) b ( j ) ( k ) = 0 ( | k | " ' ) , k ' ±» , j = 0 , 1 , 2 ,

then Ui.: a1 so have the. representation

(3.36) -^J(e.'i) = b(0)Ai(1)(n) + r, (..,•]), uit':

(3.37a) Ir^E.n)! S ell (kb) ( 2 ) ll |n" 3 / 2, ƒ«• f • 0, n - 0

(3.37b) |r,(e,Ti)| S E II (kb) ( 2 ) ll„C(n0), pi- L- - 0, r, - -nQ,

51

where C(nn) denotes the same constant as in (3.34b).

Remark:

( i) Clearly the condition b (k) =b(-k) can be omitted. We include it

because it simplifies the proof somewhat and b (k) has this

property.

( ii ) Observe that (3.33) corresponds to the second order Taylor expansion

for n fixed of the function e>-+ J(E,n) near E = 0. Let us motivate

this choice.

First of all, in view of (3.9) we are interested in the behaviour

of J(c,ri) as E 4- 0. Thus it is quite natural to look at the Taylor

expansion near c = 0. The problem is of course: which order must

we take? In the light of future estimates the answer is rather

simple: such an order n that the remainder term R (e,n) isn

integrable over (1 < n < +°°). Now, with due perseverence one can

show that R (E,H) behaves as n as n + +« for c > 0 fixed,

provided b has n bounded derivatives. Thus we must take n 5 2.

However, it is easily seen that choosing n > 2 does not improve j

the estimates much (in fact it does so only, when for some m the j

subsequent derivatives b J (0), j = 0,1,2,...,m vanish). Hence the

choice n = 2 is singled out. j

Proof: The relation b (k) = b(-k) enables us to rewrite (3.31) as (

R i(3.38) J(e,n) = - Hm Re J b(cs)e1't'ds, <j> = ns + - . I

71 R^° 0J J I

Hence, it is a direct consequence of lemma 3.1 that for any e N 0 the •

function J(e,n) is a well-defined member of C2 (—<= < n < +«>) satisfying j

(3.32). ;

Let us prove the representation (3.33-34).

To this end we insert into (3.31) the Taylor expansion

(3.39a) b(k) = b(0) + kb(1)(0) + b2(k)

(3.39b) b2(k) = Qf (k - k)bv '(k)dk.

In view of (3.3-29) this yields (3.33) with

52

(3.40a) R,(e,n) = 4~ lim f b,(es)eX<t>ds = - Re L( E,n)2 2vr R-x» -RJ 2 it 2

(3.40b) R (£,n) = lim [R b (es)e1(t>ds.

Next, fix E > 0. Set g(s) = b„(es). Then for s £ K

(3.41a) |g(s)| < is^ e2 JH>C2> l)

(3.41b) |g O )(s)| < ( s ^ l l b ^ n ^

(3.41c) |g(2)(s)| s £2llb(2)llM.

Moreover, (3.30) implies that g satisfies condition (3.10) of lemma 3.1.

To prove (3.34b) let i; > 0 be a constant. Put s. = /1 + ru. Then for

n £-n0, ^(e,!-)) n a s t'ie representation (3.15-16-17). Using (3.41a) we find

from (3.16a)

. I s f °1 ' 0J

Furthermore, applying (3.41a-b) and (3.20) we obtain from (3.16b)

(3.42a) 11. I s f ° is2e2|lb(2)ll ds = £2llb(2)ll | s*

1 ' J •» » 6 0

(3.42b) |I2| < cillb^'ll^ (sQ + is^ + s^).

Next, combining (3.41) and (3.20) we get from (3.16c-17)

(3.42c) |I3| S £2|lbU;Ij(s0) with

Together (3.15), (3.40) and (3.42) imply that for n i -r\Q the estimate

(3.34b) holds with

(3.44) C(nn) = (sn + is' + s' t y(sn))/it.U U U o 0 U

To proceed with, let us prove (3.34a).

Let n > 0. Then integrating by parts twice we arrive at (3.21) with sQ

replaced by -R and g(s) = b,,(es). Taking R -+ «° and exploiting (3.30)

we obtain

(3.45) 2irR2(E,n) = - [ G(n,s)e1<!)ds

53

with G(n,s) given by (3.17). An application of (3.41) now gives us

(3.46) |R2(£,n)| i ~ E2llb(2)|

„ . ( 2 ) . . 7 - 3 / 2lib II -sfl

CO 8

which p roves ( 3 . 3 4 a ) .

Lastly, let us show that under the condition (3.35) the representation

(3.36-37) is valid. To do so, observe that for fixed c > 0 the function

g(s) = b(es) easily meets the requirement (3.13) of lemma 3.1. Combining

(3.14) with (3.38) we therefore have

(3.47a) |^(f-,n) = t:"1J(. ,n)

(3.47b) J(e,n) = -j- lira ƒ* b(f.s)ex4>ds

(3.47c) b(k) = ikb(k).

Evidently b satisfies (3.30). Hence J(e,n) has the representation

(3.33-34) with b(k) replaced by b(k). Substituting back we immediately

find (3.36-37), which completes the proof of this lemma. a

Now is the time to apply the above results to the integral (3.9a), to

translate from J(c,n)-language into S.' (x+y;t) —language and to see what we

have got. Doing so we arrive at

Theorem 3.3. Assume that the right reflection r».;c? •"ƒ'icier;r b (k) aatisfi,?^

(3.48) b is of class C2 (B) and the X'ficvzr.MVt.: b ^ ( k ) , j = 0,1,2

are bounded on E.

Then il is iM r->'Ljl ;i differential)! e in V with }'es^e<yt : j x •.;.' .';'.'.",; :\';"»:*

(x,t), x £ E , t > 0. Let t.hf derivation be denoted iv/ .,"•'.

Furthermore, let y > 0, x £ K, t • 0.

Let \i and v denote arbitrary nonnvgative otinstaiitti.

I'ut

( 3 . 4 9 a ) w = x + y + M, b ( k , u ) = b r ( k ) e " 2 i k l ' , b ( j ) = ( ^ ) j b

(3.49b) T = ( 3 t ) 1 / 3 , Z = w(3t)"1/3.

Then one has the representation

(3.50) syx+y;t) = f1br(0)Ai(Z) - JiT~2b(1 } (0,n)Ai(1) (Z) + R(Z, T,y)

with

(3.51a) |R(Z,T,M)| S ï"3»b ( 2 )« t o^ z"3/2 for T > 0, Z -0

(3.51b) |R(Z,T,u) | S T~3llb(2)lloo j C(v) / o r T > 0, Z * -v

wheve C(v) denotes the constant ('6.44) with n replaced by u.

If we make the additional assumption

(3.52) b^j)(k) = (KM"'), k •* +», j = 0,1,2,

then we also have

(3.53) !^(x+y;t) = T % (0)Aiu-" (Z) + r(Z,T,u) wit^

(3.54a) |r(Z,T,u)| < I"3» (kb) ( 2 ) II m •— z"3/2 /or T - 0, Z > 0

(3.54b) |r(Z,T,n)| S T"3« (kb) ( 2 ) II m j C(v) /or T - 0, Z > -v

C(v) as in (3.51b).

Remark. If (3.52) holds then it follows from (3.51-54) that the functions

R(Z,T,u) and r(Z,T,p) can be estimated simultaneously by

(3.55) max(|R(2,T,y)|, |r(Z,T,p)|) S pT~3(1 + u+Z)~3/'2 for T - 0, Z Ï -v,

where the constant p is given by

(3.56a) p = DN2

(3.56b) D = D(v) = — h + (1+v)(| C(v)) 2 / 3j 3 / 2

(3.56c) N2 = H2(ii,br) = maxfjllb^ll^, II (kb) ( 2 ) II V

Proof: Let p S 0 be arbitrarily fixed. Then (3.48) obviously implies that

b(k) = b(k,p) satisfies condition (3.30) of lemma 3.2.

55

Now, let J(e,n) for n £ E , e > 0 be defined by (3.31). Reasoning as in

the beginning of this section we then obtain for y > 0, x £ E, t > 0

(3.57a) «c(x+y;t) = 2eJ(e,n)

(3.57b) e = 2 T , n = 2

with T and Z as in (3.49).

The assertions of this theorem are now easily proven by combining (3.57)

with lemma 3.2.

Specifically, the representation (3.50-51) is merely a transcription of

(3.33-34). Furthermore, lemma 3.2 tells us that ^c(^;t) belongs to

C2 (-00 < £ < +») with

(3.58) lira (-^-)jBc(5;t) = 0 , j = 0,1,2.

Plainly, this implies that for all a £ E

(3.59) N(a,t) = sup | (-^)2«c(5;t) I < +».

With the help of (3.59) it is not hard to show that fi is strongly x-

differentiable. Indeed, let (x,t), x £ E , t > 0 be an arbitrary but

fixed point. Then one has for h £ E, 0 < |h| < 1

a (x+h+y;t) - V. (x+y;t) 3ft(3.60) sup C C

sup0<y <+o it

h nJ

(h - fi)—-r- (x + h + y;t)dh

S jN(x-1,t)|h| = OC|h|) as h + 0.

Consequently, fl is strongly x-differentiable at (x,t) with derivative

(3.61) ft'(x+y;t) = 4E 2 |^ (e,n).c on

Finally, if b satisfies (3.52), then b fulfills condition (3.35) of

lemma 3.2. Together (3.36-37), (3.61) and (3.57b) yield the desired

representation (3.53-54) and so the proof is done.

56

The results laid down in theorem 3.3 are remarkable for two reasons.

Firstly, they display in a strikingly explicit way the structure as well

as the magnitude of the functions y»-»- Si (x+y;t), H'(x+y;t) in the1/3 C C

parameter region x a - u - v(3t) . This will be extensively used below,

when we start estimating in the norms (3.65). Secondly, the conditions imposed

on the right reflection coefficient b (k) are only very weak. In this

respect, let us note that since any initial function un(x) considered in

this paper is supposed to satisfy at least the conditions stated at the

beginning of subsection 2.1, we already know from that subsection that

b r £ L fi C Q C E ) such that b (k) = o(]k]~ ) for k -* ±». Hence, in view or

an interpolation argument, the only extra condition imposed on b by

(3.48-52) is

(3.62) b r £ C2(K) such that b^ 2 ) (k) = (KIM~ 1) f°r k * ±°°-

In general, initial functions satisfying (1.2) will fulfill condition

(3.62). A mild algebraic decay of u„(x) and a number of its derivatives

is already sufficient. Specifically if un satisfies (1.3) then b belongs

at least to C (R) with b J (k) = 0(|k| ), k -»•+<», j = 0,1,2 6, so

that (3.62) is amply fulfilled.

We now turn to the construction of bounds for fl , fi' in variousc' c

settings. To-avoid any misunderstandings and to have an easy reference we

stress the following:

From now on we shall assume that the reflection coefficient b (k) fulfills i

the requirements of theorem 3.3, i.e.: 1

(3.63) b is of class C' (E) and the derivatives b ( j )(k), j = 0,1,2 j

satisfy i

b^j)(k) = Od k f 1 ) , k •* ±».

With the help of theorem 3.3 it is now an easy matter to show that in the

parameter region

(3.64) T = (3t) 5 1, x ? - u - vT, where \t and v are nonnegative

constants,

57

one has the estimates

(3.65a) llficll = sup | « c ( x + y ; t ) | S Y"T ' ,

.00

(3 .65b) II fi II = \il ( x + y ; t ) | d y % |b (CC L1 O-1 C r

(3 .65c) Ihl'll = sup | f ! ' ( x + y ; t ) | e yT ,

f™ - 1

(3.65d) lla'll = !si ' (x+y;t) |dy > ,-T ' ,

C L1 0J " C

where the constant y i s given by

(3.66a) Y = AN + BN , with N as in (3.56c) and

i | d „ )

sup |Ai(n)|,sup |Ai(1)(n)|, [ |Ai(i) (n) |d-i\(3.66b) A = kM = max( |

(3.66c) B = B(v) = iC(v)(1+u) + g- with C(v) as in (3.51b),

(3.66d) N = N1(y,br) = max(2|br(0)|, |b(1} (O.ti) |) with

b(k,ti) = br(k)e"2lku.

Note that (3.65b) hinges on property (3.3) of the Airy function.

Next, let us apply the above results to investigate the mapping T in

the parameter region (3.64). By theorem 3.3 and (3.7-55) there is in this

region a function c(t) such that

(3.67) |iic(x+y;t)| + |^(x+y;t)| •• c(t) (i+y)"3/2.

As a result, the function

r(3.68) (Tg)(y)= o (x+y+z;t)g(z)dz

C 0J °

is continuous in y, since the integrand is dominated by c(t)(1+z) ~!'s'l.

On the other hand

(3.69) llTcgll <- llgll [ | ï ! c ( x + y ; t ) | d y .

Hence, in view of (3.65b), T is a continuous mapping of V into Y with a

norm that satisfies

h f° \ -1(3.70) llTcll 5 l b

r ( ° ) ( K + |Ai(r,)|dr,J + YT ' .

Finally, reasoning as in Chapter 1, we extract from (3.67), that T

is strongly x-differentiable in V with derivative

r(3.71) (T'g)(y) = s:'(x+y+z;t)g(z)dz.

c 0J °From (3.65d) we find in the region (3.64) the following estimate

(3.72) IIT'II S rT~1.

For future reference, lot us note that, in addition to (3.65),

theorem 3.3 gives us useful bounds containing both x and t. In particular,

fixing 6 € (0,1) in (3.7), we obtain in the region (3.64)

(3.73) maxjV^HI, lloMlj < yoT~2exp[- |o(^v + ' )

T-3/', xn

+ PT V1 + v + —

where the constant f( is given by

(3.74) , = AN., A = A(0,v) = max(an (v), a, (v))

with the constants a, N-, •<. (\>) as in (3.56a-66d-7) respectively.

Of course, in the degenerate exceptional case (2.20) the estimates

(3.65-70-72-73) can be improved. Specifically, then theorem 3.3 tells us

that in the parameter region (3.64)

(3.75a) 11:11 ';: fl~2 IITJI • vT"1

(3.75b) ll:;'ll i YT" 3 IIT'll S VT~2,

c c_9

with i still given by (3.66a). Moreover, (3.73) holds with the factor T

in front of the exponential function on the right replaced by T .It is

easily verified that further simplifications of this type occur when also

b (0) vanishes.

As mentioned in subsection 2.1 it rarely occurs that b (0) = 0. However, in

the usual case b (0) * 0 one can still simplify the discussion somewhat

by working in the specially selected parameter region (3.64) with u

chosen to be u = b (0)/(2ib (0)), in which case the derivative of the

Airy function disappears from (3.50). In the present discussion

however we shall stick to (3.64) with u arbitrary.

4. Solution of the KdV initial value problem in the absence of solitons.

In this section we study the asymptotic behaviour of the solution

u(x,t) of the KdV equation evolving from an initial function u,.(x),

which generates no bound states in the Schrödinger scattering problem.

It is assumed that the condition (3.63) is fulfilled.

We shall work in the coordinate region t U , x ^ -;, ^ = p + «I,1/3 .

T = (3t) , where u, v and t are nonnegative constants, with v and

t = t (n,v,b ) to be specified presently.

Clearly, in the absence of bound states the Gel'fand-Levitan equation

reduces to

(4.1) (I + T )3 = -° •c c

For notational convenience we introduce the number v > 0, uniquely

determined by

(4.2) _v f° |Ai(n)|dn = | .

c

The existence of v is guaranteed by the fact (see [14]) that Ai('i) is not

absolutely integrable over -<» < n < 0. As for the numerical value of v ,

we obtain Crom [3], p. 478

(4.3) vc = 1.39.

Moreover, [3] tells us that

(4.4) Ai(n) •• 0 for M ? -v^

Let us now work out our specification procedure.

Firstly, we select: j such that

60

(4.5) O s v < vc.

As a consequence of (4.2) and (4.4) one then has

(4.6) aQ ,(v) s -j + _J |Ai(n)|dn < 1.

Secondly, we fix u 6 0 independently of v.

Thirdly, bearing in mind that |b (0)| s 1, we select tc such that

(4.7) tc > •

with Y = y(lJ,v,b ) as in (3.66a).

After the above specification it is clear from (3.70) that in the

ameter region t s t , x £ -£, £ = u + vT, T =

T occurring in (4.1) can be estimated as follows

parameter region t s t , x £ -£, £ = u + vT, T = (3t) the operator

(4.8) IIT II i a < 1 with a = |b (0) |aQ (v) + Y ( 3 t c ) ~1 / 3 .

This implies that I + T is invertible on the Banach space V.

As a result, (4.1) has a unique solution S € V satisfying

( 4 . 9 ) llgll S io1llficll, w i t h u 1 = (1 - o ) ~ 1 .

Furthermore, (4.1) implies that 3 is strongly x-differentiable with

derivative

(4.10) S' = -(I + T )"'(T'g + £!').c c c

From (4.3-9-10) and (3.72) we obtain the estimate

(4.11) lie1» S w0T~'lls: II + u> llfl'll, with w- = :.;2y.

Recall that

(4.12) u(x,t) = - -£- 8(0+;x,t).

Since

(4.13) | -^ R(0+;x,t)| ; sup | ö(y;x.t) | = IK-MI,0-;y<+t»

we arrive at the following result

61

(4.14) | u ( x , t ) | s U()T~1IISÏcll + n^hï'J.

In particular, it follows from (3.65a-c) that

(4.15) |u(x,t)| < io2T~2, with u 2 = (uQ + w^-y.

The above results are summarized in

Theorem 4.1. Let u(x,t) he the solution of the Xor;,e,.>ej-de Vries problem

( -co . x • +<», t • 0

(4.16) { t X XXX

u(x,O) = u n(x),

where the. real initial function u.(x) is sufficient-1 y smooth and decays

sufficiently rapidly for |x| -> •» for the uhoie of the inVrrae scait-~'riny

method to work and to guarantee the regularity and decay :;ropi-rf-y (;'•.•!/•)

of the reflection coefficient b (k). Amamc that, aa a poten1 la? in :he

Sohrödinger scattering •.n'obl.cm, u„(x) jenei'ate.:; no bound siates. Lei

\>, v and t be nonnegatioe constant a, uith \> and t sat.isf.iina (•>.!•) wi 1

(4.7) respectively.

Then, in the coordinate region t •'• t_, x -• -r,, r, ~ \i + \>T, T= (3t)

one has the foi I owing cstitn.üt- f 'he solution

(4.17) |u(x,t)| • .^T' 1 sup |;:c(x+y;t)| + ,^ sup ! : ' c

O-.y.+o» o-y-+»

w h e r e u a n d ...i. a r e t h e c o n s t a n t s i n t r o d u c e d i n ( - ' . I I I a n : ,'•;..".'

r e s p e c t i v e l y a n d '.I ( x + y ; t ) (';; j i v e n .•>./ ( / - . I ) .W i t h t h e a m i ; U n i t :><„ a s i n ( - l . l h ) : j e ::ane- Cor t - C

2 " c

( 4 . 1 8 ) sup | u ( x , t ) ! • . . ,T~ 2 .x — C

Let us mention some implications of the preceding theorem.

As a consequence of (4.17) and (3.73) tlie solution u(x,t) of (4.16)1 /3satisfies, in the coordinate region t t , x • -!., r, = u + vT, T = (3t) ,

the following x aud t dependent bound

(4.19a) |u(x,t)| ' al~2ey->\- 0f

(4.19b) a = (u)Q + u^Yg, b = (ui0 + w^,,

with 6 £ (0,1), y , p the constants introduced at the end of section 3.y

Hence, for t £ t

(4.20) j |u(x,t)|dx < aT"1 [ exp -^0s 3 / 2]ds + 2bT~2.

Combining (4.18-20) with the formulae (see [18])CO |..„

(4.21a) u(x,t)dx = - - log(1 - |br(k)|2)dk

(4

we

(4

U

21b)

obtain

22a)

22b)

Remark.

fJ U2

- j

f u

(x,

(x,

2 (x

t)dx =

t)dx =

,t)dx =

_ 2

r

f? or

k2 logd

logd -

k2logd

- |b r (k ) |

|b r(k) ')

- |b r(k)

2)dk,

dk + Q(t

! 2 ) d k + 0

- 1 / 3 ) as t

(t ) as t

( i) Observe that (4.22a) can also be derived from (2.28), since by

(3.65a) and (4.9)

(4.23) [ u(x,t)dx = p(O+;-(;st) = 0(t~'/'3) as t > -.

( ii) If u0 satisfies (1.3) then all of the conditions of theorem 4.1 are

fulfilled.

(iii) In the degenerate exceptional case (2.20) the above estimates can~2 ~3

be improved. For instance, in (4.18-19) we can replace T by T

( iv) For a physical interpretation of (4.21-22-23) we refer to the

discussion in section 7.

5. The operator (I + T ) .

When, as a potential in the Schrodinger scattering problem, un(x)

produces bound states, then the operator I + T, makes his entrance in the

Gel' fand-Levitan equation. The following lemma shows that this operator

has a nice inverse.

Lemma 5.1. For any value of the parameters x £ R, C > 0, the operator

I + T. is invertible on the Banaah apace V with inverse S = (I + T.)d d

given by

N -2K.yJ(5.1a) (Sf)(y) = f(y) - . | 1 A^e J

N / f ™ ~2K.Z v(5.1b) A. = i J 1 Bi-(2 e l f(z)dzj ,

2K.xwhere ( g . . ) i s tóe inverse of the matrix A = ( [ c . ( t ) ] e S. . + (K ,+K . ) ) .

Furthermore, S and i is strong ^.-derivative S' satisfy the bounds

(5.2)

(5.3a)

(5.3b)

IIS» S a Q ,

1 + .i

N . . = 2 ( K , K . )

I I S ' x £ t > 0 ,

, Nj - IT . , a( = 2aQ i ?=

N;;>

i

£=1£*i

Nn

P=1

K . +KJ P

K . —KJ P

Thus, IISII and IISMI a r e uniformly bounded for x £ K, t > 0 and t^e bounds

are explicitly given in terms of the K. only.

Proof: For any fixed x £ E, t > 0 we may solve the equation

(5.A) (I + Td)g = f, f,g £ V

to find

N -2K.y(5.5) g(y) = f(y) - X. A.e J ,

where the A. satisfy

(5.6a) a. .A./•» -ZK . Z

J = 2 0J 6 '

f(z)dz,

(5.6b) a.. = a.6. . +lj j IJ

i = 1,2,....N

2K .x2

We shall show that the matrix A = (a..) has positive determinant

and thus has an inverse A = (6..), so that I + T, is an invertible6 . .

)operator on the Banach space V with inverse S = (I + T.) given by (5.1).

Furthermore, we shall prove that the matrix elements fi.. are uniformly

bounded for x £ E, t > 0, where the bound is explicitly given in terms of

the K• by

N(5.7) |B..| , 2 ^ . ) ^

NII

p=1

K . +K

_J PK . — KJ P

s N. . .IJ

To achieve our goal, let us first introduce some notation. By X we

denote the real Hubert space L2 (O,») with inner product f,g> = _

= nf f(y)g(y)dy. In JC we consider trie ele.nents e. defined by e.(y) = e

^ 1 1 'We write A for the Gram matrix of the vectors e.,e_,...,e , i.e.

aij ' aij < ei' ej' i KjSince the vectors e. ,e->•>•9eN are linearly independent, it is clear [5],

that det A > 0.

Let us write (A) = (g.-).

Next, select f ,f.,..., f € 3C such that <i^,e.> = 0 and <f. ,f .•» = a. 6. .

s

the

for i,j = 1,2,...,N. Put g. = f. + e.. Then a.. = <g-,g->, which shows

that A = (a.-) is the Gram matrix of the vectors g.,g9,...,g . Since

vectors e. ,e,,. . • .e,. are linearly independent, the same holds for

g, ,g„,... ,g,,. Hence we have proven, that det A •> 0 and the existence of

A~' = ((3..) is guaranteed. N

To obtain the estimate (5.7) we introduce the vectors h. = .£ B--e..

Clearly, <h.,e.> = fi.. and <h.,h.-- = ?>...

Now let P be the projection of Jf onto span ( g1 ,g_,. .. ,g,.f. Then

N N(5.3) Ph. = . f B. .-;h.,g >g. = -E, P-.g-9

so that <Ph.,h.> = 8.. and

I i j I i* i"~ j * i " ~ i i j j "

By d i r e c t c a l c u l a t i o n ( see [ 5 ] ) we o b t a i n

(5.10)

det

and so t h e proof of ( 5 . 7 ) i s c o m p l e t e .

Note , t h a t by ( 5 . 1 b - 7 )

(5 .11) | A . | % llfll . £ , ^ N . . .

which implies the bound (5.2-3) for IISII.

It remains to estimate the strong x-derivative of S which by (5.1)

is given by

N -2K.y

(5.12a) (S'f)(y) = - ^ Aje J

N 2K

(5.12b) A! = -2K.A. + . £ . g.. — - 2 - A .J J J i,P=1 iJ <i

+Kp P

Using (5.7-11-12b) one gets

/ N N 2K

(5.13) |A!| < IIfII 2K. . I . - S . . + . .1 . — N» —-r 2- N. .' j' V J 1 = 1 Ki XJ i»£,P=1 <£ £p K^+Kp U

from which the bound (5.2-3) for IIS'II is an immediate consequence.

D

Remark. From the above proof it is clear that lemma 5.1 is still valid

(with the same bounds (5.2-3)) if the time evolution of the normalization

coefficients is not given by (2.24) - as prescribed by the KdV equation -

but is instead completely arbitrary.

Corollary to lemma 5.1. let x £ I and t • 0. Then the equation

(5.14) (I + T,)0 = -a,a a

admits a unique solution B, £ V and we have

(5.15a) Sd(y;x,t) = - 2 , J = ) p..e ^

N f -2< y N -2> .y(5.15b) B'(y;x,t) = 4 £ | < ^ ^e P - ? - ^ - 3 e J'

Remark. Let us recall that B, produces the pure N-soliton solution of the

KdV equation associated with un(x) through the formula

(5.16a) u,(x,t) = - f- 6,(0 ;x,t)

N

£,p=1 p

, N N

1 - . ? , — ! — 3. . I.V i,j = 1 K.+K u/

Since

66

i_ 2K x N

e p ^

we find for u, two additional representations that are useful as well:a

N r -2 2 % V 2 \2

(5.16b) ud(x,t) = -4 g1 K [r (t)] e v { ^ B£ ) ,

N 2 -2K X/ N N2

(5.16c) ud(x,t) = -4 Z K [cr(t)] e P M - i 5 = 1 +>; Bi- 1 .

Consequently

(5.18) 0 Èud(x,t) £-4a 0 . J = ) K.N..,

so that u,(x,t) is uniformly bounded for x € B., t > 0 and the bound does

not involve the c. but depends only on the K. in a simple explicit way.

Combining (5.7-16) and (2.6-24) we obtain

(5.19a) |ud(x,t)|dx = 0(e ) as t -> »

(5.19b) ud(x,üdx = OCe ' ) ast-*».

Recall that (see [10])

=, N(5.20a) ud(x,t)dx = -4 ^ K

(5.20b) _ ƒ u|(x,t)dx-f p^ K».

Starting from (5.16b-c) it is shown in [15] that as t approaches infinity

the pure ïl-soliton solution decomposes into N solitons uniformly with

respect to x on 1R. More precisely one has

N +(5.21a) lira sup |u (x,t) - ^ (-2K * sech2 [< (x-x -k^ t) ]) | = 0 ,

67

6. Solution of the Gel'fand-Levitan equation in the presence of bound

states.

Under the condition (3.63) we now proceed to investigate the full

Gel'fand-Levitan equation

(6.1) (I + T d + Tc)3 = -a

' /3in the parameter region t 5 t ,, x S -<;, c. = M + uT, T = (3t) , whe^-e

u, \> and t , are nonnegative constants, with \> satisfying (4.5) and with

t c d = tcd(n,v,br,K1,K2,...,KN) > j to be specified shortly.

Applying the operator S = (I + T.) we can rewrite (6.1) as

(6.2) (I + STc)p = -Sn.

To ensure the invertibility of the operator I + ST , it suffices to prove

that ST has norm smaller than 1, as was the case in our comovins»c

coordinate analysis [8]. However, in the present situation the operator

ST is less manageable. Let us circumvent this difficulty and consider

the operator T S instead.

From (5.1-11) and the estimate-2v:. z

J(6.3)

we obtain

r«> - 2 v : . z| H ( x + y + z ; t ) e J d z | •; — Ils2

Q J C ^ C

i

( 6 . 4 ) UT Sll < IIT II + II£2 II . ? . —•— N . . .c c c i , j = 1 2K .K . i j

Hence, by (3.65a-70)

(6.5a) 11X Sll a jb (0)|a_ , (v) + yï"1, where an , (v) is given by (4.6)C IT U ) » U y I

and

(6.5b)

We now se lec t t , such thatcd

(6.6) t_A > i i n a x | i , Y 3 / i - |b r (O) |a f t »

For t < t . w e then havecd

(6.7) llTcSH t 5 < 1, with 5 = |br(O)|aQ , (v) + ï O t ^ ) 1/3.

This shows, that the operator I + T S is invertible on the Banach space

V. Consequently, the same holds for I + ST and we have

(6.8a) (I + S T c ) ~1 = I - S(I + T cS)"

1T c,

but also

(6.8b) (I + S T c ) ~ ' = S(I + TcS)~1S~'.

We conclude that, in the parameter region t -• t ,, x " -c,, i, = \j + vT,

1 /3 C

T = (3t) , the equation (6.1) has a unique solution 3 6 V . This

solution can be represented in terms of S, T and V. by means of a Neumann

series:(6.9) B = m|0 (-STc)

m(-SS2) .

Note that, while in general this series converges rather slowly, the

convergence in the degenerate exceptional case (2.20) is rapid for large

t by virtue of (3.75a).

7. Decomposition of the solution of the KdV problem when the initial

data generate solitons.

Let us put

(7.)) (S = Sd + 3 C, with

(7.2) e, = -SS2,.

d d

Introducing the decomposition (7.1) into (6.2), we find

(7.3) (I + ST c)@ c = -S(JJc + T c 3 d ) .

From (6.8b) it is clear that (7.3) has a unique solution p £ V,satisfying

(7.4) lig II s (1 - IIT Sll) II:

By (5.7-15a) and (6.3) one has

N ,(7.5) l lT

cSa" s llfJc" i ?=1 K V* i

so t h a t , i n v iew of ( 5 . 2 - 3 a ) and ( 6 . 7 ) ,

( 7 . 6 a ) IIS II < ü J f i H, w i t h

(7.6b) üj = (1 - 5)"1a20.

Let us keep in mind that the solution of the KdV equation is given by

(7.7) u(x,t) = u,(x,t) - -^- B (O+;x,t),O aX C

where u,(x,t) denotes the pure N-soliton solution introduced in (5.16).

Therefore, we need estimates of the derivative of B with respect to x.

From (6.8) and (7.3) it is clear that (3 is strongly x-differentiable,

the derivative 3' being uniquely determined by

( 7 . 8 ) ( I + S T ) 0 f = - S { T ' ( 6 + 3 , ) + fi' + T R j } - S'{V. + T ( p + S , ) } .c c c c d c c d c c c d

Using (5.3a-7-15b) and (6.3) one gets

N

(7.9) IIT 6'II S 2ajn II . ? . N. ..c d 0 c i,j=1 ij

Furthermore, (3.71) and (5.7-15a) imply j

(7.10) DT'g.ll s llfiMl . ? , — N. . . !cd c i,j = 1 K. i j i

t'

Combining ( 3 . 7 2 ) , ( 5 . 2 - 3 a ) , ( 6 . 7 - 8 ) and ( 7 . 5 - 6 - 9 - 1 0 ) we o b t a i n from ( 7 . 8 ) 1the following estimate

(7.11a) II6Ml S üo l lnc l l + i , IIf!Ml, w i t h

-U~ ~ N \ -1

(7.11b) uQ = M^aQ fuijO + 2aQ i t f r r . If+ a^aQ (aQ + u ^ ) 2 .

Evidently

(7.12) 1 - ^ - 6 ( 0 + ; x , t ) | S IIBMI.oX C C

70

By virtue of (3.65a-c) this yields

(7.13) |- ~ @c(0+;x,t)| < ^ T " 1 , with u 2 = (~iQ + w

Summarizing the above results we obtain

Theorem 7.1. Let u(x,t) be the solution of the Korteweg-de Vries problem

u - 6uu + u = 0 , - o o < x < + » , t •• 0

u(x,0) = UQM

where the real initial function un(x) is sufficiently smooth and decays

sufficiently rapidly for |x| •+ «> for the whole of the inverse scattering

method to work and to guarantee the regularity and decay property (3.6?)

of the reflection coefficient b (k). Assume that, as a potential in the

Schvödinger scattering problem, u~(x) produces N 5 1 bound states.

Let u, v and t , be nonnegative constants, with v and t , satisfying (4.S)

and (6.6) respectively.

Then, in the coordinate region t ? t ,, x 5 -£, r, = y + \>J, T = (3t)

one has the following decomposition of the solution

(7.15a) u(x,t) = ud(x,t) + uc(x,t),

(7.15b) |uc(x,t)| £ ~ Q sup |ïïc(x+y;t)| +.7^ sup | — i^ (x+y; t) | ,

where u , ( x , t ) is the puve Ksot.iton solution (é.16), .o and ~ :rc .'./:,-

constants introduced in (7.11b) and (7.6b) rusvectioelij ?';.,' ;: ( x + y ; t )

Is given by (3.1).

With the constant ,.>,. as in (7.13) we !iaut? <\>r t • t .2 • cd

( 7 . 1 6 ) sup | u c ( x , t ) | -, T^T" 1 .

Evidently, in the degenerate exceptional cuse (2.20) the estimate

(7.16) can be improved, since T can be replaced by T ~. Although similar

remarks apply to the estimates below, they are omittod to avoid

interference with the reasoning.

Let us emphasize that all of the requirements of theorem 7.1 are

fulfilled if uQ satisfies (1.3).

71

Theorem 7.1 has a number of interesting consequences.

Firstly, by combining (7.16) with (5.21) it is found that the

solution u(x,t) of (7.14) splits up into N solitons as t + » in the

following way:

Corollary to theorem 7.1. Let {b (k), K. > * •• ... > r.„, c , c , ..., c }

be the right scattering data associated with un(x). Then the solution of

(7.14) satisfies

N(7.17) lira sup |u(x,t) - | 1 •{-!>

2 sech2 [r. (X-X +-4K 2 t) ]) | = 0t-H» X?-C. ^ P P P

with x as in (5.21b).P

Furthermore, it follows from (7.15) and (3.73) that the nonsoliton

part of the solution satisfies, in the coordinate region t 3 t ,, x s — c;,1 /3

C = U + vT, T = (3t) the x and t dependent bound

(7.13a) |uc(x,t)| S IT"'exp[ - 10 (^-^j J + bT"2(i + ?-LA"

(7.18b) a = (SQ + ü,)Yö, 6 = (5Q + I O ^ P ,

with 0 € (0,1), Y , p the constants introduced at the end of section 3.

The estimate (7.18) is to be compared with the estimate (4.19) obtained in

the absence of solitons. Clearly, by contrast with (4.20), the bound

(7.18) does not permit us to conclude that the L ' -r, > x •- +»)-norm of

u (x,t) tends to zero as t + ».

However, we obtain from (2.28), (3.65a), (7.6)

(7.19) ! uc(x,t)dx = ec(O+;-C,t) =Q(t" 1 / 3) as t - ».

Hence, in view of (5.19a-20a)

(7.20a) j u(x,t)dx = -4 Z, v + 0(t ) as t -• «•.

From the formula (see [18])

r 2 r N

(7.21) I u(x,t)dx = - log(1 - |br(k)|2)dk - 4 ^ K

we find for the complementary integral

f-C 2 f"" -1/1(7.20b) ! u(x,t)dx=-^ I log(1 - |br(k)|

2)dk + 0(t )

as t ->

In particular, in the case of a nonzero reflection coefficient, it follows

from (2.12), (7.20) that there exists a t such that for t • t~

(7,0 r

.22) u(x,t)dx • 0 and u(x,t)dx -• 0.-J oJ

Thus, in the light of (5.18), the reflectionless solutions of the KdV

equation can be characterized as the only nontrivial solutions that do not

assume positive values.

Let us mention, that the time independent quantity (7.21) is usually

referred to (cf. [17]) as the total momentum associated with the

solution u(x,t) of (7.14). For nonzero b , (7.20) suggests that as time

goes on there is a definite positive momentum associated with the disper-

sive wavetrain given by (7.20b), as well as a definite negative

momentum associated with the pure N-soliton solution given by (7.20a).

Incidentally, observe that (7.21) gives an immediate proof of the

following result which is partly known from quantum mechanics [11]:

Let U Q ( X ) be an arbitrary potential in the Schrb'dinger scattering problem,

satisfying the conditions of subsection 2.1.i

If ƒ u_(x)dx 0 thei iin has at least one bound st:itc; if tu # 0 and {

ƒ u„(x)dx = 0, then u„ has at least one bound state and a nonzero ;'U ° „, !

reflection coefficient; if x/ u (x)dx • 0 then n. has a nonzero i

reflection coefficient.

We continue our list of consequences of theorem 7.1 by considering

Lz-estimates. From (7.18) we find

'.23) j u2(x,t)dx = 0(t"'/3) as t > -.(7.

Since by (5.20a) and (7.16)

r -1 N

(7.24) j |uc(x,t)ud(x,t)|dx • 4~,T ^ ,y

we conclude from (5.19b-2Ob) that

(7.25a) | u*(x,t)dx = -y p I , -p + 0(t"1 / 3) a s t

Using the formula (see [18])Ju 2 (x , t )dx = - - k2 log(1 - jb (k ) | 2 )dk + - £ I , • 3 ,

Ti Q J r j p-i pwe o b t a i n as a c o u n t e r p a r t t o ( 7 . 2 5 a )

J - C o f" _ 1 i-,

u 2 ( x , t ) d x = - - k M o g ( l - !b ( k ) ! J ) d k + Q ( t ' J )oJ r

a s t > ••'.

In the literature [10] the time independent quantity (7.2b), sometimes

[17] with a factor J in front of it, is referred to as the energy

associated with the solution u(x,t) of (7.Ü). For nonzero b , (7.25)

suggests that as t •* »• the dispersive wave tra in moving to the left, though

it may decay asymptotically to zero amplitude, still carries a finite

amount of energy given by (7.25b), while on the other side of the line

the N-soliton solution, falling apart into N solitons moving to the right,

carries the energy given by (7.25a). It is interesting to compare the

0(t ) term in (7.25b), due to the interaction between the dispersive

wavetrain and the N solitons, with the Q(t ) term in (4.22b) caused by

the self-interaction of the dispersive wavetrain, in the absence of

solitons.

Finally, let us remark that (7.16) improves Tanaka's result ([16],

Theorem 1.1), which can be reformulated as

(7.27) lim sup |u (x,t)| = 0 for v 0 arbitrarily fixed.

Moreover, it is not difficult to derive more precise versions of (7.27).

As a first example, let us make the additional assumption

74

(7.28) There is an integer tt S 2 such that b Ê c"(K) and all

derivatives b (k), j = 0,1,...,n satisfy

b^j)(k) =0(|kf 3) k -> ±».

Then, it follows from a slight modification of Chapter 1, Appendix B, that,

given the positive constant v, there is a constant yn such that

(7.29) \ncUlt)\ • | |-i! c(ut)| • vo(~n for ' - vt • 0.

Hence, by (7.15b) we can choose t , such that

(7.30) |uc(x,t)| Ï pQx"n, t i t c d, x < vt,

where y is some constant. Thus we arrive at

(7.31) sup |uc(x,t)j = 0(t~") as txjvt

Note that, at the same time, (7.31) improves some of the results obtained

in Chapter 1, since it is easy to show that (7.29-30-31) with slightly

different constants u„, ü-, t , are still valid if the condition (7.28)

is relaxed to that stated in Chapter 1, Appendix B, with n = n.

Incidentally, we can apply the above results to estimate the decay

rate of the solution u(x,t) of (7.14) as x v +•" for fixed t '• t ,. Since,

in view of (7.5-16c), u, (x,t) decays exponentially as x • +•••, we find

from (7.15a) and (7.30) that for t • t ,cd

(7.32) u(x,t) = 0(x~") -is x - +...

If uQ satisfies (1.3), then we obtain from (2.22) that (7.2S) holds with

n = ii;i.i + 2 - (,/2). Hence, for t • t

(7.33) u(x,t) = 0(^('/2)"3Ml1"2) as s - +-.

Herewith, for t t , the estimate

(7.34) u(x,t) = 0(x ' ' ') as x • +•,

which was obtained in [4] for any fixed t 0, is improved.

As a second example, let us suppose that u.-. satisfies (1.3) and

furthermore, that there exists an (.„ • 0 such that u„(x) = 0(exP(~2i:,,x))

as x •+ +<».

Since un satisfies (1.3), we can appLy theorem 7.1 to find a constant

t d such that (7.15) holds for t 2 t d, x > 0. Now, fix v > 0. Then it

follows from combining the last remark of Subsection 2.1 with Chapter 1,

Appendix A, that, given c with 0 < r.. •' min(c ,»: ), there exists a

constant y. such that

(7.35) |flc(f.;t)| + I •£- »c(f.;t)| ' ^,exp(-2i ,r + 8t>t), t • t^, C • v

Hence, by (7.15b)

(7.36) |uc(x,t)| - >1exp(-2L1x + 8» ]t), t - t^, x • vt

where y. is some constant.

Firstly, (7.36) gives us the decay rate of the solution u(x,t) of (7.14)

as x •> +°° for fixed t s t ,. Since, by (2.6) and (5.7-16c), u (x,t) =

= 0(exp(-2r s)) as x -> +<», we conclude from (7.15a) and (7.36) that for

t - t ,cd

(7 .37 ) u ( x , t ; = 0(exp(-2> x ) ) as x •> +«• fo r any • w i th

0 • : , - m i n ( . . 0 > - N ) .

References

t 1] M.J. Ablowitz and 11. Segur, Asymptotic solutions of the Kort owe i;--deVries equation, Stud. Appl. Math. 57 (1977), 13-44.

iS e c o n d l y , c h o o s i n g >. s u c h t l i a t 0 • • > •- rain(t: ,» , J > v ) , we o b t a i n f r o m >

( 7 . 3 & ) |

j

( 7 . 3 8 ) s u p | u ( x , t ) | = 0 ( e x p ( - ( l ) t ) ) a s t ^ - I

with a = 2c (v-4(z) •• 0. '•

And so we have obtained another more precise version of (7.27). •

t 2] M.J. Ablowitz and H. Segur, Solitons and the Inverse ScatteringTransform, Philadelphia, SIAM, 1981.

[ 3] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions,National Bureau of Standards Applied Mathematics Serie1-., No. 55,U.S. Department of Commerce, 1964.

[ 4] A. Cohen, Existence and regularity for solutions of the Korteweg-deVries equation, Arch, for Rat. Mech. and Anal. 71 (1979), 143-175.

[ 5] P.J. Davis, Interpolation and Approximation, Dover, New York, 1963.

[ 6] P. Deift and E. Trubowitz, Inverse scattering on the line, Comm.Pure Appl. Math. 32 (1979), 121-251.

[ 7] W. Eckhaus and A. van Harten, The Inverse Scattering Transformationand the Theory of Solitons, North-Holland Mathematics Studies 50,1981.

[ 3] W. Eckhaus and P. Schuur, The emergence of solitons of the Korteweg—de Vries equation from arbitrary initial conditions, Math. Meth. inthe Appl. Sci. 5 (1983), 97-116.

[ 9] C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Method forsolving the Korteweg-de Vries equation, Phys. Rev. Lett. 19 (1967),1095-1097.

[10] C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Korteweg-deVries equation and generalizations VI, Comm. Pure Appl. Math. 27(1974), 97- 133.

[11] L. Landau and E. Lifschitz, Quantum Mechanics, Nonrelativistic Theory,Pergamon Press, New York, 1958.

[12] P.D. Lax, Integrals of nonlinear equations of evolution and solitarywaves, Comm. Pure Appl. Math. 21 (1963), 467-490.

[13] J.W. Miles, The asymptotic solution of the Korteweg-de Vries equationin the absence of soiirons, Stud. Appl. Math. 60 (1979), 59-72.

[14] F.W. Olver, Asymptbtics and Special Functions, A'.ademic Press, NewYork, 1974.

[15] S. Tanaka, On the N-tuple wave solutions of the Korteweg-dc Vriesequation, Publ. R.I.M.S. Kyoto Univ. 8 (1972), 419-427.

[16] S. Tanaka, Korteweg-de Vries equation; asymptotic behavior ofsolutions, Publ. R.I.M.S. Kyoto Univ. 10 (1975), 367-379.

[17] N.J. Zabusky, Solitoiis and bound states of the time-independentSchrSdinger equation, Phys. Rev. 168 (1968), 124-128.

[18] V.E. Zakharov and L.D. Faddeev, Korteweg-de Vrics equation, acompletely integrable llamil tonian system, Funct . Anal. Appl. 5 (1971),280-287.

77

CHATTER THREE

MULTISOLITON PHASE SHIFTS FOR THE KORTEWEG-DE VRIES EQUATION IN THE CASE

OF A NONZERO REFLECTION COEFFICIENT

We study multisoliton solutions of the Korteweg-de Vries equation in ]

the case of a nonzero reflection coefficient. An explicit phase shift

formula is derived that clearly displays the nature of the interaction of

each soliton with the other ones and with the dispersive wavetrain. In '

particular, this formula shows that eacli soliton experiences in addition S

to the ordinary N—soliton phase shift an extra phase shift to the left !

caused by fhe collision with the dispersive wavotrain. I

I. Introdurtion.

We consider the Korteweg-de Vries (KdV) equation u - i;ui +

+ u = 0 with arbitrary real initial conditions u(x,0) = U „ ( N ) , whichxxx 3 ' 0

are sufficiently smooth and decay sufficiently rapidly for x • • for the

whole of the inverse scattering method to work and to guarantee certain

regularity and decay properties of the scntterini; data, lo be stated

further on. The long-time behaviour of rlie solution U ( X , L ) of the KdV

78

problem has been discussed by numerous authors. The general picture is,

that as t •* +00 the solution decomposes into N solitons moving to the

right and a dispersive wavetrain moving to the left. As t • - the

arrangement is reversed. The emergence of the N solitons as t • +"> for

rather arbitrary classes of initial conditions was demonstrated

rigorously in [8], Chapter 1 in this thesis (see also the discussion in

[7]). Earlier - but less detailed and not widely known - results in that

direction were given in [11]. Further extensions of the asymptotic

analysis and improvements of results were recently presented ir [10],

Chapter 2 of the present thesis. In the literature many attempts were made

to calculate the phase shifts of the solitons as they interact both with

the other solitons and with the dispersive wavetrain. Many incorrect

results were given (cf. [11] and [12]), until finally the question was

settled by Ablowitz and Kodama [1], who presented a correct phase shift

formula.

In this paper we rederive this phase shift formula, starting from our

asymptotic analysis of the solution given in Chapter 2. l.'o next sliov how ;i

simple substitution produces .1 more transparent, formula th.it cleirly dis-

plays the nature of the interaction of each soliton with the other ones .md

with the dispersive wavetrain. Fron our phase shift formula it is evident,

that each soliton experiences, in addition to the ordinary "-soliton phase

shift, an extra phase shift to the ' ft, the so-called continuous phase

shift, caused by the collision with the dispersive wavetrain. Thus, the

presence of reflection causes a delay in the soliton motion. Furthermore,

our formula shows that the total phase shift is completely determined bv

the bound states and the ri;T,ht reflet'.on coefficient. Hence, tlK-re is no

dependence on the ri;;ht normalization coefficients.

From the original formula the above facts are hard to see. i

The composition of this paper is as follows. In .section 2 we briefly 1

discuss the left and right scattering data associated with u-(x) and show

how the left scattering data can be expressed in terms of the right

scattering data in a convenient way. In section 3 we recall a result known

from Chapter 2, concerning the asymptotic behaviour of u(x,t) as t • +•".

By a symmetry argument we derive from this result the asymptotic behaviour

of u(x,t) as t ->• -». Next, in section 4, the two asymptotic results are

74

combined to give the Ablowitz-Kodama phase shift formula. The

representation of the left normalization coefficients in terms of the

right scattering data, which was obtained in section 2, then enables us

to write the phase shift formula in a more transparent form. Finally,

as an exercise, we calculate in section 5 the continuous phase shifts

arising from a sech2 initial function.

2. Scattering data and their properties.

For Ira k 2 0 we introduce the Jost functions tf/ (x,k) and i(i„(x,k), two

special solutions of the Schrödinger equation

(2.1) ty + (k2 - u (x))i); = 0 , -co < x < +~

determined by

(2.2a) 'i, (x,k) = e~lkxR(x,k), lim R(x,k) = 1, lim R (x,k) = 0

x>

(2.2b) <|<»(x,k) = e L(x,k), lim L(x,k) = 1, lim L (x,k) = 0.X-*-+<» X->-+»

We set j

(2.3a) r (k) = 1 - (2ik)"' [ u (y)R(y,k)dy k 6 C \{0} j

- i(2.3b) r+(k) = (2ik)"' j e~2lkyuQ(y)R(y,k)dy k t E\iO} j

-1 r - i(2.3c) £+(k) = 1 - (2ik) uQ ;L(y,k)dy k £ C+V0} i

jU.3d) £_(k) = (2ik)"' j e2lkyu0(y)T,(y,k)dy k € R\iO(.

Note that r_(k) = £+(k), whereas r^(k) = -C_(-k). It is well known [7],

that r_(k) is analytic on C with at most finitely many zeros, all simple

and on the imaginary axis. Let us denote them by it , m = 1,2,...,N and

order

(2.4) ^ - «, ... > K N •- 0.

80

Bearing in mind that if'„(x,iic ) and \|i (x,i< ) are both real-valued and

square integrable, we introduce

(2.5a) c = ! j ^(x.i/c )dx , the right normalization coefficients,m L_J l m J

(2.5b) c = ty2 (x,i< )dx , the left normalization coefficients.m L_J r m JFurthermore, we introduce the following quantities for k £ E\{0}

(2.6a) a = r , the right transmission coefficient

(2.6b) a.» = L , the left transmission coefficient,

(2.6c) b = r+r_ , the right reflection coefficient

(2.6d) b» = £_•£" , the left reflection coefficient.

Assuming that uQ(x) decays sufficiently rapidly (see [7]) we can extend

a , a,, b , b„ in a natural way to continuous functions on all of K.

We shall call the aggregate of quantities {a (k),b (k),K ,c } the right

scattering data of the potential un. Similarly we refer toI

{a„ (k) ,b»(k) ,K ,c } as the left scattering data associated with u„. In a

different, but equivalent way the ~ight scattering data were already

introduced in Chapter 2, section 2 (see also [7], Ch. 4).I

We claim that a.,b» and c can be expressed in terms of the right

scattering data in the following way

a (k)(2.7a) a£(k) - ar(k) , b£(k) = - & 5_k) br(-k),

r

o - i f fKm t°° log(1-|b_(k) | 2) T-, N IK +K

(2.7b) cl = [cr] 12»c exp -^ x- dk I!, -"—p-

m mJ ml v\ n QJ k2 + ƒƒ p=1 \*^v

Indeed, the relations (2.7a) are obvious. To derive (2.7b) we combine

certain familiar facts from [6], [7]. Firstly, from [7], p. 110 we know

(2.8) i> (x,iK ) = a t|y„(x,iK ), with a £ K M O } ,r m m t_ m m

Hence, by (2.5)

31

(2.9) I a \cZ.1 m' m

Next, by r 7 ] , '(4.3.18) one has

dr_ ,-1k=i.K

m

Eliminating a from (2.9) and (2.10) we findm

(2.11) c c dk lk=i< = 1.

Lastly, from [6], p. 154 we obtain the representation

( f , f°° l o g ( 1 ~ | b (in) | 2 ) -i-j N k-itr( 2 .12 ) r (k) = UyiP\~ : dM I' —^-E . , lm k •• 0 .

Consequently

dr_(2.13) dk k=i:c 2K f

0J

m p

ra p

where we have used that b (k) = b (-k).

Combining (2.11) and (2.13) we arrive at the desired formula (2.7b).

3. Forward and backward asymptotic o

Once the right scattering data of u (x) are known, the solution u(x,t)

of the forward KdV problem

r u - 6uu + u = 0 , t - 0(3.1) f * x xxx

1 u(x,0) = uQ(x)

can in principle be computed by the inverse scattering method [7].

Concerning the asymptotic behaviour of the solution we have obtained the

following result in Chapter 2, section 7.

Lemma 3 .1 • Assume that

(3.2) b (k) is of class C2 (H) and the derivatives b J ( k ) , j = 0 ,1 ,2

satisfy

v

Then one has

( 3 . 3 ) l i m S U P / | u ( x , t ) - t;. ( - 2 K 2 s e c h 2 [»- ( x - x - V 2 t ) ] ) ( = 0 ,t->«> x ï - t '

f [ c ] 2

Xm 2K i o g \ 2K p=1 \K +Km l m ^ N p

f [ c ] 2 m-1 ,K -K \ 2 1

Let us now consider the backward KdV problem, starting from the same

initial function u~(x), i.e.

I- u - 6uu + u = 0 , t O(3 .5 ) C x xxx

L u(x,0) = u Q (x ) .

Clearly , i f u ( x , t ) s a t i s f i e s ( 3 . 5 ) , then w(x , t ) = u ( - x , - t ) s a t i s f i e s

r w - 6ww + w = 0 , t O

(3.6) { l X XXX

<• w(x,0) = u Q ( - x ) ,

so that w(x,t) satisfies the forward KdV problem with initial function

uo(-x). To solve (3.5) it is therefore sufficient to determine the right

scattering data n^-TociaLed with uQ(-x) and apply the inverse scattering

method to (3.6). . >wever, it is readily verified that the right scattering

data associated with u„(-x) are equal to the left scattering data

associated with uQ(x), which were studied in the previous section. Thus,

to find the asymptotic behaviour of the solution u(x,t) of (3.5) for

t •* -oo we merely a^piy lemma 3.1 to problem (3.6) and perform the

transcription u(x,t) = w(-x,-t). This yields

Lemma 3.2. Assume that

(3.7) b„(k) is of class C2 (E) and the derivatives b^ (k), j = 0,1,2

satisfy

odkf1), k

Then one has

(3.8) lira sup |u(x,t) - T, (-2K2sech2[K (X-X -4K2t)])| = 0,• • •• '3 ' m = 1 m m m m '

where

r[c ] 2 t H - 1 /K - K v 2 i

<3-9> x« = - 27- M ^ F - pï ifeV1)}-m l m ^ N p m / J

4. An explicit phase shift formula.

Let us assume that b and b, satisfy the conditions (3.2) and (3.7).

Then the convergence results (3.3) and (3.8) display clearly how the

solution u(x,t) of the KdV equation evolving from u(x,0) = Un(x) splits

up into N solitons as t + ±«>.

In particular, we find for the m-th soliton the following phase shift

r ?c c \2 m-1 /K -< \i

=1 K^rj ĥin in in p

This formula was first derived by Ablowitz and Segur [2] for the N = 1

case and by Ablowitz and Kodama [1] for the N > 1 case (see also the

discussion in [3]).

It is a remarkable fact that the formulae (3.9) and (4.1) become

both more transparent and more meaningful if one inserts the representation

(2.7b). Summarizing, this leads to

r r -i, .

(4.2a) xm - 2K ^ S V 2K I ' K p=1

IT ~

84

0

(4.3a) S = Sd + SC

m m m

( 4- 3 b )

(4.3c)

Tn S we recognize the pure N-soliton phase shift (caused by pairwise

interaction of the m-th soliton with the other ones). The quantity S

(which is negative for nonzero b ) can be seen as the shift caused by

the interaction of the m-th soliton with the dispersive wavetrain. Note

that the phase shift S is completely determined by the bound states <

and the right reflection coefficient b and is thus independent of the

right normalization coefficients, a fact not in the least suggested by

the original formula (4.1). For nonzero b we obviously have

(4.4) 0 > SC. > S^ > ... > S?l.

Thus, the collision with the dispersive wavetrain causes a delay in the

motion of the solitons and the effect is most heavily felt Sy the

smallest one, corresponding to K,,.

Using the formula (see [3])

N

p=1 Kp(4.5) I uQ(x)dx = " f J 1 OS(' " |br(k)|*)dk - 4

cwe obtain for the continuous phase shift S the following estimate in

terms of the initial function un(x) and the bound states t: :0 p

(A-6) 10 s sr (_ƒIn estimating the size of S one has to distinguish two cases, the

"generic case" and the "exceptional case" (see [5], [6], as well as

Chapter 2, subsection 2.1). In the generic case, the Jost functions

ijj (x,0) and i|;„(x,0) are linearly independent, whereas in the exceptional

case they are not. In the exceptional case one has

(4.7) B = sup |b (k)| - 1,. k£ K

whence

85

(4.3) |S^| S ~ 27m

In the generid case there is an a z O with

(4.9) b (k) = -1 + ak + o(|k|) as k >• O,

so that in the integral defining S the contribution of k = 0 becomes

important. In particular, fixing |b |, we find for K + 0

(4.10a) SC ~ - — log(1 - |b (0)|2) in the exceptional casem

c 1(4.10b) S ~ — log K in the generic case,

m < mm

c dClearly, in general the sizes of S and S are incomparable.

On the other hand one can cr>sily construct examples in which one of the

two dominates. For instance, consider a generic case with two bound states

K and K ? = |K.. Then, for fixed |b |, the discrete phase shifts Sc

dominate for K. •* +», whereas the continuous phase shifts S dominate for

K. I 0; in the K. + 0 case the familiar picture of a KdV soliton over-

taking a smaller one, where the smaller one is shifted to the left and the

larger one to the right, changes, since now both are shifted to the left.

5. An example: the continuous phase shifts arising from a sech2 initial

function.

To illustrate the previous discussion let us compute the continuous

phase shifts arising from the initial function

(5.1) uQ(x) = -A(\ + 1)sech2x, A 0.

From [9] we find

,«. , , ,. . r(a)r(b) . ,,, r(c-a-h): (.-i)r(b) .(5.2a) ar(k) = r ( c ) r ( a + b_ c ) , \(k) = y^=^nc.h)r(a+b.cy . «"h

(5.2b) a = 1 + A - ik, b = - \ - ik, c = 1 - ik,

where r denotes the gamma function ([4], p. 253). Clearly, a is analytic

86

on C \ { K ,K ,..,,< } with simple poles at the bound states K ,K„ , ... ,>:,,•

Here N Ï 1 is the unique integer such that N-1 < \ i N and the r are

given by

(5.3) K = 1 + A - p, p = 1,2,...,N.

Note, that ur.(x) is reflectionless (i.e. b a 0) if and only if

A = 1,2,..., in which case N = A. For the other values of .'• we find that

b (0) = —1 so that we are in the generic case.

To compute the continuous phase shifts S we notice that by (5.2)

On the other hand, by (2.12)

r log(1-|b (k)|2) n N v-i-

7 J -FT7 dk}} P"I ' • °-Equating both expressions we obtain, after repeated use of the recurrence

formula r(z+1) = zT(z), the following identity

(5-6) 7 J k^-r^ dk - - 7 losIr(i+v^-N)r(,-;+J

where B refers to the beta function ([4], p. 258).

Finally, combining (5.3) and (5.6), we find that the continuous phase

shifts S are given by

. c 1 ƒ B(2-nH-A,1-m+Q^ m 1+A-m °S{B(2-m+2A-N,1-ni+N

However, to get an idea of the magnitude of S it is much simpler to

employ the estimate (4.6) which gives us immediately

(5.8) Is!I S -

a 7

References

[ 1] M.J. Ablowitz, Y. Kodama: Note on asymptotic solutions of theKorteweg-de Vries equation with solitons. Stud. Appl. Math. 66 (1982)No. 2, 159-170.

f 2] M.J. Ablowitz, H. Segur: Asymptotic Solutions of the K&tteweg-deVries Equation. Stud. Appl. Math. 57 (1977), 13-44.

[ 3] M.J. Ablowitz, H. Segur: Solitons and the Inverse ScatteringTransform, Philadelphia, SIAM, 1981.

[ 4] M. Abramowitz, I.A. Stegun: Handbook of mathematical functions.National Bureau of Standards Applied Mathematics Series, No. 55.U.S. Department of Commerce, 1964.

[ 5] A. Cohen: Existence and Regularity for Solutions of the Korteweg-deVries equation. Arch, for Rat. Mech. and Anal. 71 (1979), 143-175.

[ 6] P. Deift, E. Trubowitz: Inverse scattering on the line. Comm. PureAppl. Math. 32 (1979, 121-251.

[ 7] W. Er'.haus, A. van Harten: The inverse scattering transformation andthe theory of solitons. North-Holland Mathematics Studies 30, 1981.

[ 8] W. Eckhaus, P. Schuur: The Emergence of Solitons of the Korteweg-deVries Equation from Arbitrary Initial Conditions. Math. Meth. in theAppl. Sci. 5 (1983), 97-116.

[ 9] G.L. Lamb Jr.: Elements of soliton theory. Wiley-Interscience, 1930.

[10] P. Schuur: Asymptotic estimates of solutions of the Korteweg-de Vriesequation on right ' Lf lines slowly moving to the left, preprint 330,Mathematical Institute Utrecht (1984).

[11] S. Tanaka: Korteweg-de Vries Equation; Asymptotic Behavior ofSolutions. Publ. R.I.M.S. Kyoto Univ. 10 (1975), 367-379.

[12] V.E. Zakharov: Kinetic equation for solitons. Soviet Phys. JETP 33(1971), 538-541.

83

CHAPTER FOUR

ON THE APPROXIMATION OF A REAL POTENTIAL IN THE ZAKHAROV-SHABAT SYSTEM BY

ITS REFLECTIONLESS PART

In this paper Che inverse scattering algorithm associated with the

Zakharov—Shabat system with real potential is simplified considerably.

Exploiting this simplification we derive an estimate which clearly

displays how well the potential is approximated by its reflectionless

part.

1. Introduction.

The inverse scattering method associated with the Zakharov-Shabat

system with real potential [14] can be used to solve a rich class of

integrable nonlinear evolution equations, counting the modified Korteweg-

de Vries equation and the sine-Gordon equation among its most distinguished

members (cf. [2], [9], [13]). However, the only solutions of these

equations that can be computed in explicit form are the so-called

reflectionless solutions, i.e. solutions whose associated right

reflection coefficient is zero. In a more general setting this situation

leads automatically to the following question: Given an arbitrary real

potential in the Zakharov-Shabat system, in which sense is it

approximated by its reflectionless part?

In this paper we shall give an answer to this question.

To this end we first simplify the inverse scattering algorithm by

showing how the Gel'fand-Levitan equation that appears in the literature

can be simplified to a scalar integral equation containing only a single

integral. The newly found Gel'fand-Levitan operator has, when considered

in the complex Hubert space L2(0,<"), the remarkable structure of the

identity plus an antisymmetric operator. Exploiting this structure we

shall derive a pointwise estimate of the difference between the potential

and its reflectionless part, which is remarkably simple in form and depends

only on the bound states and the right reflection coefficient associated

with the potential.

Let us emphasize that in applications of the inverse scattering

method (cf. [3], [9]) the scattering data are usually known in explicit

form. Therefore this estimate has immediate consequences in a practical

case.

In Chapter 5 and Chapter 7 we shall use our estimate for an asymptotic

analysis of the modified Korteweg-de Vries and the sine-Gordon equation

respectively.

The composition of this paper is as follows.

In section 2 we review the direct scattering problem for the Zakharov-

Shabat system with real potential. In section 3 the inverse problem is

discussed and simplified. Next, in section 4 we state our main result,

which, after the introduction of a convenient notation and the derivation

of a useful lemma in ssction 5, is proven in section 6.

2. Construction and properties of the scattering data.

Let us briefly discuss the direct scattering problem for the

Zakharov-Shabat system

90

(2.D 1) -( r , • = £ ,

where q = q(x) is a real function and z, a complex parameter.

For details and proofs we refer to [1], [2], [6], [12]. Our notation is

similar to that used in [6].

Following [6] we assume that the potential q satisfies the

hypotheses:

(2.2a) q 6 c'(R)

(2.2b) lim q(x) = lim q'(x) = 0| x [ •+» | x | -M°

(2.2c) j (|q(s)| + |q'(s)|)ds <. +».

In addition we shall need some conditions on the zeros of the

Wronskian of the right and left Jost solutions, to be specified

presently in (2.13).

For lm C Ï 0 we define the (right and left) Jost solutions >JJ (x,r,)

and v,(x,r,) as the special solutions of (2.1) uniquely determined by

(2.3a) v (x,Ü = e~1CXR(x,,;), H m R(x,r.) = ! )

irx A(2.3b) v„(x,rj = e ' L(x,t), lim L(x,rJ =

x^+m ^U

The vector functions R and h are continuous in (x,r.) on E^C and analytic

in r, on € + for each x £ K. Furthermore their components satisfy

(2.4) max sup |R.(x,rj[, sup !L.(x,rJ!J- exp{ [ !q(s)jds[,LI.'-C + 1R-C+

X J I~«J J

i = 1,2.

For Im .", ' 0 we set

(2.5a)

(2.5b) h ( ) ( ft

-i|>„ (x,-r.) -I|J„ (x,c )1 1

It is readily verified that i(i and ifi„ are solutions of (2.1). Moreover,

for x, K £ E one has

(2.6a) J

(2.6b) W(J £,^) = JL, (x,C> jz + |L2(x,f;)|

2 = 1,

where W(IJJ,<|>) = ip,c(> — tp cfi denotes the Wronskian of ij) and $. Hence, for C real,

the pairs IJJ ,<\> and iK>, ipp constitute fundamental systems of solutions

of equation (2.1). In particular, we have for x, z, £ E

(2.7a) <l>r(x,<;) = r+(c)^(x,c) + v_(.O^^(xtrJ

(2.7b) r+(c) = W(^,*r)

(2.7c) rJO = Wdi^,^)-

The representation (2.7c) makes it possible to extend r_(c) to a function

analytic on Im (; '• 0 and continuous on Im r, r> 0. The following properties

are easily demonstrated:

(2.8a) |r+(?)|2 + |r_(t)|J = 1 , C 6 R

(2.8b) r*(c) = r+(-c), f, € E

(2.8c) r*(c) = r_(-?*), Im C ~: 0.

Furthermore one can derive the integral representations

(2.9a) r+(c) = - j q(s)e~2lfsR] (s.rjds, r. e E

(2.9b) r_(c) = 1 + | q(s)R2(s,Ods, Im c - 0.

In combination with (2.6a) these yield

(2.10) max sup jr. (<;) I , sup |i - r (c)|j - [ |q(s)|ds.Lc£R r, € m -I -J

92

At £ = 0, the scattering problem (2.1) can be solved in closed form. One

has

(2.11) R1(x,0)=cosj I q(s)dsj, R2(x,O) = -sinf [ q(s)dsj

so that by (2.9)

(2.12) r+(0) = -sinf f q(s)dsL r_(0) = cos[ f q(s)ds|.

In terms of r we make our final assumptions:

(2.13a) r_(c) * 0 for r, £ Ü.

(2.13b) All zeros of r in <C+ are simple.

Let us emphasize that, strictly spoken, condition (2.13b) is not

necessary. In fact, a very elegant direct and inverse scattering

formalism using only (2.2) and (2.13a) has been developed by Tanaka in

[12]. Our motivation for requiring (2.13b) is that it simplifies the

reasoning considerably.

Condition (2.13a), on the other hand, cannot be omitted. It poses an

implicit restriction on the potential q and forms therefore a weak point

in the Zakharov-Shabat scattering theory, as developed so far. An obvious

consequence of (2.13a) is, in view of (2.12),

(2.14) [ q(s)ds * (k + J)*, k f. 7L.JOther explicit consequences for the potential q are still to be found.

Note, however, that for small potentials no problems arise.

Specifically, if

(2.15) |q(s)|ds < 1

then (2.10) shows that (2.13a) is fulfilled.

Moreover, if

(2.16) j |q(s)jds < 0.904

then (2.13a) and (2.13b) are trivially fulfilled since r_(O * 0 for

h ? ? 0 (see [2] for a proof).

93

We now proceed with the construction of the scattering data associated

with q(x). As a result of (2.13a) the function r_(O has at most finitely

many zeros t,. ,{,„,... ,£„, Im t,. > 0. By (2.8c-13b) they are all simple and

distributed symmetrically with respect to the imaginary axis. Let P be the

number of purely imaginary zeros and set M = (N - P)/2. We order the

in such a way that

(2.17) j = 1,2,...,N,

where a denotes the permutation among integers between 1 and N defined by

(2.13) o(j) = j

= j

= j

+ 1

- 1

j

j

odd

even

j

5 2M

•= 2 M

• 2 M

It is a remarkable fact, that the r,. are precisely the eigenvalues of (2.1)

in the upper half plane (the so-called bound states). The associated

L2-eigenspaces are one-dimensional and spanned by the exponentially

decaying vector functions n> „ (x,£,.), j = 1,2,...,N. Note that by (2.7c)

there are nonzero constants a(^.) such that

(2.19) ^ ( x , ^ ) = a(Cj)^(x,?j).

One can now derive the representation

(2.20) ^ = (;.) --2i.*(t,> f *, ,i,-)v,, (s,r )ds.J l2 J

Bearing in mind that the integral on the right does not vanish because of

(2.13b), we define the (right) normalization coefficients by

r ! C°° 1-1(2.21) C. = Ji ,|;„ (s,c.)>Jj. (s,c.)ds .

It is easily seen that they satisfy the same symmetry relation as the r,. ,

i.e.

(? ??} r = —(c >"

Next, we introduce the following functions of c € K

(2.23a) a (J;) = 1/r_(c), the (right) transmission coefficient

94

(2.23b) b (c) = r (c)/r_(O, the (right) reflection coefficient.

By (2.3) one has for C € K

(2.24a) a (4) = a (-5), b (c) = b (-£)

(2.24b) | a r ( c ) | 2 " | b r ( O | 2 = 1.

In [6] i t is shown that b^ i s an element of C fl L1 0 L2 (3Ü, which

behaves as o( |?

of (2.12) that

behaves as o ( | c | ) for £ -> +°°. Furthermore, i t i s an obvious consequence

(2.25) b (0) = - tan { q(s)ds ,r L-coJ -I

which shows that in general the reflection coefficient may assume

arbitrarily large values.

Clearly, by imposing stronger regularity and decay conditions on q(x)

in addition to (2.2-13), one can improve the behaviour of b (f,) . For

instance, if q(x) has rapidly decaying derivatives, then b (r;) has

rapidly decaying derivatives as well. The converse also holds. In

particular, q is in the Schwartz class if and only if b is in the

Schwartz class (see [12]).

Moreover, if, for a potential q satisfying (2.2-13), there exists an

t. • 0 such that q(x) = Q(exp(-2f»x)) as x + +<*, then it follows from

(2.9) that for any s: . •- 0 with i- • i.Q and f • Im r, . , j = 1,2,...,N,

the function b (c) is analytic on 0 •- Im c •• >• . a"d continuous and

bounded on 0 •; Ira r, •" i , .

Uc shall call the aggregate of quantities {b (') ,L. . ,C.} the (right)

scattering data associated with the potential q. Their importance lies

in the fact, that a potential is completely determined by its

scattering data.

In concluding this section, let us point out that, as in the

Scliröd int;er case (see Chapter 2, subsection 2.1), it is usually not

possible to obtain the scattering data in closed form. Also here, there is

an interesting exception: for the potential q(x) = .< sech :•;,

< € K\;k+i;k £ Z] one can solve the scattering problem (2.1) in closed

form. This is shown in Chapter 6, section 5 of this thesis.

r>5

3. Simplification of the inverse scattering algorithm.

Let q be any potential satisfying (2.2-13). Then q can be recovered

from its scattering data {b (<;),£;.,C.} by solving the inverse

scattering problem.

For that purpose one defines the following functions of s £ E

(3.1a) .ft(s) = S2d(s) + <2c(s),

K 2i5.s(3.1b) £2d(s) = -2i .£ C.e J ,

(3.1c) u (s) = - ( b (rje2lcsdr,.c TI r

-co-'

Because of (2.17-22-24a) both ;:,(s) and ;'• (s) are real functions. Sinced c

b is in C n L'(E), the integral in (3.1c) converges absolutely and ',".

belongs to CQ n L2(H).

Next, introduce the 2x2 matrix

, o -n(s(3.2) g(s) = {

\:(s) o

and consider the Gel'fand-Levitan equation (see [1], [2], [6], [12])

(3.3) tó(y;x) + s;(x+y) + g(z;x)L!(x+y+z)dz = 0

with y :• 0, x £ E. In this integral equation the unknown £(y;x) is a

2x2 matrix function of the variable y, whereas x is a parameter.

Observe that some authors use a slightly different version of the

Gel'fand-Levitan equation which can be transformed into (3.3) by a

change of variables (see [6], p. 46).

In [2] it is shown that for each x € E there is a unique solution9 x 0

<j(y;x) to (3.3) in (L2 ) (0 < y • +»). it has the form

(3.4) j.where a(y;x) and p(y;x) are real functions belonging to C 0 L1 fl L2

(0 •' y •' +<*>), which vanish as y •+ +». The inverse scattering problem is

now solved, since the functions a and ;•• are related to the potential q

in the following way

(3.5a) q(x) = 3(0+;x)

(3.5b) j q2(s)ds =-a(O+;x), x £ R.

Using (3.4), the matrix integral equation (3.3) can be reduced to a

scalar integral equation involving only I'

-TO -OD

(3.6) S(y;x) + Q(x+y) + (? (z;x)s;(z+s+x):,:(s+y+x)Jsdz = 0.

This is the form of the Gel'fand-Levitan equation that appears in the

literature (cf. [1], [12]) and is frequently used in the asymptotic

analysis of nonlinear equations solvable via the Zakharov-Shabat inverse

scattering formalism.

It has, in our view, a number of disadvantages. Firstly, the information

about a, which was still present in (3.3), is now lost. Secondly, the

equation contains a double integral which is of course harder to analyse

than a single one. The third objection is of an algebraic nature: the

structure of g, which is simply the matrix representation of a complex

number, is violated.

Let us try to mend these shortcomings, starting with the last one.

Set

(3.7) y(y;x) = m(y;x) + ifi(y;x).

Then it is straightforward to deduce the following integral equation

from (3.3-4)

r(3.3) ,(y;x) + i;:(x+y) + i ::(x+y+z), (z;x)dz = 0.

Clearly, equation (3.3) has none of the disadvantages mentioned

above. In particular, the information about « and 3 is obtained by taking

real and imaginary parts. This yields

(3.9a) q(x) = 1m r(0+;x)

(3.9b) qJ(s)ds = -Re f(0+;x).

x-"

Ivote, that the equations (3.3) and (3.3-4) are equivalent. In fact, let

ij) denote the mapping that identifies a complex number with its matrix

representation in the following way

(3.10) <K£ + in) = ( ) C, n £ K.n f.

Then (3.3-4) is the image of (3.8) under *.

4. Statement of the main result.

At the moment no method is known that produces explicit solutions of

the Gel'fand—Levitan equation (3.8), associated with arbitrary scattering

data. However, if b = 0 , then the equation gets degenerated and reduces

essentially to a system of N linear algebraic equations, which can be

solved in explicit form by standard procedures.

If q is a potential with scattering data {b (c,),t-»C.} then ther

potential q with scattering data {0,£.,C.} is called the reflectionless

part of q. The structure of the reflectionless part has been discussed

by several authors (see [2], [9], [10]). In the next section we shall

derive a rather elegant representation of q, in terms of determinants.

In applications of the inverse scattering method the following

situation is generic: by some procedure the scattering data of a potential

are known in explicit form. The potential itself is predicted to exist by

the general theory, but its explicit form is unknown. The only thing one

can calculate explicitly is its reflectionless part. Thus, a natural

question to ask is the following:

In •jShich sense is l,>ir potent-la', appyccimi4^ i hy f.'r, ;•.•/*',•.-• :.".•'..-.• ••;."•."

Our main result is the next theorem which gives an answer to this question.

T h e o r e m 4 . 1 . L"l q l-c .i i:'t.ci;.ia' in ' •';• .";-•<'.••?(••• v-."-1 ;•>,•• .•,/.•••<••: :".

:jhu;h .;jtijfC.\i (S.'>!/•) .rii h.i.*. > -V .••;:.• -.'fit.j :::i ib (.O,',- ,Cr.}.

L>:<- q dcnove r.h.c jv;7t.v>' i. •;/••.:.; pr.ri .'? q. '."<•."•'. ;".J." •• i "': x £ F

(4.1) jq(x) ~q d(x)! • a.2! ! ;..c(x+y)[»dy + sup : (x+y) '^0 0- y • + ••

• j i : h ."• j i v e ^ b y ( S . l ? ) m d a . :k*- ;"•)• ',j:j:'nj • .- '.'•'• ^'•.-.'•' •

93

bound states z,.

(4-2a) ao = f

(4.2b) * - 2 U . 11k=1

Herewith, q - q, is estimated completely in terms of the scattering Jala.

More oreai.-ely, the bound -jiven by (4. 1-2) depends only on the-

re flection coefficient b (O and the bound states t,. and not on ther

normalization coefficients C.

Corollary to theorem 4.1. Under the conditions of '.heoreti 4.1 -,v hajc :-.'.?

a priori bound

a 2 r°"

(4.3) sup |q(x) - q(j(J<)| - — (|br(t)| + !t>r(rj |

2 )d^.

Let us mention an important application of theorem 4.1. Consider a

family of potentials q(x,t), depending on a parameter t • 0 referred to

as time. Suppose that q(x,t), which is assumed sufficiently smooth and

rapidly decaying for |xj -* ™, satisfies the initial value problem for a

suitable nonlinear evolution equation of AKNS class [2], e.g. the

modified Korteweg-de Vries equation. Then the bound states i,. do not

change with time, whereas the associated normalization coefficients and

reflection coefficient change in a simple way. The estimate (4.1-2) now

tells us how well the solution q(x,t) is approximated by its soliton

part q,(x,t). Since a„ is invariant uith time, only the behaviour of

:. (x+y;t) is of importance. In particular, for those nonlinear

evolution equations, whose linearized version has a negative sroup

velocity associated witli it (cf. [7], or Chapter 1, Appendix C) one can

construct coordinate regions of the form

(4.4) t • tQ, x • ,(t)

with t„ a nonnegative constant and i(t) a function of time characteristic

for the problem, such that (x+y;t) is small in \? fl \T (0 y • +.-••)

(see [11], or Chapter 2, section 3, for an example of such a

construction). In that case, (4.1—2) shows that in the region (4.4) the

solution q(x,t) is given by qd(x,t) plus a small correction term, which

dies out as C + ».

We shall prove theorem 4.1 in section G. Before doing so we introduce

some notation and derive a useful lemma in section 5.

5. Auxiliary results.

From now on we shall assume that the conditions of theorem 4.1 are

fulfilled.

To start with, let us give our reasoning an appropriate abstract setting.

To this end we introduce the Banach space G of all complex-valued,

continuous and bounded functions g on (0,»0, equipped with the supremun

norm

llgll = sup |g(y)|.0<y <+c«

Furthermore, l e t JC^denote the complex Hube r t space L2 (0,m) with inner

product < f , g > = ! f(y)g (y)dy and corresponding norm II II,.0J

From sect ion 3 we know that for each x € E the functions y >-<• :. (x+y),

Sid(x+y) belong to S 0 K.

Next, keeping x 6 E fixed, we formally wri te

(5.1) (Tdg)(y) = ! Hd(x+y+z)g(z)dz

(5.2) (T g)(y) = } s; (x+y+z)g(z)dz.

0 J

Evidently, T, can be considered as a mapping from 3 into S, but

equally well as a mapping from 3f into 3f. On the other hand, T is not

necessarily a mapping from u into B. However, an obvious adaptation of

formula (4.5.10) in [6] shows that T maps ?f into JC with a norm that

satisfies

(5.3) IIT II < sup jb <c)|.

100

In view of (2.25) we do not expect this norm to be particularly small.

Since both ft, and Q are real-valued, the operators T, and T are self-d c ' ' d c

adjoint on Jf. This fact will play a crucial role in our analysis. Actually,

to such an extent that the size of UT IL is irrelevant.

In the above abstract language, the Gel'fand-Levitan equation (3.8) takes

the form

(5.4a) (I + iTc + iTd)y = -in

(5.4b) tt = fi + SJ ,

where I is the identity mapping.

A first advantage of this formulation is readily seen. Since T + T, is

self-adjoint, the operator I + iT + iT is invertible on K and so we

know at oa"e that (5.4) has a unique solution y £ Jf. Note that this fact

was already mentioned in section 3, from which we recall that, moreover,

V £ B n jff.

For the proof of theorem 4.1 the following lerana is basic.

Lemma 5.1. For any value of the parameter x € K, the opct'ator I+iT, is

invertib'le on the Banaah space B with inverse S = (I + iT.) niocn b'j

N(5.5a) (Sf)(y) = f(y) - .]

0 J(5.5b) A. = p I , (Jpj(2 J f(z)e l C p Zdz) ,

( [cf] e J -hnïv (b •) i-s the inverse of the matrix A = ( [cf] e

ufih^rmjre, the operator S satisfies the bound

_.+i(? +c.) j'.

(5.6) USII i a0, x € K,

(3.7a) (Im r,

J, i ü _j> 'S_( 5 . 7 b ) !1 . = 2 ( l m £ . p ) 2 ( I m ? j ) 2 ^

r,:u», I1SII -ïiï un if om !^ oounded for x e

' ; j r-k'

101

Proof: Let x £ 1R he arbitrarily chosen.

We first consider T. as an operator from the Hubert space ')( into itself.

Since T, is sél^-adjoint, I + iT is an invertible operator on Jf. From the

relation

(5.8) II (I + il'd)gll2 = «3«2 + «Tjgll^, 8 E 3f,

we obtain tiie following bounds for the inverse S = (1 + iT,)

(5.9) IISII2 : 1, HTJSII., • 1.

Next, let us consider T, as an operator from 3 into 5 and show that

I + iT is invertible on 3. Suppose that (1 + iT.)i; = 0 for some ", £ S.

Then g = -iT g € Jf and thus g is identically zero by the preceding

argument. This shows that I + iT is one to one on 8. However, T, is an

operator of finite rank and hence compact. It follows that 1 + iT, is an

invertible operator on the Banach space S.

Furthermore, solving the equation

(5.10) (I + iTd)- = f, f,g € D

we find

N 2 L\ .y(5.11) g(y) = f(y) - X} A.e J" ,

where the A. satisfy

N (•»• 2 if. z

(5.12a) .rittp.A. = 2 j f(2)e «* Jjs. p.,,2,....N

(5.12b) u . = n.b . + i(r, +;,.) , .i. = [C.] e J .PJ J PJ P J J J

Since the operator I + iT, is one to one on S, the matrix A = (i -) is

invertible with inverse A = (H . ) . We conclude that the inverse

operator S = (I + iTj) is given in explicit form by (5.5).

We shall now prove that the matrix elements ,- . are bounded is

functions of x € K, where the bound is explicitly given in terms of the

.',. by

(5.13) |Bpj| < 2 (lm Cp

tJ"•lelil M—^j = n ..

To this end we first introduce some notation. In 3f we consider the elements2ic.y

e. defined by e.(y) = e J . Let A denote the Gram matrix of the vectorsJ J * _]

e 1 ' e 2 ' " " V i-e- A = '•"pj^ ^pj = <ep'ej ' = (2i(r'j " S } ) ' S l n C e t h e

vectors e.,e„,...,e are linearly independent, it follows that der A • 0

(see [5]). Let us write (A) = ($ .) and introduce the vectorsN PJ

h = .'L, 3 .e.. Evidently, <h ,e."P J=1 PJ J P J

combination with (5.5) this gives

h . and h ,h.PJ P J

(•; .. InPJ

N(5.1A) Sh , . = h , s - 2 X, 6 .e.,ci(p) n(p) j = 1 pj j'

where o is the permutation defined in (2.13).

Using the identity I - S = iT,S, we get

(5.15) 20 . = <iT.Sh , ,,h.•.PJ d o(p) J

Hence, in view of ( 5 . 9 )

(5.16) 4 If-; .]' pj '

II h , Jl^llh.ll^ = I . . , A...o(p) 2 j 2 o(p)o(p) jj

liy direct calculation (see [5]) we obtain

(5.17)"it

which completes the proof of (5.13).

By (5.5b-13) we have

= B

W( 5 . 1 3 ) :• « f l l (Im 'N p j l

from which the bound (5.6-7) for IISII is obvious.

Corollary to lemma 5 . 1 . Few each x £ IR the. equal i'on

(5.19) (I + iTd)y = ~iild

103

admits a unique solution v , € 8 and ws havea

N 2ic.y(5.20) Yd(y;x) = -2 p ?=1 Bpje

J .

Remark. Let us recall that y, produces the reflectionless part of the

potential q through the formula

+ ?(5.21) q,(x) = Im Yj(0 ;x) = -2 Im ? . B ..

d d p, j = I pj

Clearly, by (5.13) we have the a priori bound

N(5.22) sup |qd(x)| < 2 ?l=] N p j,

which does not involve the C. but depends only on the ?. in a simple

explicit way.

Starting from (5.21) it is easy to obtain a more elegant representation

of q.. Indeed, introducing the matrices

(5.23a) V( , l(^ +^ - )x\ / i£.x >r •/ . \~lv-.r P J \ „ ƒ \ r.

PJ P J J /' 1 \ PJ;

(5.23b) D, =fc^e J 6 .V E = (c . ) , c . = 12 V J PJ/ PJ PJ

we find that

N -1 -1 -1(5.24) 5 ts . = Tr(EA ) = Tr(ED,V D.) = Tr(D.ED„V ),

p,j-1 pj I l 1 I

where we used the fact that V = D AD., as well as the invariance of the

trace under cyclic permutation. Clearly, D.ED = - — V. Therefore,

using the familiar formula (cf. [4], p. 28, (7.17))

(5.25) 4- (det V) = (det V)Tr(^- v"1)dx dx

we arrive at

(5.26) yd(0+;x) = 2 -^ log det V.

An alternative derivation of formula (5.26) is given in [9], pp. 103-105.

Transforming back one gets

i04

(5.27) Yd(O+;x) = 2 -^ logfdetCD^det A]

N= 4i I, ? + 2 -f- log det A.

p=1 ^p dx ö

NNote that ^i ? i-s Purely imaginary. We conclude that

(5.28a) q,(x) = 2lm -~ log det A

or equivalently

(5.23b) qd(x) = 2 Im T r ( ^ A~1) =

N _1= Im £ (-4U a )g with a = [Cr] e

p=1 p P PP P P

Observe how this simplifies (5.21). For time-dependent potentials the

representations (5.28) form a good starting point for the asymptotic

analysis of q,(x,t) as t + t« (cf. [6], 2nd ed., Appendix A1).

6. Proof of theorem 4.1.

After the preparatory work done in section 5 the proof of theorem

4.1, which we present in this section, is comparatively easy.

Let x £ E be arbitrarily fixed. As a first step, let us write the

solution y of (5.4) in the following form

(6.1) Y = Yd + Y » with

(6.2) Yd = -iS«d.

By (3.9a) and (5.21) we plainly have

(Ó.3) q(x) - qd(x) = Im Yc(0+;x).

From the previous section it is clear that both y and y, belong to B 0 3f.

Hence, we already know that y £ 8 Cl ÏC. It remains to find a concrete

estimate of y in the supremum norm. For that purpose we insert the

decomposition (6.1) into (5.4), thereby obtaining

105

(6.4) (I + iT + iTjy = -iT y, - is; .c d c c d c

Consider (6.4) as an equation in the Hubert space 3f. Since T + T. is

self-adjoint, the operator I + iT + iT, is invertible on JC.

Furthermore, the relation (5.8) holds with T^ replaced by T c + T .

Consequently, (6.4) has a unique solution y E 3( satisfying

(6.5) h II, •-• IIT y.ll, + IIi: II,.c J. c d I c /

An application of the generalized Minkowski inequality (see [8], p. 148]

gives us

(6.6) 111

Hence

(6.7) 111

where by (5 .

(6.8) II y

We conclude

( 6 . 9 ) II Y

W!2

c>d»2

13-20)

d ' , =

that

II -

0

£ l l

ro J

|,SJ

with a. given by (4.2a). The trick is now to rewrite equation (6.4) as

(6.10) (I • iT d), c = -iTcvc - i l V , d - i,:c

and to realize that the a priori estimate (6.9) paves the way to

estimate the right hand side of (6.10) in the supremum norm. In fact,

we have by Schwarz' inequality

(6.11) IIT f :: sup f f |\i (x + y + Z)|2dzW f |r (Z;x)[

2dZ)i

"'c 2 * c 2 'c 2 ^ A 1

Moreover, invoking again the generalized Minkowski inequality, one «e

(6.12) llTcïd

11 ' ll!.lcllllvd»r

iOó

Together, (6.11) and (6.12) lead to the estimate

(6-13) ll-i V c - i V d - iu ( 2ucl

Applying lemma 5.1 we obtain from (6.10-13) the following estimate for

r in the supremum norm

( 6 . 14 ) II y II £ a ^ f l l n II * lift l l ^ \c 0 \ c c 2 /

The desired estimate (4.1-2) is now a direct consequence of (6.3-14) and

the obvious fact

(6.15) |lm7c(0+;x)| <: sup | >c<y;x) | = IIY II.

Herewith the proof of theorem 4.1 is completed.

Remark. Note that we have proven more, since by (6.1-14) and (3.9b) the

estimate (4.1-2) is still valid if one replaces the left hand side of

(4.1) by

(6.16) maxhq(x) - qd(x)|, | j (q2(s) - qjj

References

[ 1] M.J. Ablowitz, Lectures on the inverse scattering transform, Stud.Appl. Math 58 (1978), 17-94.

t 2] M.J. Ablowitz, D.J. Kaup, A.C. Newell and :i. Segur, The inversescattering transform-Fourier analysis for nonlinear problems,Stud. Appl. Math. 53 (1974), 249-315.

[ 3] M.J. Ablowitz and H. Segur, Solitons and the Inverse ScatteringTransform, Philadelphia, SIAM, 1981.

[ 4] E.A. Coddington and N. Levinson, Theory of Ordinary DifferentialEquations, Me Graw-Hill, 1955.

[ 5] P.J. Davis, Interpolation and Approximation, Dover, New York, 1963.

[ 6] W. Eckhaus and A. van Harten, The Inverse Scattering Transformationand the Theory of Solitons, North-Holland Mathematics Studies 50,1981 (2nd ed. 1983).

[ 7] W. Eckhaus and P. Schuur, The emergence of solitons of the Korteweg-de Vries equation from arbitrary initial conditions, Math. Meth. inthe Appl. Sci. 5 (1933), 97-116.

107

[ 8] G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities, 2nd. ed.,Cambridge, 1952.

[ 9] G.L. Lamb Jr., Elements of Soliton Theory, Wiley-Interscience, 1980.

[10] M. Ohmiya, On the generalized soliton solutions of the modifiedKorteweg-de Vries equation, Osaka J. Math. 11 (1974), 61-71.

[11] P. Schuur, Asymptotic estimates of solutions of the Korteweg-de Vriesequation on right half lines slowly moving to the left, Preprint 330Mathematical Institute Utrecht (1984).

[12] S. Tanaka, Non-linear Schrödinger equation and modified Korteweg-deVries equation; construction of solutions in terms of scattering data,Publ. R.I.M.S. Kyoto Univ. 10 (1975), 329-357.

[13] S. Tanaka, Some remarks on the modified Korteweg-de Vries equations,Publ. R.I.M.S. Kyoto Univ. 8 (1972/73), 429-437.

[14] V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one—dimensional self-modulation of waves in non-linearmedia, Soviet Phys. JETP (1972), 62-69.

108

CHAPTER FI1/E

DECOMPOSITION AND ESTIMATES OF SOLUTIONS OF THE

MODIFIED KORTEWEG-DE VRIES EQUATION

ON RIGHT HALF LINES SLOWLY MOVING LEFTWARD

We consider the modified Korteweg-de Vries equation q +6q2q +q =C X XXX

= 0 with arbitrary real initial conditions q(x,0) = q_(x), sufficiently

smooth and rapidly decaying as |x] -*• °°, such that q„ is a bona fide

potential in the Zakharov-Shabat scattering problem. Using the method of

the inverse scattering transformation we analyse the behaviour of the

solution q(x,t) in all coordinate regions of the form T = (3t) > 0,

x a - u - vT, where u and v are arbitrary nonnegative constants. We

derive explicit x and t dependent bounds for the non-reflectionless part

of q(x,t). If all bound states are purely imaginary these bounds make it

possible to derive a convergence result, clearly displaying the

emergence of solitons. Furthermore, if the reflectionless part of the

solution is confined to x s 0 as t -> », then the bounds help us to

establish some interesting energy decomposition formulae.

109

1. Introduction.

We focus our attention on the modified Korteweg-de Vries (tnKdV)

problem

d . i a ) q t + 6q 2 q x + q x x x = o, -••• • x • +«", t •• o

(1.1b) q(x,O) = qQ(x),

where the initial function qn(x) is an arbitrary real function on R,

such that

(1.2a) Intx) satisfies the hypotheses (2.2-13) in [8] (i.e. Chapter 4)

and is therefore a bona fide potential in the Zakharov-Shabat

scattering problem.

(1.2b) ^n^x^ *"S su^ficiently smooth and (along with a number of its

derivatives) decays sufficiently rapidly for Jx| -> m:

( i) for the whole of the Zakharov-Shabat inverse scattering

method [1] to work,

(ii) to guarantee certain regularity and decay properties of

the right reflection coefficient to be stated further on.

Note that the method of L2-energy estimates yields uniqueness of

solutions of (1.1) within the class of functions which, together with a

sufficient number of derivatives vanish for |xj * •<> (cf. [3]). Let us

refer to this class as the "modified Lax-class" (after [4]).

In [10] it is shown by an inverse scattering analysis that condition

(1.2a) and a special case of condition (1.2b) (namely q„ in Schwartz

space) guarantee the existence of a real function q(x,t), continuous on

K'[0,"0 , which satisfies (1.1) in the classical sense. Whenever, in this

i.aper, we speak of "the solution" of (1.1) we shall refer to the solution

obtained by inverse scattering (unique within the modified Lax-class).

Let us recall that by the inverse scattering method the solution

q(x,t) of (1.1) is obtained in the following way. First one computes the

(right) scattering data {b (r),r.,C.J associated with qQ(x). For their

definition and properties we refer to Chapter 4. Next one puts (see [1],

[10])

1 10

(1.3a) Cr(t) = cTexp{3i(;?t}, j = 1,2,...,N

(1.3b) br(5,t) = br(c)exp!8ic,3t}, — -- r, •• +«-.

Then by the solvability of the inverse scattering problem, there exists

for each t > 0 a smooth potential q(x,t) satisfying the hypotheses

(2.2-13) in Chapter 4 and having {b (t,, t) ,r,. ,C. (t) } as its scattering

data ([10]). The function q(x,t) is the unique solution of the mKdV

initial value problem (1.1).

Explicit solutions by the above procedure have only been obtained for

b 3 0. The solution q,(x,t) of the mKdV equation with scattering data

(0,r,. ,C. (t)} is called the reflect ionless part of q(x,t).

The long-time behaviour of the solution q(x,t) of (1.1) in the

absence of solitons is discussed in [2], [9].

In [2] the existence is claimed of three distinct asymptotic regions

I. x > 0(t) II. |x| ' 0(t1/3)

III. -x ,' 0(t).

Here Q denotes positive proportionality. Within each region, the

solution q(x,t) has an asymptotic expansion, characteristic for that

region. However, as stated in [2], p. 68 proofs are yet to be given.

Apparently, the asymptotic structure of the solitonless solution is

simpler for mKdV than for KdV, where four asymptotic regions were found.

When solitons are present the degrees of complexity are reversed. The

reason is the asymptotic structure of the reflectionless part q (x,t).

First of all, depending on the location of the bound states r., it may

happen that solitons do not separate out as t • +«>. Furthermore, if they

do separate out, one can expect both sech-shaped solitons as well as

breathers (see the discussion in section 2). The sech-shaped solitons

move nicely to the right, but the breathers are unpredictable: depending

on the r,. , the breather envelope may move to the right, to the left, or

even be at rest. Moreover, if one relaxes condition (1.2a) to Chapter 4,

(2.2-13a) and uses the inverse scattering formalism developed by Tanaka

in 110], the asymptotic structure of the associated reflectionless part

of q(x,t) is even more complicated, since now multiple-pole solutions can

occur, as calculated in [11]. In the sequel, however, we shall stick to

1 1 1

(2.2-13). In summary, only under additional restrictions on the location

of the bound states can we expect q, (x,t) to decompose into a rinite

number of solitons moving to the right.

However, we prefer to consider the general situation. In that case,

the only thing about q(x,t) one can expect (in view of the negative

group velocity associated with the linearized version of (1.1)) is the

leftward motion of the dispersive wavetrain.

Thus we are confronted with the question: Confining ourselves to the

regions I and II how well is the solution q(x,t) approximated by its

reflectionless part q,(x,t)?

In this paper we give a complete answer to this question by analysing

behaviou

of the form

the behaviour of the function q(x,t) - q,(x,t) in all coordinate regions

(1.4) T = (3t)1//3 > 0, x a -a, a = u + vT,

where y and v are arbitrary nonnegative constants. Plainly, (1.4) covers

all of the regions I and II. It is proven that

(1.5) sup |q(x,t) - qd(x,t)| = 0(t"'/3) as t ->• ».xS—a

If all bound states are purely imaginary, then (1.5) leads to a con-

vergence result (see (5.5)) which clearly displays the emergence of

solitons. Moreover, we construct several explicit x and t dependent

bounds for q(x,t) - q,(x,t) valid in the region (1.4). In the special

case that the reflect ionless part is confined to x -- 0 as t -> ™, we

derive the energy decomposition formulae

r N -i/i(1.6a) q2 (x.t)dx = 4 Z. Im c, + 0(t ) as t • •

_aJ P P

(1.6b) | q2(x,t)dx=| j iog(i + |br(i;)|J)dC + 0(t"1/3) as t * ».

The results obtained in this paper are of a similar nature as those

obtained for KdV in [7], i.e. Chapter 2 in this thesis. However, there

are differences. For instance, in the KdV case we could improve (1.5) in

the absence of solitons (i.e. q, = 0). The analysis in this paper

indicates that no such improvement is likely for mUdV. Furthermore, the

112

KdV estimates in Chapter 2 are only valid for 0 :; v v where v is

some fixed number connected with properties of the Airy function, and

for t 2 t , where t , is some critical time. The mKdV estimates obtainedcd cd

here are valid for any value of \> • 0 and for all t • 0. A. third

difference is found in the structure of the energy decomposition formulae

(1.6) versus Chapter 2, (7.25). Finally, we refer to the remarks about

the asymptotic structure of the reflectionless part made above.

The organization of this paper is as follows.

In section 2 we isolate certain properties of the reflectionless part

q,(x,t). In section 3 we recall two essential results obtained previously

in Chapter 4 and Chapter 2. The first is a general theorem in which

q(x,t) - q,(x,t) is estimated in terms of V, (x+y;t). The second is a

lemma revealing the structure of '.1 (x+y;t). In section 4 we apply this

lemma to estimate Q. (x+y;t). Then in section 5 the estimates of

ii (x+y;t) and the theorem are combined to give estimates of

q(x,t) - qd(x,t).

2. Some comments on the asymptotic structure of the reflectionless part.

The reflectionlest: part of q(x,t) is ™iven in explicit form (see

Chapter 4, (5.28)) by

(2.1) qd(x,t) = 2 Im — log det A

where A = (u .) denotes the N*N matrix with elementsPJ

_ -2ir, .x

(2.2) ,tp. = [ j - 1 J r '

If all the bound states are purely imaginary, say c. = in.,

0 ii ... n - n., then the asymptotic structure of q,(x,t) is

relatively simple. In that case, corresponding to M = 0 in Chapter 4,

(2.13), the normalization coefficients are purely imaginary as well, say

C- = iu-t u• £ E\IO}. It is shown in [6], that as t approaches infinity

the reflectionless part of q(x.t) decomposes into N solitons uniformly

1 1)

with respect to x on E. Specifically

i + ^1( 2 . 3 a ) lira sup q^ ( x , t ) - ^Z1 j — 2ri s g n ( u r i ) s e c h [ 2 n r i ( x - x ~ 4 r i 2 t ) ] j [ = 0 ,

f*» x£R

( 2 . 3 b ) x + = -^-

Note how closely this resembles the corresponding formula Chapter 2,

(5.21) for the KdV case.

If M > 0 in Chapter 4, (2.13), then the structure of q,(x,t) is more

complicated. Let us consider the simplest case: M = 2, {,, = r, + i;i,* r r

[,2 = ~C + in = "(;,> with f=, n > 0. Then C = • + ip and C. = - A + in,

where A and JJ are real constants that do not vanish simultaneously.

Using (2.1) one gets (see [5], [11])rn , % / „i / .... [sin <f + (ii/>')cos <> tanh S'l(2.4) q.U.t) = 4ri seen V , , . ' ' —• -,—- ,

d L (n/v cos2 * sech2 fj

with

(2.5a) * = 2f,x + 8f,(-"2 - 3n2)t + *

(2.5b) f = 2nx + 8,i(3fz - n2)t + v,

where the constants $ and ^ satisfy

(2.6a) exp(-v) = J (>-./n) ( \2 + |.2)^(f2 + n2 ) ~ J

(2.6b) sin $= (-\n+ nO(X2 + u 2 ) " * ^ 2 + n2 ) " *

(2.6c) cos ,), = (<<•+ i„^(.^ + ^n~hr* + n 2)"-.

Thus q (x,t) has the structure of an oscillating function that is

modulated by an envelope having the shape of a hyperbolic secant. The

envelope and phase velocities are found from (2.5) to be v = 4(M2 - 3 "2)

and v , = 4(3')2 - ' , 2 ) , respectively. Because of the undulations in its

profile, this solution is usually referred to as a breather. Note that

the sign of v is undetermined. Hence, the breather envelope may

propagate to the right, to the left or be at rest.

If M •• 0 in Chapter 4, (2.13) and M 2, then, generically, q,(x,t)

will decompose into M breathers and (N - 2M) sech-shaped solitons.

114

However, it is easy to construct examples (e.g. N = 3, r,. = 5 + in,1

r,2 > 3f,2 , ?_ = i(n2 ~ 3C2) ) , in which no complete decomposition into

breathers and sech-shaped solitons takes place.

Looking at the envelope velocity v , one would expect that for large t

the function q, (x,t) will be concentrated on the positive x-axis,

provided that the breather bound states c = f, + in satisfyP P P y

(2.7) r,p - 3e» > 0, P = 1,2,....2M.

Let us confirm this mathematically. By Chapter 4, (5.28b) we have

M / -2i£

(2.3) qd(x,t) -H.pl, (-4iS[cJ(t)]-1e * J3pp,

where B = (g .) is the inverse of the matrix A given by (2.2). Hence,

using Chapter 4, (5.13), one gets

(2.9a) |qd(x,t)|

(2.9b) iji (x,t) = 8t(Im r, )((Im c ) 2 - 3(Re t )2) - 2(lm z, )x,VP P P 'P ^P

where the constants N ., introduced in Chapter 4, (5.7b) reappear in

this paper in (3.4b). Now, suppose (2.7) holds, then by (2.9)

(2.10a) sup |q (x,t)| = 0(e '>t:) as t + «

r° •> t-(2.10b) qd(x,t)dx = 0(e ) as t -. »,

-co-'

where .; is the positive constant defined by

(2.11) . = mini8(Im c. )((lm r,^)1 - 3(Re j, ) 2); p = 1,2,...,M|.

Thus, indeed, (2.7) implies that, as c ime goes on, qf)(x,t) is confined

to the positive x-axis.With regard to (2.10b), recall that (see [2])

(2.12) J qd(x,t)dx = 4 pS, In. r,p.

1 15

3. Two useful results obtained previously.

To start with let us benefit by the work we have done in Chapter 4.

Since for each t -• 0 the solution q(x,t) of (1.1) satisfies the

hypotheses (2.2-13) in Chapter 4, required for a bona fide potential in

the Zakharov-Shabat scattering problem, we immediately have the

following result, which is established in Chapter 4 in the form of theorem

4.1 and its corollary.

Theorem 3 . 1 . Let q ( x , t ) be the solution of the '•luJtfied Kor: e^ej-de Vries

problem

(3.1)q + 6q2q + q = 0 , —" • x • +»', t • 0

q(x,0) = qQ(x),

where the initial function q „ ( x ) ia an arbitrary real '"iviciijn ''i E ,

isatisfyinrj (l.P,a-b(U). hcl {b ( O ,S • . c f } be the a.ia:.le.>'in-j .tati

associated with q n ( x ) . Then fjr each x £ E .m.i t • 0 •;>:<• ::ar,

( 3 . 2 ) | q ( x , t ) - q d ( x , t ) | • a ^ [ |- (x+y; t ) ! = dy + sup | :.• (x+y; t ) I ) ,

y»

where q , ( x , t ) is the rofLc.ation't-us part >ƒ q ( x , t ) ;::'i\~n by

(3.3) ac(stt) - J- ,-., s €

( 3 - 4 a ) ao

( 3 . 4 b ) N . = 2 (lm c, )PJ P

r..)N N

c, - r J k=1j 'k

Furthermore?, the folloüing a priori, Ixmn.i in ihi'i\'

a 2 (•"• /

(3.5) sup | q ( x , t ) - q d ( x , t ) | • - ^ | M b r ( r . ) | + ! b r ( ü | 2 W ,

Note that a is invariant with time. Hence, to get nn idea of the

magnitude of q(x,t) - qj(x,t) in the region (1.4), only the behaviour of

. (x+y;t) is important. Fortunately, we can now once again benefit from

1 lo

our previous work. In fact, a direct quotation of Chapter 2, theorem 3.3

gives us

Lemma 3.2. In the situation of theorem i. 1, assume that the r'i-yhi

reflection coefficient b (;) satisfies

(3.6) b is of class C2 (») and the derivative:-, b J (rj, j = 0,1,2

are bounded on E.

Let y •• 0, x E R, t > 0. Furthermore, Let \i, v Jrnoti' jrbilv.T.r-j non-

n&jative constants. Put

(3.7a) w = x+y+y, b(c,p> = b r ( r , ) e~ 2 i r ' J , b ( j ) = (~^) jb

(3.7b) T = ( 3 t ) 1 / 3 , Z = w ( 3 t ) " 1 / 3 .

Then one has the representation

(3.3) iic(x+y;t) = T"'br (O)Ai(Z) - Jiï"2b(1 } (0,;,)Ai( ' } (Z) + R(Z,T,n),

(3.9a) |R(Z,T,u)| '1 ï"3llb(2) 11 ~ Z~3/2 ."•..- T • 0, Z • 0,

(3.9b) |R(Z,T,u)| :' T"3llb(2)llto C(v) ;'.,. T • 0, Z - -v,

with C(\) M: in ::haptcr 'i, ('A.hibS and with Ai(n) the Airy function

Chapter i', (ö.j).

Note that the first term in the representation (3.8) is directly connected

with the initial function qQ(x), since by Chapter 4, (2.25)

(3.10) br(0) = -tan j qQ(s)ds .

4. Estimates of i (x+y;t).

Let us assume that the requirements of lemma 3.2 are fulfilled. Then

it is readily verified that in the parameter region

1 17

(4.1) T = (3t) ï 1 , x i: -u, u = u + oT, where u and v arenonnegative constants,

the following estimates hold

(4.2a) sup |£lc(x+y;t)| s YT~'0<y<+co

(4.2b) f |j2 (x+y;t)|dyï |b CO) | f'-l + f | Ai(n) ]dr,) + 7T~'

where the constant >• is given by Chapter 2, (3.66a), with N? replaced by

ilb(2)l..

In addition to the bounds (4.2), which depend only on t, lemma 3.2 also

gives us useful bounds containing both x and t. In particular, fixing

6 E (0,1) in Chapter 2, (3.7), we obtain in the region (4.1)

(4.3) sup |n (x+y;t)| s V W - 1 *&r>0<y<+oo ••

where p denotes the constant Chapter 2, (3.56a) with No replaced by(2)

Jllb II and y is given by Chapter 2, (3 .74) .

If q,,(x) enjoys the specia l property

(4.4) j qQ(s)ds = kn, k € 2 ,

then, in view of (3.10), the estimates (4.2-3) can be improved. For instance,

by lemma 3.2 one has in the parameter region (4.1)

(4.5) sup \<l (x+y;t)| < >ï"2,

with i as in (4.2). Moreover, (4.3) hoLds with the factor T in front of-2

the exponential function on the right replaced by T .Of course, further

simplifications of this type occur when also b (0) = 0. If, as usual,

(4.4) is not fulfilled, one may simplify the representation (3.3) by

working with u = b (0)/(2ib (0)), thereby removing the derivative of

the Airy function.

Though in the present discussion we keep v > 0 arbitrary, our choice of

the bound (4.3) is motivated by this property.

Let us point out that (3.6) is only a weak condition on the

reflection coefficient, which can easily be fulfilled. Actually, as noted

in Chapter 4, if qn(x) 'las rapidly decreasing derivatives then the same

1 18

is true for b (Ü.

In particular, if q„ is in the Schwartz class, then so is b . In that

case, there is not a shadow of a doubt that (3.6) holds.

5. Estimates of q(x,t) - q,(x,t).

Combining theorem 3.1 with the estimates of '.} (x+y;t) obtained in

section 4 we arrive at the main result of this paper, which can be

stated as follows

Theorem 5.1. Let q(x,t) be the solution of Ike modified Kortoweg-i<? Vries

problem

- q + 6 q 2 q + q = 0 , -«• •; x < +« t • 0

(5.1) J t x xxx1 q(x,0) = qQ(x),

bihe.re the initial function qQ(x) is an arbitrary rea' function on E,

satisfying (1.2) in ouch a way thai (S.ii) ic, fulfilled. Let (b (r.),r,.,C.l

be the serattaring data associateJ uiih q_(x). Then for each x € K and

t :> 0 one har,

(5.2) |q(x,t) - q d(x,t)j •= a2J j |s:c (x+y; t) 1

2dy + sup |:Jc (x+y; t) | ) ,

J-ilh q , ( x , t ) the reflectiovlese, part C'..i) of q ( x , t ) , a_ .'/;,- con^ia'U

rive'! :>./ (i'.-l) and a 'he ranmion ini ro.huie-J in l^.,:).- c

:VV'.r; , • ' ; J , y be. arbitrary nemnei.i: ivo i-ove'm: c. i'u'. .i = u + v T ,

T = ( 3 t ) ' / 3 . -hen the followr:j e..-' CH-.I:,, holdü

( 5 . 3 a ) sup j q ( x , t ) - q d ( x , t ) | : A for t • 0X •"-;l

( 5 . 3 b ) s u p | q ( x , t ) - q d ( x , t ) | • > T ~ ' ; ' .•• t • ~x —-i

(5.4a) A = —

1 19

f°(5.4b) Y = a*Yj '

+ Y + |b(O)|(-i + |Ai(„)|dn

with the constant y os in (4.2).

Clearly, if in addition qn satisfies (4.4), then we can improve the-1 -2

estimate (5.3-4) since T can be replaced by T . Although similar

remarks apply to the estimates below, they are omitted to avoid inter-

ference with the reasoning.

Note the similarity between (5.3-4) and the corresponding estimate

Chapter 2, (7.16) in the KdV case. There are, however, two important

differences. Firstly, the estimate (5.3-4) holds for all t 0, whereas

Chapter 2, (7.16) is only valid for t : t ,, where t , is a certain

critical time. Secondly, the results of Chapter 2, theorem 7.1,

including (7.16), hold only for restricted v-values, whereas theorem 5.1

is valid for any value of v "' 0.

Let us discuss some consequences of theorem 5.1.

Firstly, by combining (5.3-4) with (2.3) it is found that, if all

bound states are purely imaginary, the solution q(x,t) of (5.1) splits

up into N solitons as t + « in the following way

Corollary to theorem 5.1. Support' that q„(x) •'«.•.• / •>.' .h.hi; wri'ij .hi;

{b (c),C-,cT} with c. = ii|., 0 * i) < ... • n • w 3<ui cT = i ;i. , u . £ 1R\' 01.

f'nen the. solution of (h.l) r,.U. Cafics

( 5 . 5 ) l i m s u p | q ( x , t ) - ^ ( - 2 n s g n ( u ) s e c h [ 2 : i ( x - x - 4 n 2 t ) ] ) ! = 0

t-*» x > - v T P P P P P Pijith x an in (",ji>).

P

Furthermore, besides the bound (5.3b), which depends only on t, we

obtain from (4.2-3) and theorem 5.1 in the coordinate region

T = (3t) '•-• 1 , x ' -a, a = u + \>T, the x and t dependent bound

(5.6a) | q ( x > t> - ,d(x,t)| • aT-'exp[ § •l

(5.6b) a = vY i(l» b = n ,

120

with ü £ (0,1), Y„, P the constants introduced in section 4.

u

It is a direct consequence of the last remark of Chapter 4, section 6,

that the estimates (5.2-3-6) remain val id if on the left one replaces

|q(x,t) - qd(x,t)| by

(5.7) J (q2(s,t) - qd(s,t))ds .XJ \ /

In particular, this yields

(5.8) q2(x,t)dx = q2(x,t)dx

_aJ _aJ

Note that by (5.6)

(5.9) ! |q(x,t) - q,(x,t)|2dx = Q(t"1/3) ast

ast

! |q(x,t) - qd(x,t)|2dx = 0(t"1/3)

—a

This gives us another way to derive (5.8) (see Chapter 2, pp. 73, 74).

In the special case that the breather bound states satisfy (2.7) we

find from (2.10b-12) and (5.3)

(5.10a) q2(x,t)dx = 4 I, lm c + 0(t~1/3) as t -> -.—(r

Using the formula (see [2])

(5.11) j q 2 (x , t )dx = | j logM + |b (r.)|2]dr, + 4 ï Ia f

we obtain for the complementary integral

(5.10b) j q2(x,t)dx = | J logM + |br(f,)|2 jdf. +0(t~'/3) as t ->• -.

It is interesting to compare (5.10) with the formulae Chapter 2, (7.20-25)

obtained in the KdV case.

Finally, let us remark that, as in Chapter 2, we can apply theorem 5.1

to obtain estimates in subregions of (1.4), e.g. x ' vt • 0, with v • 0

arbitrarily fixed.

Specifically, if we make the additional assumption

I21

(5.12) There is an integer n £ 2 such that b £ Cn(E) and a! 1(' \ ^

derivatives b (5), j = 0,1,...,n satisfy

then, reasoning as in Chapter 2, p. 75, we obtain

(5.13) sup |q(x,t) - q (x,t)| = 0(t ") as txgvt

Likewise, one can make the extra assumption

(5.14) There is an £» ~> 0 such that 1n(x) - (Kexp(-2t;,.x)) as x >

and combine the remark at the end of Chapter 4, section 2, with the

reasoning in Chapter 2, pp. 75, 76. Choosing c > 0 to be strictly

less than t;_, b/v and Im c., j = 1,2,...,N one then arrives at

(5.15) sup |q(x,t) - q (x,t)| = 0(exp(-u t)) as t •> »xïvt

with a. = 2c (v - 4c2) > 0.

References

L 1] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inversescattering transform - Fourier analysis for nonlinear problems, Stud.Appl. Math. 53 (1974), 249-315.

[ 2] M.J. Ablowitz and H. Segur, Solitons and the Inverse ScatteringTransform, Philadelphia, SIAM 1981.

[ 3] Y. Kametaka, Korteweg-de Vries equation IV. Simplest generalization,Proc. Japan Acad., 45 (1969), 661-665.

[ 4] P.D. Lax, Integrals of nonlinear equations of evolution and solitarywaves, Comm. Pure Appl. Math. 21 (1968), 467-490.

[ 5] G.L. Lamb Jr., Elements of Soliton Theory, Wiley-Interscience, 1980.

[ 6] M. Ohmiya, On the generalized soliton solutions of the modifiediiorteweg-de Vries equation, Osaka J. Math. 11 (1974), 61-71.

[ 7] P. Schuur, Asymptotic estimates of solutions of the Korteweg-de Vriesequation on right half lines slowly moving to rho left, preprint 330,Mathematical Institute Utrecht (1984).

122

[ 8] P. Schuur, On the approximation of a real potential in theZakharov-Shabat system by its reflectionless part, preprint 341,Mathematical Institute Utrecht (1984).

[ 9] il. Segur and M.J. Ablowitz, Asymptotic solutions of nonlinearevolution equations and a Painleve transcendent, Proc. Joint US—USSRSymposium on Soliton Theory, Kiev 1979, V.E. Zakharov andS.V. Manakov eds., North-Holland, Amsterdam, 165-134.

[10] S. Tanaka, Non-linear Schrödinger equation and modified Kortewejj-deVries equation; construction of solutions in terms of scatteringdata, Publ. R.I.M.S. Kyoto Univ. 10 (1975), 329-357.

[11] M. Wadati and K. Ohkuma, Multiple-pole solutions of the modifiedKorteweg-de Vries equation, J. Phys. Soc. Japan, 51 (6) (1982),2029-2035.

CHAPTER SJK

MULTISOLITON PHASE SHIFTS FOR THE MODIFIED KORTEWEG-

DE VRIES EQUATION IN THE CASE OF A NONZERO REFLECTION

COEFFICIENT

We study multisoliton solutions of the modified Korteweg-de Vries

equation in the case of a nonzero reflection coefficient. Confining

ourselves to the case that all bound states are purely imaginary, we

derive an explicit phase shift formula that clearly displays the nature

of the interaction of each soliton with the other ones and with the

dispersive wavetrain. In particular, this formula shows that each soliton

experiences, in addition to the ordinary N-soliton phase shift, an extra

phase shift to the right caused by the interaction with the dispersive

wave train.

1. Introduction.

We consider the modified Korteweg-de Vries (mKdV) equation

q + ó<j2<j + q = 0 with arbitrary real in i t i a l conditions q(x,0) =t X XXX

= q„(x), such that q_ is a bona fide potential in the Zakharov-Shabat

scattering problem, sufficiently smooth and rapidly decaying for

|xj ->• <" for the whole of the Zakharov-Shabat inverse scattering method

124

[1] to work and to guarantee certain regularity and decay properties of

the scattering data, to be stated further on.

The long-time behaviour of the solution q(x,t) of the mKdV problem

has been treated by several authors. For details we refer to [9], i.e.

Chapter 5, where some of the idiosyncrasies are discussed.

In the asymptotic part of this paper (sections 3, A and 5) we

confine ourselves to the case that all bound states are purely imaginary.

Then it is found that as t -> +•» the solution decomposes into N sech-shaped

solitons moving to the right and a dispersive wavetrain moving to the left.

As t + -« the arrangement is reversed. One can now ask for the phase

shifts of the solitons as they interact both with the other solitons and

with the dispersive wavetrain.

In this paper, starting from our asymptotic analysis of the solution

given in Chapter 5, we derive a phase shift formula that closely resembles

that found by Ablowitz and Kodama [2] for the KdV case. We next show that

- as in the KdV case - a simple substitution produces a more transparent

formula. From the latter formula it is evident, that each soliton

experiences, in addition to the ordinary N-soliton phase shift, an extra

phase shift to the right, the so-called continuous phase shift, caused by

the interaction with the dispersive wavetrain. Thus, the presence of

reflection causes an advancement in the soliton motion. Note that in our

KdV analysis [8], i.e. Chapter 3, we found the opposite. There the inter-

action with the dispersive wavetrain causes a delay in the soliton

motion, since the continuous phase shifts are to the left.

The composition of this paper closely resembles that of Chapter 3. In

section 2 we briefly discuss the left and right scattering data associated

with q„(x) and show how, in the general case that the bound states are

distributed symmetrically with respect to the imaginary axis, the left

scattering data can be expressed in terms of the right scattering data in

a convenient way. In the rest of the paper we assume that all bound states

are purely imaginary. In section 3 we quote a result from Chapter 5,

describing the asymptotic behaviour of q(x,t) as f v +1. By the same

symmetry argument as in Chapter 3, we derive from this result the

asymptotic behaviour of q(x,t) as t >• —". Next, in section 4, the two

asymptotic results are combined to give a phase shift formula of Ablowitz-

125

Kodama type. The representation of the left normalization coefficients

in terms of the right scattering data, which was obtained in section 2,

then enables us to write the phase shift formula in a more transparent

form. To illustrate our results, we calculate in section 5 the continuous

phase shifts arising from a sech initial function.

2. Left and right scattering data and their relationship.

For Im z, ~~ 0 we introduce the Jost functions g> (x,0 and ï>„(x,r.),

two special solutions of the Zakharov-Shabat system

(2.0 ( ) = ( )( ). ' = 4-> -~ -* • +->V / \ • J\ J dxZ U i

uniquely determined by

(2.2a) Yr(x,r.) = e~1CXR(x,f;), H m R(x,r.) = ^

(2.2b) i|< (x,<-) = e 1 C XL(x,O, H m I x,-.) = ( )-

Here R and L .irt> vector functions with two components. Ue set

(2.3a) r_(r.) = 1 + | qo(x)R?(x,f.)dx, Im r, 0

(2.3b) r+(rj = - q()(x)e~2l'-XR1 (x,,.)dx, : £ E

(2.3c) C+(r.) = 1 + qo(x)L](x,r,)dx, lm : O

(2.3d) t_(O =- f qQ(x)e2lrxL2(x,f )dx, r. € R.

Note that r_(r) = Z+(.r.) and r+(r.) = ('_(-,-). It is si own in [5] thai

r_(O is analytic on Im r. 0 and continuous on Im c. ' 0.

Following [5] and Chapter 5, we shall assume throughout that r lias nc

zeros on the real axis and that all zeros of r in C are simpK'. As a

result the function r (,-) lias at most finitelv many zeros,» T '

I2o

lm r, •» O, distributed symmetrically with respect to the imaginary axis.

We shall call them the bound states associated with q„. From [5], p. 156

we know that there are nonzero constants n such thatm

(2. A) g»r(x.Cm) = a A ( x , r , m ) .

Furthermore, by [5], (5.3.2) one has

Observe that the integral does not vanish since the r. are simple ze-os of

r_. This enables us to introduce the right normalization coefficients

(2.6a) <-li[J ^^(x.^^

as well as the left normalization coefficients

(2.6b) c f -

Next, we introduce the following functions of r. £ E

(2.7a) a = r_ , the right transmission coefficient,

(2.7b) a,, = £ , the left transmission coefficient,

(2.7c) b = r r_ , the right reflection coefficient,

(2.7d) b,, = H_Z , the left reflection coefficient.

We shall call the aggregate of quantities (a (r,),b (c),t ,C } the righc

scattering data associated with the potential q_. Similarly we refer to

(a„ ((,) ,b , (r) ,r, ,C I as the left scattering data associated with q„.

Note that the right scattering data were already introduced in Chapter 4,

where some of their properties were discussed.

We claim that a,,, b,, and C can be expressed in terms of the right

scattering data in the following way

a (r,)(2.3a) af (O = a^c) bf(O = a\_o br(-r.),

(2.8b) c£ -[«ft"1

p=1pffin

-2c f«> log(i + |b (c) |2 )mi r

•n 0 J ? Cm

Clearly, only formula (2.8b) deserves a proof.

To provide it, note first that by (2.4-6)

Cr = ctJC .m m m

( 2 . 9 )

Next, it follows from (2.5-6b) that

(2.10) d r"

Eliminating a from (2.9) and (2.10) we find

Lastly, from [3], p. 57 we obtain the representation

(2.12) r_(c) = jexpj Im c, - 0.

Differentiating and using the symmetry relation b (c) = b (~O, we find

(2.13)dr-

* 1

p=m ra p

T o g e t h e r , ( 2 . 1 1 ) and ( 2 . 1 3 ) y i e l d the d e s i r e d formula ( 2 . 8 b ) .

3 . Forward and backward a s y m p t o t i c s .

Once t h e r i g h t s c a t t e r i n g d a t a of q (x) a r e known, t he s o l u t i o n q ( x , t )

of t h e forward mKdV problem

(3.1) j qt

qxxx = tO

q(x,0) = qo(x)

123

can in principle be computed by the inverse scattering method [1]. An

asymptotic analysis of the solution was presented in Chapter 5. It was

found there that the asymptotic structure of the reflectionless part of

q(x,t) is rather complicated when the location of the bound states r, is

not suitably restricted.

To avoid any unpleasantness of this kind we shall assume from now on

that all the bound states are purely imaginary. Let us denote them by

(3-2> ?m = inm' With ° *] \ < ••• < n2 '; V

Since ui ( x , i n ) and iJjp(x,in ) a r e r e a l we then have t h a t t h er m cm

normalization coefficients are purely imaginary as well, say

(3.3) Cr = ip r , C = i y \ with u r, \il € KMOhm m m m tn m

The asymptotic behaviour of the solution q(x,t) of (3.1) is now

enlightened by the following lemma, which was established in Chapter 5,

section 5.

Lemma 3.1. Assume that

(3.4) b (c) is of jlass C2 ( E ) and the dpiv.vativus b ^ ( c ) , j = 0 , 1 , 2

jP'j bounded on iR.

Then one hao

(3.5) lim sup1

q(x,t)- z. \-2n sgn(ur)secht2-i (x-x+-4n21) ] f| = 0,m=1 I m & m m m m J

P=1 l^TTm v s p ra

Next, let us consider the backward mKdV problem, starting from the

same initial function q_(x), i.e.

f q,. + 6q2q + q = 0 , — • x • +'••, t 0(3.7) \ t X X X X

L q(x,0) = qQ(x).

120

Plainly, if q(x,t) satisfies (3.7), then w(x,t) = q(-x,-t) satisfies

(3.3)r w. + 6w2 w + w = 0,J t x xxx

*• w(x,O) = qp(-x),

so that w(x,t) satisfies the forward mKdV problem with initial function

q.(-x). To solve (3.7) it is therefore sufficient to determine the right

scattering data associated with q..(-x) and apply the inverse scattering

method to (3.8). However, an easy calculation reveals that the right

scattering data associated with q_(-x) are identical with the left

scattering data associated with q_(x). The latter were examined in the

previous section. Thus, to find the asymptotic behaviour of the solution

q(x,t) of (3.7) for t -* -°° we merely apply lemma 3.1 to problem (3.3) and

perform the transcription q(x,t) = w(-x,-t). This yields

Lemma 3 .2 . Ai'tinni^ tint-

( 3 . 9 ) b f ( r . ) /:.-; >>f ,-l.ics C* ( R ) .v.J

Ü;V boundc\i on E.

) , j = 0,1,2

then onr. nar,

(3.10) lim supt ~ » x: | t | 1/3

• " f f - 11q ( x , t ) - X , ^ - 2 ' s m i ( i : ) s c c h f 2 ' . ( x - x - U - : ? t ) ] = 0 ,

ra=l 1 in ° m m m m f

(3.11) X,X,nV.,\2

ra v ra p m

S i n c e •<• i s r e a l i t f o l l o w s from ( 2 . 9 ) t h a t s n n ( . . ) = s n n ( . . ) . I ' l i e r e fo r im m r.i

i n ( 3 . 5 ) a n d ( 3 . 1 0 ) t h e s a m e s o l i t o n s e m e r g e , a p a r t f r o m a p h a s e s h i f t .

4 . An e x p l i c i t p h a s e s h i f t f o r m u l a .

L e t u s a s s u m e t h a t b a n d b , , s a t i s f y t i n ' c o n d i t i o n s ( i . i ) a n d ( 3 . " ) .

T h e n t h e c o n v e r g e n c e r e s u l t s ( 3 . 3 ) a n d ( 3 . 1 0 ) d i s p l a y c l e . i r l v h o w t l i e

s o l u t i o n q ( x , t ) o f t h e raKdV e q u a t i o n e v o l v i n g f r o m q ( . \ , 0 ) = q . . l x ) S ' - l i l s

up into N solitons as t -> ±«. In particular, we find for the m-th

soliton the following phase shift

;J I m-J /'I ~ i

(4.1) S = x + - x =m ra mm ra m 2r,m

b\ 4,^ P=1m

Note how closely this resembles the phase shift formula derived by

Ablowitz and Kodama [2] for the KdV case (see Chapter 3, (4.1)).

Similarly, the formulae (3.11) and (4.1) become more transparent if

one inserts the representation (2.8b), taking into account (3.2-3). In

summary, this leads to

(4.2a)

(4.2b)

1 r i / P m\+ — p i , '°gL' + ,, )

..2m

(4.3a) Sm = Sd + Si

(4.3b) S ^ f j 'm

(4.3c) S C = - .

m

In S we recognize the pure N-soliton phase shift (caused by pairwise

interaction of the m-th soliton with, the other ones). The quantity Sm

(which is positive for nonzero b ) can be seen as the shift caused by

the interaction of the m-th soliton with the dispersive wavetrain. For

nonzero b we obviously have

(4.4) S^ ... SC2 • S^ > 0.

Thus, surprisingly enough, the interaction with the dispersive wavetrain

advances the solitons in their motion and the effect is most heavily felt

by the smallest one, corresponding to •; . Recall that in our KdV analysis

(Chapter 3) we found the opposite situation. There the interaction with

the dispersive wavetrain causes a delay in the motion of the solitons.

Let us examine where this difference comes from. Apparently, formula

131

(4.3b) for the mKdV pure N-soliton phase shift S coincides with the

corresponding KdV formula Chapter 3, (4.3b), provided we identify n with

K , m = 1,2,...,N. But after this identification it is clear that,

remarkably enough, also the continuous phase shifts S are given by the

same formula, namely

p log|ar(k)|2

where a (k) denotes the right transmission coefficient. Now, suppose b is

not identically zero. Then, in the KdV case we have by Chapter 2, (2.12)

(4.6a) |ar(k)|2 = 1 - |br(k)|

2,

cso that S is negative.By contrast, in the mKdV case, Chapter 4, (2.24b) tells us

(4.6b) |ar(c)|2 = 1 + |br(c)|

2,

c c

leading to a positive sign of S . Thus the difference in sign of S stems

from the difference in sign of |b (k)|2 in the formulae (4.6).

Usin^ the formula (see [3])

r» , p N(4.7) q2j(x)dx = log(l + jb (r.)|2)d? + 4 ± Im r

we obtain for the continuous phase shift S the following estimate in termsm •'

of the initial function q,- (x) and the bound states t = i-i :0 p P

(4.8) 0 a S^S -±( { ^ W d x - 4

5. An example: the continuous phase shifts arising from a sech initial

funct ion.

To make the previous discussion less abstract let us compute the

continuous phase shifts arising from the initial function

132

:5.1) qQ(x) = a sech x, a £ K\{ k+ i; k e Z}.

[t is a remarkable fact that for the potential (5.1) one can solve the

scattering problem (2.1) in closed form. In fact by an obvious modification

}f the calculation performed in [7], section 3, we obtain for x £ ÜR,

lm C S 0

(5.2a) Rt(x,C) = F(a,-a; J-H; z)

(5.2b) R2(x,e) = o T ^ d - z ) * A.F(a,-a; t-it; z)

with

(5.3) z = JO + tanh x) .

Here F denotes the hypergeometric function in the notation of [A], p. 556.

The same argument as in [7] now gives us

Note that the assumptions about r_ made in sections 2 and 3 are fulfilled

since all zeros of r (;) in Im c, 5 0 are simple and lie on the positive

imaginary axis.

Specifically, one has the following situation.

If |a| < i, then r_(q) has no zeros at all. Note that in this case

(5.5) J |qo(x)|dx < \,

which is in agreement with the sufficient condition [1]

(5.6) |qo(x)|dx < 0.904-co-'

for the absence of solitons.

For j ex | > \, let N £ 1 denote the unique integer such that

N - J < | a. | < N + i. Then r_(^) has precisely N zeros given by

(5.7) C p = i(i + \A - P), P = 1,2 K.

Clearly, ^Q(X) is reflectionless (i.e. b a 0) if and only if • £ 72..

133

Moreover, if a £ Z\{0}, then qn(x) is reflectionless with N = |a| bound

states.

As is easily seen, q_ belongs to the Schwartz space. Hence, the

same holds for b , which equals b„, since q is even. We conclude that

(3.4) and (3.9) hold, so that (3.5) and (3.10) are valid.

Now, let us compute the continuous phase shifts S .

By (5.4a) one has

On the other hand, by (2.12), (3.2)

r« log(1+f f r«

(5.9) r (iv) = \exp{- - .: ü v . o.o=1 v+n

P

Equating both expressions and making repeated use of the recurrence formula

r(z + 1) = zln(z), we obtain

| , H 1 )1B(Uv, l + v) ƒ '

where Ë refers to the beta function ([4], p. 258). Combining (5.7) and

(5.10), we arrive at the following expression for the continuous phasec

1 , JB(1 + 2|.»i 1 1 J-O21—7-7- 1i+|'i|-m &\ B(1 + |a

shifts Sin

(5.1D s1- - ' , . J Q M ^ M M - u - N , i - u t :i){m - m, 1 + |.( | - m) ƒ'

To estimate the magnitude of S we may benefit from (4.8), which yields

(5.12) 0 ' SC -in

The lower bound in (5.12) can be improved by means of the inequality

/r i->\ B(c — 2b, c) . b 2 rt , i

(5.1 3) -57 r '—r4 " 1 + — , c • 0, b • <c,B(c - b,c - b) c

due to Gurland [6l. Together with the simple estimate lo.';(l+x) TTX for

O x - I, this tells us

3 (1 - m + N)(J + |a| - m)

References

[ 1] M.J. Ablowitz, D.J. Kaup, A.C. Newell and 11. Segur, The inversescattering transform. Fourier analysis for nonlinear problems, Stud.Appl. Math. 53 (1974), 249-315.

[ 2] M.J. Ablowitz and Y. Kodaraa, Note on asymptotic solutions of theKorteweg-de Vries equation with solitons, Stud. Appl. Math. b6 (1982),No. 2, 159-170.

[ 3] M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering;Transform, Philadelphia, SIAM, 1931.

[ 4] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions,National Bureau of Standards Applied Mathematics Series, Ho. 55,U.S. Department of Commerce, 1964.

[ 5] W. Eckhaus and A. van Harten, The Inverse Scattering Transformationand the Theory of Solitons, North-Holland Mathematics Studies 50,1981 .

[ 6] J. Gurland, An inequality satisfied by the gamma function, Skand.Aktuarietidskr. 39 (1956), 171-172.

[ 7] J.W. Jliles, An envelope soliton problem, SIAM J. Appl. Math. 41(1931), No. 2, 227-230.

[ 3] P. Schuur, Multisoliton phase shifts in the case of a nonzeroreflection coefficient, Phys. Lett. 102A (1984), No. 9, 337-392.

[ 9] P. Schuur, Decomposition and estimates of solutions of the modifiedKorteweg-de Vries equation on right half lines slowly moving leftward,preprint 342, Mathematical Institute Utrecht (1984).

CHAPTER SEl/EW

ASYMPTOTIC ESTIMATES OF SOLUTIONS OF TIE SINE-GORDON

EQUATION ON RIGHT HALF LINES ALMOST LINEARLY MOVING LEFTWARD

We consider the sine-Gordon equation q = 4 sin[2 _cJ' q(x',t)dx']

with arbitrary real initial conditions q(x,0) = q„(x), sufficiently smooth

and rapidly decaying as |x| -* °=, such that q_ is a bona fide potential in

the Zakharov-Shabat scattering problem. Using the method of the inverse

scattering transformation we analyse the behaviour of the solution q(x,t)

in coordinate regions of the form t > 0, x ? -ii - vt , where u, -.• and r

are nonnegative constants with 6 •• 1.

We derive explicit x and t dependent bounds for the non-reflectionless part

of q(x,t). Owing to the rather explicit structure of the reflectionless

part it is then a small step to obtain estimates of q(x,t) as well as some

interesting energy formulae.

1. Introduction.

iVe study the sine-Gordon problem

X

(1.1a) qt = i sin 2 q(x',t)dx'|, —• • x +-, t • O

(1. 1b) q(x,O) = qo(x),

136

where the initial function qn(x) is an arbitrary real function on K,j

such that

(1.2a) qo(x) satisfies the hypotheses (2.2-13) in [9] (i.e. Chapter 4)

and is therefore a bona fide potential in the Zakharov-Shabat

scattering problem.

(1.2b) There is an integer k_ such that

rI qo(x)dx = kQ!i.

(1.2c) 1n^x^ *s sufficiently smooth and (along with a number of its

derivatives) decays sufficiently rapidly for | :•: | > '•••:

( i) for the whole of the Zakharov-Shabat inverse scattering

method [1], [2] to work,

(ii) to guarantee certain regularity and decay properties of

the right reflection coefficient to be stated further on.

Uniqueness of solutions of (1.1) can be proven within the class of

functions q(x,t) vanishing sufficiently rapidly for |x| • *° and satisfying

(1.3) q(x,t)dx = k0.-.

Suitably adapting the procedure outlined in [11] for the raKdV problem one

can establish by an inverse scattering analysis that condition (1.2)

guarantees the existence of a real function q(x,t), continuous on Rx[0,~),

such that

(1.4a) For each t • 0 th° function q(x,t) satisfies the hypotheses

(2.2-13) in Chapter 4.

(1.4b) q(x,t) has the property (1.3).

(1.4c) q(x,t) satisfies (1.1) in the classical sense.

(1.4d) q(x,t) falls in the class of functions for which uniqueness of

solutions of (1.1) can be proven.

Whenever, in this paper, we speak of "the solution" of (1.1) we shall

refer to the solution obtained by inverse scattering. Let us add that,

157

despite the multitude of sine-Gordon papers appeared so far, we know of

no reference in which the above is spelled out in full detail.

Given the solution q(x,t) of (1.1) we put

rx rx(1.5) a(x,t) = -2 q(x\t)dx\ o Q ( x ) = - 2 q^xMdx'.

— CO* —CO-*

Plainly, a(x,t) satisfies

(I.óa) a = sin a, -°° < x -; +°°, t •• 0

(1.6b) a(x,0) - oQ(x)

and for fixed t S 0 one has

(1.7) lim o(x,t) = 0 , lim o(x,t) = -2k -i.

In discussions based on the Zakharov-Shabat inverse scattering method the

version (1.6) of the sine-Gordon problem is most frequently used (see

Li], [2], [5], [7]), the only cases allowing an easy treatment being

those in which a has the boundary behaviour displayed in (1.7).

Let us recall that by the inverse scattering method the solution q(x,t)

of (1.1) is obtained in the following way.

First one computes the (right) scattering data fb (r,),c.,C.} associated

with q„(x). For their definition and properties we refer to Chapter 4.

Next une puts (see [1], [2])

(1.3a) cT(t) = cTexp{-it/(2c.)}, j = 1,2,...,N

(1.3b) br(c,t) = br(t)exp{-it/(2c)}, -» •' c - +«•.

Then by the solvability of the inverse scattering problem, there exists

for each t • 0 a real potential q(x,t), satisfying the hypotheses (2.2-13)

in Chapter 4 and having {b (r;, t),C-,C. (t)} as its scattering data. Note

that by Chapter 4, (2.25)

r r(1.9) br(O,t) = -tan q(

(x,t)dx ,

so that property (1.3) follows from (1 . 1b-2b-3b-9) and a continuity

argument. The function q(x,t) is the unique solution of the sine-Gordon

138

initial value problem (1.1).

Explicit solutions by the above procedure have only been obtained for

b = 0. The solution q,(x,t) of the sine-Gordon equation (I. la) with

scattering data fO,C.,C.(t)} is called the reflectionless part of q(x,t).

By contrast with KdV and mKdV, the long-time behaviour of the

solution q(x,t) of the sine-Gordon problem (1.1) is not treated

extensively in the literature.

In the absence of solitons, it is suggested in [3], pp. 30, 91, that

there are three distinct asymptotic regions

I. x ; 0(t) II. |x| • 0(t)

III. -x •: Ö ( t ) ,

where 0 denotes positive proportionality, within each of which the

solution q(x,t) of (1.1) has a different asymptotic expansion. Mote that,

in view of the negative group velocity associated with the linearized

version of (1.1), one may expect that the solution will evolve into a

dispersive wavetrain moving to the left.

If solitons are present, then, generic-ally, as :. ime goes on, q.(s,t)

will desintegrate into breathers and sech-shapod solitons (see the

discussion in section 2 ) . But the sech-shapcd solitons as well as the

breather envelopes propagate to the left. Consequently, a decomposition of

q(x,t) into a dispersive and a soliton part will not easily be

demonstrated in this case. The only thing beyond doubt is that for large

t the solution q(x,t) will be confined to the negative x-axis. This raises

the question: How small is q(x,t) in the regions I and 11?

In this paper we give a definite answer to this question by analysing the

behaviour of q(x,t) in coordinate regions of the form

(1. 10) t - 0 , x • -,, , = u + . t'

where ,., and •• are nonnegative constants with ' • 1. Here ;, and , are

arbitrary but •• depends on properties of q .

More precisely, let q_ and a number of its derivatives decay fast enough

to guarantee that b has n . 2 derivatives decaying sufficiently rapidly

(see (4.1)).

139

Choose

(1 .11 ) 0 • X ' n - | , 0 • ö •• 1 - (^~

Then we prove t h a t

(1 .12 ) sup | q C x , t ) | = Q ( t ~ " ) as t > ~.

Moreover , we d e r i v e t he ene rgy formulae

(1 .13a ) j q2 ( x , t ) d x = 0 ( t ~ ' ) as t >••

ra ? r J

(1.13b) q 2 ( x , t ) d x = ^ - log(1 + ! b r ( . . ) | 2 )d . ' + 4 ^ Ira r. + Q(t )

a s t ->• •».

In particular, if q_ is in the Schwartz class then (1.12-13) hold far all

\ • 0 a n d a l l 0 •- 'i • 1 .

The paper runs as follows.

In section 2 we isolate certain properties of the rel'lect ionless part

q,(x,t). In section 3 we recall a theorem estahlished previously in

Chapter 4, in which q(x,t) - q.(x,t) is estimated in terms of (x+y;t).

Section 4 is devoted to the construction of some simple explicit bounds of

i! (x+y;t). Then in section 5 these bounds and the theorem are combined to

yield estimates of q(x,t) - q,(x,t). Since q,(x,t) was already explored in

section 2, we have reached our goal and found estimates of q(x,t).

2. The asymptotic structure of the reflectionless part.

From Chapter 4, (5.28) we obtain for the reflectionless part of q(x,t)

the explicit expression

(2.1) qd(x,t) = 2 Im -^ log det A,

where A = (:i .) denotes the N'N matrix with elementsPJ

-2U.x( 2 . 2 ) * . = [ C r ( t ) ] e J •• . + i ( c + ,',.) ' .

PJ J PJ P J

140

If :i = 0 in Chapter 4, (2.10), then the asymptotic structure of

q, (x,t) is relatively simple. In that case all the bound states and

normalization coefficients are purely imaginary, say r,. = in-,

0 • n.. ••• ... < i|. •• n, and C. = iu.> v • £ R\{0}. An obvious adaptation2 1 j j j

of the reasoning given in [3] shows that as t approaches infinity then

the reflectionless part of q(x,t) decomposes into f! solitons uniformly

with respect to x on R. Specifically

N(2.3a) lira sup

t-*» x£R

q,(x,t)- I. -2M sgnd. )sech[2', (x-x -v t)] = 0Md ' P=1 \ P P P P P "

Note that v has negative sign. Thus, all sech-shaped solitons propagate

to the left.

In M • 0 in Chapter 4, (2.18), then the structure of q (x,t) is more

complicated. Let us consider the simplest case: S = 2, <"., = •' + it,,* r r

C7 = -•" + in = ~C.» with r, 'i • 0. Then C. = • + in and C? = -\ + iu,

where '• and |i are real constants not vanishing simultaneously.

Using (2.1) one gets (cf. [10] (or Chapter 5), (2.4))fsin !• + (:i/: )cos •:• tanh 1'h 1']

! J( 2 . 4 ) q , ( x , t ) = 4-1 sech r —; 7—rrv; rr T5Hd I 1 + ( M / \ ) 2 C O S Z J > sech 2

w i t h

( 2 . 5 a ) v = 2^x - r,(2(.-2 + r , 2 ) ) ~ 1 t + ji

( 2 . 5 b ) ï' = 2:,x + n ( 2 ( ^ + ' , 2 ) ) ~ ' t + i>,

where the constants i and <;> satisfy

(2.6a) ^ ~ {

(2.6b)

(2.6c) cos Q = (A»; + un)(\2 + M2)~*(c2 + -i 2)" 2.

Tlius q , ( x , t ) has t he form of a b r e a t h e r ( s ee [ 6 ] ) w i t h enve lope and phasev e l o c i t i e s v = ( - 4 ( r 2 + - i 2 ) ) " ' and v . = (4(f2 + - ! 2 ) ) ~ 1 = - v ,

e ph e

141

respectively. Observe that the sign of v is negative.

If M > 0 in Chapter 4, (2.18) and N :• 2, then, generically, q,(x,t)

will decompose into M breathers and (N - 2.M) sech-shaped solitons. However,

it is easy to construct examples (e.g. N = 3, r,. = r + in, z = i(r;2 + i2)'),

in which no complete decomposition into breathers and sech—shaped solitons

takes place.

Since sech-shaped solitons as well as breather envelopes propagate to

the left one expects that, regardless of the location of the bound states,

for large t the function q,(x,t) will be concentrated on the negative

x-axis. Let us verify this. By Chapter 4, (5.21) we have

N(2.7) qd(x,t) = -2 lm ? = ] B ,,

where B = (J5 .) is the inverse of the matrix A given by (2.2).

Since

N . -2ir, x N

(2.8) 1 - „ ? = ] — ~ — 8£. = [Cr(t)]~ e P -I, .- .,

we can rewrite (2.7) as

(2.9) qd(x,t) = -2 Im Z^ Cr(t)e P ^1 - £ = ) *f i-. .V

Hence, using Chapter 4, (5.13), one gets

(2.10a) |qd(x,t)| s 2 ^ |C

(2.10b) IJ, (x,t) = 2(lm c ) (x + |2c l~2t),

where the constants N,., introduced in Chapter 4, (5.7b), reappear in this

paper in (3.4b). In particular, this shows that

(2.11a) sup |q (x,t)| =3(e"';t) as tx>0

(2.11b) | q2(x,t)dx = a(e" 2 t) as t , -,

where „ is the positive constant

(2.12) .. = min|2(lm C )|2- \~2; p = 1,2 N|.

142

Thus, indeed, we conclude that, as time goes on, q.(x,t) is confined to

the negative x-axis.

) satisf

r0 N

Since q,(x,t) satisfies Chapter 4, (2.12), we obtain from (2.11b)

(2.13) | q|(x,t)dx = h Im C + 0(e~2wt) as t •» -,

3. A useful result obtained earlier.

The fact, that for each t > 0 the solution q(x,t) of (1.1) satisfies

the hypotheses (2.2-13) in Chapter 4, required for a bona fide potential

in the Zakharov—Shabat scattering problem, immediately yields the

following result, which is established in Chapter 4 in the form of theorem

4.1 and its corollary.

Theorem 3.1. Let q(x,t) be the solution of the sine-Gordon problem

(3.1)q = { s i n 2 q ( x ' , t ) d x ' | , - » < x < +°°, t *

q(x,0) = qQ(x),

•jhere the initial function q o ( x ) is an arbiivar'j i'cal ^unction jti E ,

satisfying {1.2a-b-o(i)). Let {b ( c ) , ^ . , C . } be the cectticrin-j l i t ;

associated with q . ( x ) . Then for each x £ E and t "• 0 ori^: 'as

( 3 . 2 ) | q ( x , t ) - q ( x , t ) l S a^f f |s: (x+y; t ) | 2 dy + sup ! ( x + y ; t ) | ) ,

jlure q , ( x , t ) is the reflect to nlesa part of q ( x , t ) jivci b'j (''..!) .rid

(3.3) r:c(s;t) - 1 f ^ i ü ^ 5 ^ 1 ^ ^ , , s € R,

(3.4a) a o = 1 + p J = , < I n . C p ) - 1 N p j ,

( 3 . 4 b ) N p j = 2 ( ! m C p ) * ( l B c . ) * ^ J _ ^ | k U 1

P t

~"i(i'J','."<'."!Ji'i'', t h e f j i loui'i^ a v y i j r i b o u n d ••:•.*• df

•j"V

143

ao f" / \(3.5) sup | q ( x , t ) - q , ( x , t ) | •: — |b (O | +|b (c) I2 WC

(x,t)£]Rx[0,») — •' K ;

4. Estimates of '' (x+y; t) .

Since in (3.2) the quantity a is invariant with tine, the magnitude

of q(x,t) - q,(x,t) in the region (1.10) depends only on the behaviour of

;,: (x+y;t). This function in turn can be estimated in a simple and explicit

way, as is shown in the next lemma.

Lemma 4 . 1 . In the situation .•;ƒ i i v i v i «•;./, asnifu? ;':.it

There is an integer n : ' 2 such th.n b is . • ƒ i/ • \ r

a l l d e r i v a t i v e s b J ( c , ) , j = 0 , 1 , . . . , n s t ! : ' i \ - y

(4.1) There is an Integer n ;- 2 such ih.n b iü . • ƒ -.•' !;<•• c"(]R) nJr *

Lev t - 0, y > 0 and j

( 4 . 2 ) x •£ - M - v t , 'Sih'i'c p , v j f l . i is . f j ' i 1 ' I . I V H . - J .-.•.•; '. ' .'..'•;*! : ; • ! • , : .

Ii

Pu L I

( 4 . 3 ) i = 1 + ii + v t ' , w = x + y + i , B d \ , t ) = b ( - , ) e ~ l l ! , D = ( — ) m l i .

rhiin, for' t*l.r.t:d n In (4.1) o'ic has

(4.4a) |«. (x+y; t)| • c T V( 2 n " 3 ) / V 3 / 2 ,

( 4 . 4 b ) c = d m a x d l ^ b ( j ) l l ; j = 0 , 1 , . . . . n ) ,n n n r ™ J ' ' '

Jhere d is an

la defined bij

here d i s a c-ontitant, Lnd<*pcnd<?nt of b , ;>, •.• md • .•••;' ..••'.•;-.• • G C ( E )n r n

(4.4c) «,n(t) = max(1,C2n 2 ) , f, € E.

i ' r o o f : F o r f u n c t i o n s f ( x , y , t , r . ) p a r t i a l d e r i v a t i v e s w i t h n - s p e r t t o ." w i l l

be denoted by

144

(4.5) f(m)

Let us write

(4.6) >> =

(4.3-6) and the symmetry relation b^iO - b^i-O (see Chapter 4,

(2.24a)), we can rewrite J. asJ

(4.3) s7c(x+y;;t) = | Re [ el*Bdc.

Evidently, ü belongs to C (-•» • f; -- +•») and by Leibniz' formula

(4.9a) |v' U ) B (C,t)| 'I N i , with

(4.9b) N = 2'"mmax{(!>)j b J 11^; j = 0,1,. ..,m}.

Mow, let 0 < r < R -1" +•». Integrating by parts n times we find

R

(4.10)

r+ j ei?(iT)nBdr;,

where the operator T is defined by

(4.11) If - (sf) ( 1 ) = s ( 1 ) f + sf ( 1 ).

Induction reveals that the £-th iterate of T has the structure

(4.12a) ICf = s^ I u„ f(C"p) with t„ . = t, whereas for pp=U L,p L,U

(4.12b) ••,,, = I aL'p oï£,,.e2,..,e ez

f

where a, « n » are nonne'jative integers, independent of s and f.

Applying Leibniz' formula to the identity (4C2w+t)s = 2f.2 , we find

(4.13) (4^w+t)s(j) + Sjf.ws^"0 + 4j(j-i)ws(J"2) = (2c.2)(J),

from which it is easily seen that

145

(4.14),(j) M,

IJKMO}, j = 1,2,...,

where M. is a constant, independent of p, v and ö.

Thus, in view of (4.12) there are constants A» , independent of u, v and

6, such that for every f of class C (-» < c. •- +«•)

(4.15) |/f| 5 s£ p I Q '-£*£ \f

U~p)[, K € E\{0}, £ = 0,1 n.

Taking r + 0 and R •+ •» in (4.10) we obtain from (4.0-9-15)

(4.16) i^(x+y;t) = £ Re

where the integral is absolutely convergent,

ifete that by (4.7-9-15)

( 4 . 1 7 ) | T " B | S . V - 2 np 0 n t p n

^ a ,n L A .)n n'n p=0 n'P

V 3 / 2 with

Consequently

(4.13a) |ïïc(x+y;t)| s c^ V( 2 n - 3 ) / V 3 / 2 , with

(4.18b) c = d 2~2nN , d = 1 23n~hd,n-h Z A ,n n n,n n r? 2 2 p=0 n,p

where B refers to the beta function ([4], p. 258).

Herewith, the proof of the lemma is completed.

Let the conditions of the preceding lemma bo fulfilled.

Having fixed n in (4.1) we select * such that

(4.19) 0 * \ % n - |.

Next, we choose 5 such that

<4.20) 0 , 6 . ,

Dy virtue of lemma 4.1 we then have in the parameter region

146

i

(4.21) t S 1, x S -p - vt , where y and v are nonnegative constants,

the following estimates

—A — 3/2(4.22a) sup |Oc(x+y;t)| S Pnt (x + T) '

0+

.22b) f |fl (x+y;t)|2dy 5 |p*t 2A(x + i) 2, with-J C

(4O'

(4.23) Pn = (1 + u + v)"cn.

Here i = 1 + y + ut , while c denotes the constant introduced in (4.4).

5. Estimates of q(x,t) - q,(x,t) and q(x,t).

It is now merely a matter of combining theorem 3.1 with the

Lmates of S2 (x+y;t) derived in sect

this paper, which we state as follows

estimates of S2 (x+y;t) derived in section 4 to obtain the main result of

Theorem 5.1. Let q(x,t) be the. solution of the inc-uopdoi problem

r rx isinj2 q(x ,t)dx , -«• •' x < +», t > 0

(5.1) C:q(x,O) = qo(x),

jhfi'c the initial function qQ(x) is an arbitvacit w j i function on K,

oatisfjing (1.2) in such a mij that (1.1) is fulfilled. Let {b (r,), j ; . , C T}

;•'.,' tho <3iiatlevinj daia associated with q n (x ) . Then foi' each x £ E -i".d

t - 0 one hats

( 5 . 2 ) ] q ( x , t ) - q d ( x , t ) | 'i a d h c ( x + y ; t ) | - f l y + s u p i ^ C x + y ; t ) ] j ,

[Jith q d ( x , t ) ',he i't:fit:ct.Lorii<iss part- c:..V . ; ' q ( x , t ) , a Q :.-".• sjnuurr

(Z.4) and "

(5.3) j. • 0, v • 0, 0 • •. • n - | , 0 ^ • 1 -

147

Put a = u + vt . Then one has the estimates

(5.4a) | q ( x , t ) - q d ( x , t ) | 5 A for t > 0, x S -a

_\ — 1/9(5.4b) | q ( x , t ) - q d ( x , t ) | £ p^t (x + 1 + a) ' for t Ï 1 , x g -a

with

a 2 (••»

(5.5a) A = - ^ | ( | b r ( c ) | + | b r ( r j | 2 ) d c

(5.5b) Pn = a^pn(1 + i p n ) ,

ufaira the. constant p in -liven b:i (4.CA).n

To find the behaviour of q(x,t) in the coordinate region t • 0,

x S -a, it clearly suffices to combine the preceding theorem with the

results of section 2.

It follows from the last remark of Chapter 4, section 6, that the

estimates (5.2-4) still hold if one replaces the left hand side by

(5.6) (q2(x\t) - q,(x\t))dx'

In particular, this yields

(5.7) q2(x,t)dx = qd(x,t)dx + 0(t"X) as t - ~.

_ aJ _aJ

However, by (2.10) one has

fw

(5.3) qd(x,t)dx = 0(e ) as t •• ~

-a*

where ~i i s any c o n s t a n t s a t i s f y i n g 0 • ~i • ..i w i t h ..i as in ( 2 . 1 2 ) .

Consequen t ly

(5.9a) { q2(x,t)dx =0(t A) as t •» ~.-a'

formula (see [3])

['" 2 r :i

q2(x,t)dx = - logO + (b (?,) \!)d;. + 4 r Im .'. ,-J " 0J

r P • P

-U'

Using the formula (see [3])

(5.10)

we obtain for the complementary integral

148

jas t ->

,t)dx = - I-N

(5.9b) j qMx,t)dx = - I log(1 + |br(rJ|2)dc + 4 p|1 lm

i'tote that in addition to (5.3) it ai.-o follows from (2.10) that

(5.11) sup |qd(x,t)| = Ove"u)t) as t ->• •».

xi-a

Thus, in view of (5.4), we arrive at

(5.12) sup |q(x,t)| = 0(t"A) a s t * » ,xi-ct

Let us mention some consequences of the above results for the

solution a(x,t), given by (1.5), of the sine-Gordon problem (1.6). By

(1.3) and (5.4b) one has for t 2 1 , x è -a.

(5.13) | o ( x , t ) - o d ( x , t ) + 2 (k o -k 1 )n | g 4f>nt~A (x+1+u) , with

rX

( 5 . 1 4 a ) o d ( x , t ) = - 2 I q ( x ' . t ) d x ' and

(5 .14b) k = 7 q d ( x , t ) d x £ Z.-co - '

Using (2 .10) , we obtain

(5.15) sup | o ( x , t ) + 2k0iT | = Q(t~' ) as t -> «.

xï-a

To conclude with, let us observe that if the initial function qn(x)

in (1.1b) is in the Schwartz class, then so is b .

This implies that (4.1) holds for all n. Hence (5.7-9-12-15) are in this

case valid for all A •> 0 and all 0 i 6 < 1.

References

[ 1] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Method forsolving the sine-Gordon equation, Phys. Rev. Lett. 30 (1973),1262-1264.

149

[ 2] M.J. Ablowitz, D.J. Kaup, A.C. Hewell and '1. Segur, The invers-scattering transform - Fourier analysis for nonlinear problems,Stud. Appl. Math. 53 (1974), 249-315.

[ 3] M.J. Ablowitz and U. Segur, Solitons and the Inverse ScatteringTransform, Philadelphia, SIAM, 1981.

[ 4] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions,National Bureau of Standards Applied Mathematics Series, No. 55,U.S. Department of Commerce, 1964.

[ 5] D.J. Kaup and A.C. Newell, The Goursat and Cauchy problems for thesine-Gordon equation, SIAM J. Appl. Math. 34 (1978), 37-54.

[ 6] G.L. Lamb, Jr., Elements of Soliton Theory, Wiley-Interscience, 1380.

[ 7] G.L. Lamb, Jr. and D.W. HcLaughlin, Aspects of soliton physics, in:Solitons (Ed. R.K. Bullough and P.J. Caudrey) Topics in CurrentPhysics 17, Springer-Verlag, Mew York, 1980.

[ 8] M. Ohmiya, On the generalized soliton solutions of the modifiedICorteweg-de Vries equation, Osaka J. Math. 11 (1974), 61-71.

[ 9] P. Schuur, On the approximation of a real potential in theZakharov-Shabat system by its reflectionlcss part, preprint 341,Mathematical Institute Utrecht (1984).

[10] P. Schuur, Decomposition and estimates of solutions of the modifiedtCorteweg-de Vries equation on right half lines slowly moving left-ward, preprint 342, Mathematical Institute Utrecht (1934).

[11] S. Tanaka, Non-linear Schrödinger equation and modified Korteweg-de Vries equation; construction of solutions in terms of scatteringdata, Publ. K.I.M.S. Kyoto Univ. 10 (1975), 329-357.

CHAPTER EIGHT

CU THE APPROXIMATION OF A COMPLEX POTENTIAL IN THE

ZAKHAROV-SHABAT SYSTEM BY ITS REFLECTIONLESS PART

We consider the Zakharov-Shabat system with complex potential and

derive a pointwise estimate of the error made in approximating the

potential by its reflectionless part. As an illustration we apply this

estimate to investigate the lon^-time behaviour of the solution oc the

complex modified Korteweg-de Vries initial value problem.

1. Introduction.

In Lo], i.e. Chapter 4 of this thesis, studying the Zakharov-Shabat

system with real potential, we derived a pointwise estimate of the error

made in approximating the potential by its ref lectionless part. For t'iat

purpose we reduced the matrix Gel'fand-Levitan equation appearing in the

literature to a scalar intejral equation containing only a single

integral. Furthermore, we exploited the fact that the corresponding

scalar Gel'fand-Levitan operator, when considered in the complex Hubert

space Lz(0,™), has the structure of the identity plus an antisymmetric

operator.

lil

In this paper we take a more general standpoint by considering the

Zakharov-Shabat system with a potential that may assume complex values.

This confronts us with a matrix Gel'fand-Levitan equation that can no

longer be reduced in the above way. It has, however, one important

property in common with the scalar case: The corresponding matrix

Gel'fand-Levitan operator has, when considered in the complex Hubert2x2

space (L2(0,<*0) , the structure of the identity plus an antisymmetric

operator.

Motivated by this resemblance we present an analysis of the matrix

Gel'fand-Levitan equation that - as far as its abstract setting is

concerned - parallels the analysis of the scalar Gel'fand-Levitan equation

given in Chapter 4. To increase the similarity matrix norms and notation

are selected with due care. Although the technicalities are different (cf.

the proof of lemma 5.1 with that of Chapter 4, lemma 5.1) this analysis

leads to an estimate of the difference of the potential and its

reflectionless part, which, surprisingly enough, is almost identical

with that obtained in Chapter 4. Since the only alteration consists in some

numerical front factors due to the particular matrix norms involved, we may

truly speak of a generalization of the main result of Chapter 4.

This generalization is of practical importance, since working with a

complex instead of a real potential considerably enlarges the class of

nonlinear evolution equations solvable via the associated inverse scattering

method. For example, the complex modified Korteweg-de Vries equation, as

well as the nonlinear Schrödinger equation can be solved by the complex

but not by the real Zakharov-Shabat inverse scattering method.

The composition of this paper is as follows.

In section 2 we briefly discuss the direct scattering problem for the

Zakharov-Shabat system with complex potential. The inverse scattering

formalism is outlined in section 3. Next, in section 4 we state our main

result, which after the introduction of some convenient notation and the

derivation of a useful lemma in section 5, is proven in section 6. Finally

we apply the aforementioned result to investigate the long-time behaviour

of the solution of the complex modified Korteweg-de Vries initial value

problem.

152

2. Direct scattering.

To begin with let us tersely survey the direct scattering problem for

the Zakharov-Shabat system [11]

(2.1)

where q = q(x) is a complex function and x, a complex parameter. For

details and proofs we refer to [1], [2], [5], [10]. Our notation is

similar to that used in [5].

Following [5] we assume throughout that the potential q has the

regularity and decay properties stated below:

(2.2a) q 6 C1(E)

(2.2b) lira q(x) = lim q'(x) = 0jx | "*» Jx j -wo

(2.2c) j (|q(s)| + |q'(s)|)ds - +».

In addition we shall require some conditions on the zeros of the

Uronskian of the right and left Jost solutions, to be specified presently

in (2.11).

It is interesting to compare the results of this section to those of

Chapter 4, section 2 and to single out the symmetry relations in

Chapter 4 caused by the realness of the potential.

For Ira c ï 0 we define the (right and left) Jost solutions -j; (x,t,)

and o, (x,t) as the special solutions of (2.1) uniquely determined by

(2.3a) ,: (x,e) = e~1CXR(x,<;), lim R(x,c) = (,1)r o

(2.3b) v£(x,f,) = e1 C XL(x,O, lim L(x,O = (°).

*+

The vector functions R and L are continuous in (x,r) on R<C+ and analytic

in I, on C for each x £ "R.

Furthermore, their components satisfy

(2.4) maxi sup iR.(x,c)|, sup JL.(x,:.)' •. exp] | |q(.s)lds[,LE-<C K>-C J l— J '

i = 1,2.

153

For Im t 5 0 we set

(2.f-<!> ( x . f , ) \ /•••„ ( x , ;

*J ". (x,c y*• i L '

It is readily verified that ij* and ip„ are solutions of (2.1).

Moreover, for x, <; € R one has

(2.6a) W0|.r,<l>r) = |rM (x,c) I2 + |R2(x,r,)|

2 = 1

where W(y,cj>) = t'.'K ~ J'T 'I denotes tiie Wronskian of , and ;•. Hence, f

c, real, the pairs if/ ,y and gi„ ,g/„ const itute fundamental systems of

solutions of equation (2.1). In pa~ticular, we have for x, r € E

(2.7a) Y r(x,O = r+(rji,^(x,r,) + r_(r,),i(,(x,r.)

(2.7b) r.(c) = M(f,,, )+ -c r

(2.7c) r (fj) = W(i)/ .v'1?)-

It is not hard to show that

(2.3) |r (c)|2 + |r (O\2 = 1, ;. £ E.

Let us use (2.7c) to extend r_(r,) to a function analytic on Im r. • 0

continuous on Im c, ' 0.

Then the following integral representations hold

(2.9a) r+(r,) =- | q*(s)e~2ir'sR] (s,,;)ds, r, £ K

(2.9b) r_(O = 1 + q(s)R2(s,<;)ds, Ira C - 0.

In combination with (2.6a) these yield

(2.10) inax[ sup |r+(r,)|, sup l i - r _ ( r , ) | ] - [ |q(s)!ds.r.cK

In terms of r_ we make our f inal assumptions:

(2.11a) r (O z 0 for r. £ E

(2.11b) All zeros of r_ in C+ are simple.

Let us point out that condition (2.11b) can be circumvented by using

TanaLca's direct and inverse scattering formalism [10]. We only include

it for reasons of simplicity.

Incidentally, if

(2.12) | |q(s)|ds ,- 1

then (2.10) shows that (2.11a) is f u l f i l l e d .

Moreover, if

(2.13) f IqCs) Ids < 0.904

then (2.11a) and (2.11b) are trivially fulfilled since r_(-) * 0 for

Im ; -i 0 (see [2]).

We now turn to the construction of the scattering data associated

with q(x). As a result of (2.11a) the function r_(r,) has at most finitely

many zeros C, j^o. • • • »C»,> Im (,. • 0. They are all simple by virtue of

(2.11b). It is a remarkable fact, that the t;. are precisely the eigen-

values of (2.1) in the upper half plane (the so-called bound states).

The associated L2-eigenspaces are one-dimensional and spanned by the

exponentially decaying vector functions y»(x,c;.), j = 1,2,...,N. MoteJ

that by (2.7c) there are nonzero constants i(t-) such that

(2.14) <v (x,£.) = u(t.).jv(x,C,).

(2.15) % ( £ • > = -2ia(c,) f i|V (s,c.)^„ (s,r,.)ds.

One can derive the following representation

f i|V ( s , c . ) ^2

iiearing in mind that the integral on the right does not vanish because of

(2.11b) we define the (right) normalization coefficients by

C. = ii[ [ *„ (s,c,)». (s,c,)dsl(2.16) CT. = i i [ [ *„ ( s , c , )» . (s ,c , )ds l .l| J 2

itext, we introduce the following functions of £; £ E

(2.17a) a (c) = 1/r_(c), the (right) transmission coefficient

155

(2.17b) b (O = r+(;)/r_(c), the (right) reflection coefficient.

By (2.3) one has for c É I

(2.13) |ar(c)|* " |br(O|2 = 1.

In [5] it is shown that b is an element of C II L1 (1 L2 (K) , which

behaves as o(|c| ) for t, -* ±=>. Of course, by imposing stronger

regularity and decay conditions on q(x) in addition to (2.2-11), one can

improve the behaviour of b (5). For instance, if q(x) has rapidly

decaying derivatives, then so has b (5).

We shall call the aggregate of quantities {b (f,),t-»C'} t n e (right)

scattering data associated with the potential q. Remarkably enough, a

potential is completely determined by its scattering data.

3. Inverse scattering.

Let q be any potential satisfying (2.2-11). Then q can be recovered

from its scattering data {b (O.C-.C.l by solving the inverse scattering

problem.

For that purpose one defines the following functions of s € E

(3.1a) f;(a) = s.' (s) + :• (s),d cN 2ii;.3

(3.1b) :;,(s) = -2i .1. C^e : ,

(3.1c) ;.c(s) = | j br(r,)e2lCSd:..

Since b is in C D 1- (K), the integral in (3.1c) converges absolutely

and .: belongs to C n L Z(K).

Next, introduce the 2*2 matrix

/ 0 -."(s)^(3.2) .us; = ( I

V.(s) 0 '

and consider the Ge]'fand-Levitan equation (see [1], [2], [5], [10])

(3.3) g(y;x) + iü(x+y) + B(z;x);j(x+y+z)dz = O

0J

with y > O, x £ E. In this integral equation the unknown 3(y;x) is a

2x2 matrix function of the variable y, whereas x is a parameter. Observe

that some authors use a slightly different version of the Gel'fand-

Levitan equation which can be transformed into (3.3) by a change of

variables (see [5], p. 46).

In [2] it is shown that for each x 6 * there is a unique solution 3(y;x)

to (3.3) in (L 2) 2* 2 (0 < y •- +»). It has the form

(3.4)

*a -b

where a(y;x) and b(y;x) are complex functions belonging to C fl L1 f! V

(0 '- y < +°°) , which vanish as y •> +«>. The inverse scattering problem is

now solved, since the functions a and b are related to the potential q

in the following way

(3.5a) q(x) = b(0+;x)

(3.5b) [ |q(s) |2ds = -a(0+;x), x C R.X-"

Using (3.4), tiie matrix integral equation (3.3) can be reduced to a

scalar integral equation involving only b

(3.6) b(y;x) + ;. (x+y) + b(z;x)L (z+s+x) •.; (s+y+x)dsdz = 0.

In this form the Gel'fand-Levitan equation frequently appears in the

literature (cf. [1], [10]). However, for our present analysis the matrix

form (3.3) proves to be more convenient.

4. Statement of the main result.

If q is a potential with scattering data !b (r.),i.,C.! then the

potential q, with scattering data iO,u.,C.l is called the ref lectionless

part of q. The function qr>(x) can be o tained in explicit form (see (5.30))

by solving the Gel'fand-Levitan equation (3.3), which in that case reduces

157

to a system of Ï1 linear algebraic equations. The main result of this

paper is the next theorem which tells us in which sense the potential is

approximated by its reflectionless part.

h i c h :'••'!! CcJ'i-..: (::.;:-!]) -inJ hi.' -..!••- ;:•*/• •-• •ƒ'.'>._.• ;';;.• •. b ( r ) , ' . . , C . }.

T h e o r e m 4 . 1 . /.<•;' q .'•'•' •/ : J'i'

Qhich :'••'!! CcJ'i-..: (::.;:-!]) -inJ

q dcnoii'. ;•'.'>• f.•;~\\?t.L»ii• ::.; .•'./-"; .•;' q. ."•':• •. ;' ;• ,•_?•'. >; c 18

(4.1) iq(x) - q (x) | -,,::(, 1 f '.. (x+y) 'J dy + sup ' (x+y);),u\ 0J L 0 y + ' y

c ' 0 'n^und .sr.ii! n: i,.

J» _,

(4.2a) .,. = 1 + ? , (lm .; ) M .,0 P,J = 1 P PJ

(4.2b) M - 3(lm ,/a* ,^ j, j 72^; J, | 7^i .

;';.. t - . \ j L : . h , q - q j . ' . ! • • . ; . ' . • . ' « ; . • • • , / • • • • « • ' • • : . ;• '•. • • . " - . . • v ' " •'•• . • • . • • • • • • • . • • . . - • • • • • • .

. - - • ^ . • / / i ' - i , . ' . ' ! / b r ( r j „ • : , . ; ' ••/;,• / - - ; / . - - ; • , . • _ .- • •. • • • •" • • . • • - , ; • . • . • : . • • . ' •.

C o r o l l a r y t o t h e o r e m 4 . 1 . .'..••/' ' •. • ••.- .' '. ' .' '' '•.• .'• ' :'. .' ' • .''

2 pt' i.Jl'i. '••• •' .1 'hi

(4.3) sup |q(x) - qd(x)| -~ [ C-h^,-)' + ,1' br (• ) = )d • .

We shall prove theorem 4.1 in section h. Before doiiv; so we introducé

some notation and derive a useful lemma in section =>.

5. Tirst steps to the proof.

In the remainder of this paper it is understood that the conditions

of theorem 4.1 are fulfilled.

We begin by introducing some useful concents and notation.

Kor 2<2 matrices

133

(5.0 A - P " 3 ) . B - p "3

2 4 2 "4

with complex entries we set

( 5 . 2 ) <A,J3>E = i | 1 u . 0 . , IIAIIE = ( A , A E ) * .

By B we denote the linear space of all 2v-2 matrix functions

/&, (y) S-,(y)\(5.3) g(y) =

V,2(y) s4(>,such that each g. is a complex-valued, continuous and bounded function on

(0,a)). We turn S into a complex Banach space by equipping it with the norm

(5.4) llgll = sup ll3(y)llE.

?>:2

Furthermore, we write 3f to indicate tlie complex Hilbert space (L2(0,«>))~

with inner product

(5.5) -.f,g> = f <f(y),g(y)-.dy

and corresponding norm II II2 .

The choice of the above spaces is motivated by our wisli to exploit

analogies with the scalar Gel' fand-Levitan equation (3.8) in Chapter 4.

Returning to (3.2), let us put

(5.6a) ..,(s) = •.,d(s) + -.c(s),

/ 0 -!-\(s)(5.6b) ,.d(s) = (

(5.6c) .;c(s) = I C

From section 3 we know that for each x 6 B the functions y ••* •..< (x+y),

;.',(x+y) belong to 3 D 3C.

Next, keeping x € 1R fixed, we formally write

15')

(5.7a) (Tdg)(y) = j g(z)ü>d(x+y+z)dz

f™(5.7b) (T g)(y) = g(z)u (x+y+z)dzc

0J c

with g as in (5,3). Plainly, T, can be considered as a mapping from S in

B. but equally well as a mapping from JC into 3C. On the other hand, T is

not necessarily a mapping from B into 8. However, suitably modifying

formula (4.5.10) in [5] one can easily show that T reaps Jf into K with a

norm that satisfies

(5.8) IIT II2 S sup |b (c)|.c (;€E

It is straightforward to verify that the operators T and T are both* * " c

antisymmetric on JC, i.e. Td = ~TJ» T = -T . This fact plays a dominatin

role in our analysis.

In the above abstract language, the Gel'fand-Levitan equation (3.3) take

t he fo rm

(5.9a) (I + T + T,)3 = -•»c d

(5.9b) u = ui + ;..,c d

where I i s the iden t i ty mapping. A f i r s t advantage of t h i s formulation i

read i ly seen. Since T + T is antisymmetric, the operator I + T + T, i

i n v e r t i b l e on JC and so we know at once that (5.9) has a unique so lu t ion

3 £ JC. Hote tha t t h i s fact was already mentioned in sect ion 3 , from whic

we r e c a l l t h a t , moreover, fl £ Ó fl JC.

For the proof of theorem 4.1 the following lemma is bas i c .

Lemma 5 . 1 . FOP :m i vaiutf of ih>/ :'tiparm': et' x £ R , r .V ••:<• :•.>: JP I + T , :j" ' _ i ^

inVCPlibic on ihf. B.inaah a:>usc 8 .Jiih l<rO'i:%: S = ( I+T , ) .-.'.V' /'•'d

(5 .10a ) ( S f ) ( y ) = f ( y ) - \ . \ ( P ' ) , • (y) = e ' p ' ,p ~ ] p \ . p ( y ) o / p

(5 .10b) A

1Ó0

P J =

' j K*J U ' P . j + ^ ' ''p+N,j+N

where (g ) is the inverse of the 21I*2N matrixrs

Z(5.11a) A = f !, wi t^

V-D Z t y

( 5 . 1 1 c ) D = ( Ö . 6 . ) , 6 . = ( - 2 i C . ) e J , D = ( 6 . 6 . ) .J r J J J J rJ

Furthermore, the operator S satisfies the bound

( 5 . 1 2 ) IISU & aQ, x £ E

II( 5 . 1 3 a ) a n = 1 + Z , (lm t, ) M .

U P j j = ' P PJ

*

(5.13b) Mp. = 8(lm t/Vm K-^ & | ^ - | | ,»,

Thus, llsll is uniformly bounded for x £ E, where the bound is an explicit

function of the 5..

Proof: Let x £ E be arbitrarily fixed.

Recall that, when considered as an operator from the Hubert space X into

itself, T, is antisymmetric. Hence I+T, is invertible on X and one has

( 5 . 1 4 ) II ( I + T d ) g l l ! = llgll| + H T d g l l | , g £ 3C,

so that the inverse S = (I+T,) satisfies the boundsd

(5.15) IISII2 i 1, HTdSU2 < 1.

Shifting our gaze, let us consider T, as an operator from 3 into 3 and

show that I+Td is invertible on 3. Suppose that (I+Td)g = 0 for some g £ B.

Then g = ~Tjg £ JC and thus g is identically zero by the preceding

argument. This tells us that I+T, is one to one on S. However, Tj is ofa a

finite rank and therefore compact. It follows that 1+T, is invertible on

the Banach space S.

Next, consider in a f) iff the elements e. ,e0,.. . ,e.„ defined by

161

(5.1óa)O

,0 0 s / 0 0(5.16b) e^0 M(y) = f *

Solving the equation

(5.17) (I + Td)g = f, f,g 6 3

we find

N , o >-*<y>(5.13) g (y) = f(y) - p S , Ap( ^

where the A satisfy

" U

(5.19a) Ap )

" Up+3N

(5.19b) s S ] „ r 8 ( S ) = >{ r )

"s+2H l'r+2U

r™( 5 . 1 9 c ) 3 q = j < f ( y ) , e q ( y ) ^ d y j q = 1 , 2 , . . . . 4 N

with A = (n ) the matrix given by (5.11).

Since the operator I+T, is one to one on B, tlie matrix A = (.\ ) isv d rs

invertible. More directly, one can verify the invertibility of A by

writing it as a positive definite matrix plus an antisymmetric one.

Denoting the inverse by A = (,' ), we obtain from (5.19)r s

( 5 . 20) A =P

2N('

P

p , s

I'.,s

s

s+2N

sP+N

W,s"sbs+2N

= i - i v r J , A , P > J , P+"'J ;•

j + 2N ' j + 3M ' ' p , j + N p + N . j + N

l'ogether, (5.16), (5.13) and (5.20) imply tliat the inverse operator

S = (I+T ) is given in explicit form by (5.10).

Iö2

We shall now prove that the matrix elements R are bounded asrs

functions of x E E. In fact, we shall estimate the x dependent matrix

that occurs in the right hand side of (5.10b), in the following explicit

way

(5.21)3p+N,j+Il

I-I= M

PJ

A

lror this purpose we develop some further notation.

By A we denote the Gram matrix of the vectors e, ,e_,...,e. , introduced

in (5.16a), i.e. A = (3 r g), 5r <er,eg>.

Since the vectors e.,e„,...,e„ are linearly independent, it follows that

det A > 0 (see [3]).

Evidently

„t

(5.22) A =^0 V

(A)

2NLet us write (A) = (3 ) and introduce the vectors h = £, 3 e .

rs r s=1 rs sNote that <h ,e > = 5 and <h ,h > = 8

r' s rs r' s rsIn combination with (5.10-20) this gives

2N

(5.23) (I - S)hs = rJi 6rser.

Using the identity I - S = T,S, we get

(5 .24) S r s = <T d Sh s , h r >.

Hence, in view of (5 .15)

(5.25) | B r s | ' S llhrll|llhsll^ = B r r 8 s s , r , s = 1,2 211.From (5 .22) i t i s c l e a r t h a t for Z = 1 , 2 , . . . , N

(5.2Ó) ~ - - - 1

Jlt - ) N .PJ P»J=1

4(lmN

p=1

163

The desired estimate (5.21) is, of course, an immediate consequence of

( 5 . 2 5 - 2 6 ) .

By (5.1Ob-21) we have

N( 5 . 2 7 ) IIA IIE S IIfII •5 1 (> 2 Im {,.) H . ,

y i e l d i n g the bound ( 5 . 1 2 - 1 3 ) for IISll.

Corol lary to lemma 5 . 1 . For each x € E the eqwttion

(5.23) (I + Td)3 = -u>d

admits a unique solution f> £ 8 and 'j>j haos

(5.29) ,d(y;x) - - J = , (^ ^p , j P+-V

-Vb

, a ,d d

Remark. Let us recall that 3. produces the reflectionless part of the

potential q through the formula

(5.30) qd(x) = bd(0+;x) = ?=1 i3 . + r

Clearly, by (5.21) we have the a priori bound

N(5.31) sup |q,(x) I < ? , M . ,

x G ^ lqd P,J=1 P J'

which does not involve the C. but depends only on the ",. in a simple

explicit way.

j. Proof of theorem 4.1.

The nature of the results obtained in the previous section enables us

to provide a proof of theorem 4.1 which is remarkably similar in form to

that given in Chapter 4 for the corresponding scalar case.

164

Let x £ E be arbitrarily fixed.

To start with, let us write the solution 6 of (5.9) in the form

(6.1) ö = Sd + Bc, with

(6.2) Sd = -Sud.

By (3.4-5) and (5.29-30) we plainly have

b ac c

w i t h a = a - a , and b = b — b . , such t h a tc d c d

(6.4a) q(x) - qd<x) = bc(0+;x)

(6.4b) [ (|q(s)|* - |qd(s)|2)ds =-ac(0

+;x).

From section 5 it is clear that both fi and S, belong to B H X. Hence, we

already know that •/, € 8 0 5f. It remains to find a concrete estimate of

3 in the norm (5.4). For that purpose we insert the decomposition (6.1)

into (5.9), thereby obtaining

(6.5) (I + T + T )3 = -T Ë - ,-. .c d c c d c

Consider (6.5) as an equation in the Hubert space X. Since T + T, isc d

antisynimetric, the operator I + T + T is invertible on X. Furthermore,

the relation (5.14) holds with T d replaced by T + Tj. Thus (6.5) has a

unique solution r. € 3f satisfying

(6.6) «.•: II, • IIT .-; II + il. II, .c 2 c d 2 c 2

Using the generalized Minkowski inequality (see [6], p. 148) we obtain

(6.7) , V d l , • ƒ ( J ' i ! , d ( Z ; x ) . c ( x + y + z ) i U d y ) ^ƒ (Jo

Hence

(6.3) IT ,•.„!!, •

where by (5.21-29)

r N -1

( 6 . 9 ) «ÉSd«i = j l lR d (z ;x) l lEdz < p ? = 1 (/2 Im C p ) ' M ? J .

We conc lude t h a t

( 6 . 1 0 ) II3CU2 'i ilu. llj(1 + Il6dlli) < a o l lu c « 2

w i t h a g i v e n by ( 5 . 1 3 a ) .

The trick is now to rewrite equation (6.5) as(6.11) (I + T,)3 = -T 6 - T 0 - u

d c c c c d c

and to realize that the a priori estimate (6.10) paves the way to estimate

the right hand side of (6.11) in the norm (5.A). In fact, since

(6.12) II (T 3 )(y;x)ll„É [ «3 (z;x)IIJu> (x+y+z)ll dz,

we have by Schwarz' inequality

(6.13) IIT 3 II = sup ( f II3 (z;x)IIJ,dz) { [ ILo (x+y+z) l|2,dz )c c 0<y<+~ V C / \QJ C '

s He « J U I L < iiu i i | ( i + O R , i i , ) .c 2 c 2 c 2 d

Moreover, invoking again the generalized Minkowski inequality, one gets

(6.14) llT 0,ll £ «io 1111(4,Hi.c d c d

T o g e t h e r , ( 6 . 1 3 ) a n d ( 6 . 1 4 ) p r o v i d e t h e e s t i m a t e

( 6 . 1 5 ) II-T 8 - I B , - u I S (II io II + II us II | ) < 1 + l l 3 . H i ) .c c c d c c c 2 d

Applying lemma 5.1 we obtain from (6.11-15) the following estimate for ;i

in the norm (5.4)

(6.16) «3 II i a*(IU II + llu I In .c 0 c c.

Insert ion of (5.6c) and (6.3) then leads to

(6.17) sup ( | a ( y ; x ) | ' + |b (y ;x ) | * ) ï a ' f / I [ | ;• (x+y) | *dy +0<y<+=o ^ C / U \ 0J C

sup |ft (x+y) | ).0<y<+oo C '

166

Combining (6.17) with (6.4a) we arrive at the desired estimate (4.1-2),

wherewith the proof of theorem 4.1 is completed.

Remark. Actually, we have proven more, since by (6.4b) and (6.17) the

estimate (4.12) still holds if one replaces the left hand side of (4.1)

by

(6.18) max[|q(x) - qd(x)|,

7. An application: cmKdV asymptoties.

ds j

The importance of theorem 4.1 stems from the fact that it is a useful

tool to investigate the asymptotic behaviour of solutions of certain non-

linear evolution equations solvable by the Zakharov-Shabat inverse

scattering method (see the discussion in Chapter 4, section 4).

As an illustration let us consider the complex modified Korteweg-de

Vries (cmKdV) problem

(7.1a) qt + 6|q|2qx + q ^ = 0, -» < x < +», t > 0

(7.1b) q(x,0) = qQ(x),

where the initial function qQ(x) is an arbitrary complex-valued function

on E, such that

(7.2a) 1r/x^ satisfies the hypotheses (2.2-11) and is therefore a bona

fide potential in the Zakharov-Shabat scattering problem (2.1).

(7.2b) 1o^x^ ^s su^^iciently smooth and (along with a number of its

derivatives) decays sufficiently rapidly for |x| -+ =>:

( i) for the whole of the Zakharov-Shabat inverse scattering

method [2] to work,

(ii) to guarantee certain regularity and decay properties of the

ri-rht reflection coefficient to be stated further on.

167

Uniqueness of solutions of (7.1) can be established within the class

of functions which, together with a sufficient number of derivatives vanish

for |x| -> « (of. [7]).

Suitably adapting the procedure outlined in [10] for the real raKdV

problem one can prove by an inverse scattering analysis that condition

(7.2) guarantees the existence of a complex-valued function q(x,t),

continuous on Kx[0,«>), such that

(7.3a) For any value of the time t :? 0 the function q(x,t) satisfies

the hypot-heses (2.2-11).

(7.3b) q(x,t) satisfies (7.1) in the classical sense.

(7.3c) q(x,t) falls in the class of functions for which uniqueness of

solutions of (7.1) can be proven.

Whenever, in the sequel, we speak of "the solution" of (7.1) we refer to

the solution obtained by inverse scattering.

Uniqueness implies that if q~(x) is a real-valued function then so is

q(x,t) for all t • 0. Thus, in that case, the cmKdV problem (7.1) reduces

to the mKdV problem (1.1) in [9], i.e. Chapter 5 of this thesis.

Let us point out that by the inverse scattering method the solution

q(x,t) of (7.1) is obtained as follows.

Having computed the (right) scattering data {b (t),r_.,C.t associated with

q.,(x), one puts (see [4], p. 307)

(7.4a) cT(c) = cjexp{8ir,?t}, j=1,2,...,H

(7.4b) br(r,,t) = b (r.)expl 3 L ;3 1 } , — • -. •. +....

Then by the solvability of the inverse scattering problem, there exists '"or

each t -• 0 a smooth potential q(x,t) satisfying the hypotheses (2.2-11)

and having \b (r., t ) , r,. ,C. (t) ) as its scattering data. The function q(x,t)

is the unique solution of the cmKdV initial value problem (7.1).

Plainly, the reflectionless part q (x,t) of q(x,t) can be obtained in

explicit form by substituting (7.4a) into (5.30).

!f>3

r i <hIn the N = 1 case, setting t,. = f, + in, C = |ie ', with ii, •,> • O,

?, J) c K, this yields

(7.5) q d(x,t) = 2'ie"1*sech

with

(7.óa) •;• = 2.-,x + 8 .•-,(•-.* - 3n2)t + .> + j

(7.6b) i' = 2nx + 8n(3f;2 - M2)t + v

(7.6c) v ^

Thus the one-soliton solution is a single wave packet modulated by an

envelope having the shape of a hyperbolic secant. The envelope and phase

velocities are found from (7.6) to be v = UO 2 - 3''2) ande

v , = <4(3n2 - t,2 ), respectively. According to the sit>n of v the envelope

may propagate to the right, to Ihe left or be at rest.

If Ï1 ' 1, then, generically, for large time q (x,t) will decompose into

N distinct solitons of the structure (7.5). However, it is easy to

construct examples (e.g. N = 2, c, = •. + i-i, v2 • 3'2, c,n = K M 2 - 3r2)*)

in which no such decomposition takes place, but a more complicated

structure is found instead.

ïSecause of the negative group velocity associated witli the linearized

version of (7.1) we expect that for large t, when considered on the

positive x—axis, the solution q(x,t) of (7.1) is approximated with

reasonable accuracy by its reflectionless part q.(x,t).

We shall use theorem A.1 to verify this.

Note first that, if qQ(x) satisfies (7.2a-b(i)), then by (7.3a) the

function q(x,t) satisfies the conditions of theorem 4.1.

Consequently, for each x € R and t • 0 one has

(7.7) !q(x,t) -q d(x,t)| ' ajlti j | (x+y ;t) |2 dy +

+ sup ]:t (x+y;t)|),0- y<+« '

wi th

169

Jt)-l | \ r(7.3) fic(8Jt)-l | \ r ( ü e 2 i C S + 8 U I t d ; , s £ K

and et, the constant given by (4.2).

Next, suppose that q«(x) satisfies (7.2) in such a way that

(7.9) b is of class C2 (E) and the derivatives b '((;), j = 0,1,2

are bounded on E.

Then, reasoning as in Chapter 2, but taking into account that the symmetry

relation b (?) = b (~c) is no longer guaranteed, it is easily seen that

in the parameter region

(7.10) T = (3t) 2 1, x £ -u - \)T, where p and v are nonnegative

constants

one has the estimates

(7.11a) sup |fic(x+y;t)| < YT~'

0J c x+y>t y = r \1 + _ J x n y +

where y is some constant.

Combining (7.7) with the estimates (7.11) we arrive at

Theorem 7.1. Let q(x,t) be the solution of the complex modified Korte-jeg-

de Vries problem

<- q -•- 6|qj2q + q = 0 , -<» < x < +», t * 0(7.12) ' x xxx

L q(x,0) = q Q(x),

where the initial function qn(x) is an arbitvapu aomplejr-valujJ function

on E, satisfying (7.2) in such a way that (7.9) is fulfilled. Let

{b (5),c.fc5} be the scattering data associated with qQ(x). Then fj?

each x 6 E and t y 0 one has

(7.13) |q(x,t) - qd(x,t)| S ad/2 J |r>c(x+y; t) \^ dy +

(x+y;t)|),

0J

1 _ » » i \

+ sup c

170

with q, (x,t) the reflectionless part (5.30), (7.4a) of q(x,t), a~ the

constant given by (4.2) and Q the function introduced in (7.8).

Next, let v and \> be arbitrary nonnegative constants. Put a = u + vT,1 /3

T = (3t) . Then the following estimate holds

(7.14a) sup |q(x,t) - qd(x,t)| i A, for t > 0xS-a

(7.14b) sup |q(x,t) - qd(x,t)| S yT~', for t ^

with

a2 f»(7.15a) ^

(7.15b) Y = afrh + fiy + v^"|br(0)|^|+ j

where y denotes the constant appearing in (7.11).

Clearly, theorem 7.1 generalizes theorem 5.1 in Chapter 5 obtained for

the real case. Generalizations of other results can be derived in a similar

way, but they fall outside the scope of this paper and are therefore left

to the interested reader.

References

[ 1] M.J. Ablowitz, Lectures on the inverse scattering transform, Stud.Appl. Math. 53 (1978), 17-94.

t 2J M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inversescattering transform - Fourier analysis for nonlinear problems, Stud.Appl. Math, 53 (1974), 249-315.

[ 3] P.J. Davis, Interpolation and Approximation, Dover, New York, 1963.

[ 4] R.K. Dodd, J.C. Eilbeck, J.D. Gibbon and H.C. Morris, Solitons andNonlinear Wave Equations, Academic Press, 1982.

[ 5] W. Eckhaus and A. van Harten, The Inverse Scattering Transformationand the Theory of Solitons, Worth-Holland Mathematics Studies 50,1981 (2nd ed. 1983).

171

[ 6] G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities, 2nd ed.,Cambridge 1952.

[ 7] Y. Kametaka, Korteweg-de Vries equation IV. Simplest generalization,Proc. Japan Acad., 45 (1969), 661-665.

[ 8] P. Schuur, On the approximation of a real potential in theZakharov-Shabat system by its reflectionless part, preprint 341,Mathematical Institute Utrecht (1984).

[ 9] P. Schuur, Decomposition and estimates of solutions of the modifiedKorteweg-de Vries equation on right half lines slowly moving leftward,preprint 342, Mathematical Institute Utrecht (1984).

[10] S. Tanaka, Non-linear Schrödinger equation and modified Korteweg-deVries equation; construction of solutions in terms of scatteringdata, Publ. R.I.M.S. Kyoto Univ. 10 (1975), 329-357.

[11] V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linearmedia, Soviet Phys. JETP (1972), 62-69.

172

CHAPTER WINE

INVE11SE SCATTERING FOR THE MATRIX SCI1RODIMGER EQUATION WITH NON-HERJUTIAN

POTENTIAL

We develop an inverse scattering formalism for the nxn matrix

Schrödinger equation with arbitrary, in general non-Hermitian potential

matrix, decaying sufficiently rapidly for |x| •+ •».

1. Introduction.

In 1974, Wadati and ICamijo [5] developed an inverse scattering

formalism for the matrix Schrödinger equation with Hermitian potential

matrix U(x). Assuming si.iilar results in the non-Hermitian case the

authors and later on Calogero and Degasperis [1] found several classes of

solvable nonlinear evolution equations.

To our knowledge this assumption has nowhere been justified in the

literature. Indeed, in 1930 Wadati [4] writes:

"At the moment, it is an open question as to what are the most general

conditions for which the inverse scattering problem can be solved.

I have considered three cases:

173

I. ü(x) is diagonal (and can be complex),

II. U(x) = J'J(x)J; J a constant matrix, J2 = I,

III. Ü (x) = JU(x)J; J a constant matrix, J2 = I.

The first crucial point in the discussions is the definition of the

Wronskian. Although we need extra assumptions, similar arguments seem

to be valid".

The present paper is an attempt to settle this question. In section 2 we

consider the Schrödinger scattering problem for any continuous potential

matrix U(x) decaying sufficiently rapidly for |x| -• °>. Superimposing some

rather natural regularity conditions on the right transmission coefficient

that are partly familiar from the Zakharov-Shabat scattering problem

(see [3]), we show that - apart from the unique solvability of the

Gel'fand-Levitan equation - the inverse scattering problem can be solved

completely. Though initially the approach parallels that of [5], our ways

split up in subsection 2.2. The reason is that the Wronskians employed in

[5], (2.17) to extend the scattering coefficients to the upper half plane

are of no use in the non-Hermitian case. To circumvent this difficulty

we perform the extension by means of integral representations in terms of

the potential and the Jost functions (see (2.20)).

Throughout we shall use the following notation:

If A = (a. .) is an n*n matrix with a.. £ -, then we write:J-J ij

nI A. | = max .£ la. . I .

The determinant of A will be denoted by det A. Furthermore, \»e write

I for the nxn identity matrix.

Our notation for the Jost functions, scattering coefficients, etc. closely

resembles that used in [3]. This notation is different from the one in

[5]. For convenience we specify here the relation bet-ween these different

notations:

F1 = V F2 = V C11 = R+' C12 = R-' C22 = '-'

C2) = L+, K = Br, K^x.y) = ;Ux,y-x).

174

2. The inverse scattering problem for the matrix Schrödinger equation.

We consider the differential equation

(2.1) -i'" + (k2 " U(x))Y = 0 , ' = "fj » -» < x < +»,

where U(x) is a complex n*n matrix and if is a vector function with n

components. Equation (2.1) is called the nxn matrix Schrödinger equation.

Accordingly, the complex parameter k2 and the matrix U(x) are referred to

as energy and potential, respectively.

We shall impose the following conditions on the potential:

(2.2a) U(x) is continuous on -•*> < x < +«,

(2.2b) lira U(x) = lim U(x) = 0,

(2.2c) I (1 + jx|) |U(x)|dx < +», C = 0 or 1.— CO''

We call (2.2) a growth condition of order I and write [[£]] to indicate

which growth condition is meant. For many results it suffices to

a.-3urne [[0]]. However, we need [[1]] to prove existence and regularity of

the Jost functions for k = 0 (see subsection 2.1), a result which is

essential in the derivation of the Gel'fand-Levitan equation in sub-

section 2.4.

Clearly, any system of n solutions of (2.1) can be represented as an

nxn matrix ï satisfying the equation

(2.3) T" + (k2 - U(x))'i = 0, — < x • +••-.

Conversely, the columns of any solution of (2._>> are solutions of (2,1).

Following [5] we shall not study (2.1) directly, but instead focus our

attention on (2.3). This enables us to exploit analogies 'Jith the scalar

Schrödinger equation with real potential, of which the inverse

scattering mechanism is now perfectly understood (see [2] and [3]).

2.1. Jost functions.

In this subsection we generalize some results that are familiar from

the n = 1 case with real potential as described in [2] and [3]. Since

the proofs of these generalizations are analogous, we omit them.

?or l i k H we introduce the Jost functions Ï (x,k) and ^«(x.k), two

special solutions of (2.3) satisfying plane wave boundary conditions at

infinity, i.e.

(2.4) Vr(x,k) = e"l k xR(x,k), H^Cx.k) = e l k xL(x,k),

(2.5) R" = 2ikR' + UR, lim R(x,k) = I, lim R'(x,k) = 0,

(2.6) L" = -2ikL' + UL, lira L(x,k) = I, H m L'(x,k) = 0.

x-«° x-«°

The problems (2.5-6) for R and L can be reformulated as integral

equations solvable by iteration and yielding useful information concerning

regularity, asymptotic behaviour and Fourier representation of the Jost

functions. The essential results are as follows.

Lemma 2.1. If k £ cAfO} and [[0]] oi< if k = 0 and [[1]], : hrv w !,\ti>e

( i) R it; J olasaicai o! it ion of (2.!>) »

R is tjcmr.tnuou.-i in x, bounded for x + -"• nut s,i: l.t^U'r

(2.7a) R(x,k) = I + J G(x,y,k)R(y,k)dy

r(2ik)~ 1ie 2 l k ( x" y ) - 1lU(y), k f C \{0)(2.7b) G(x,y,k) = < +

l(x-y)U(y), k = 0.

(ii) L is a nlausical solution of fl'.f'J «=»

L is continuous in x, howidcd j'o.- x • ™ .;':.:' (i,f• iii'':c.'

(2.8a) L(x,k) = I + J H(x,y,k)L(y,k)dyx^

f(2ik)"1le 2 l k ( y" x ) - i)U(y), k € C \>-0]

(2.3b) iKx.y.k) = j +

l(y-x)U(y), k = 0.

17b

Theorem 2.2. If [[0]] and Im k > 0, k * 0, tWi o«e tea:

( i) The problem (2. 7) for R has a unique solution that i.; continuous in x

and bounded for x -+ -<». ftw solution satisfies (2.i>) in classical

sense and is given by the- Neumann series

(2-9) R = nlo Gm' G0 = l' G m + 1C x ' k ) = _ ƒ C5(x,y,k)Gin(y,k)dy, m - 0.

The G 's satisfy the estimate:

|(2.10) |Gm(x,k)| É (m!)"'{uo(x)/|k|}m, uQ(x) = | |U(y)|dy.

(ii) The problem (2.8) for L has a unique solution that is continuous in x

and bounded for x -»• <». 'phis solution satisfies ('.',.G) in i-'iassical

sense and is given by

( 2 . 1 1 ) L = f 0 H , H o = I, H m + 1 ( x , k ) = [ H ( x , y , k ) H ( y , k ) d y , m > 0 ,xx

(2.12) |Hm(x,k)| i (m!)-'{vo(x)/|k|}m, vQ(x) = [ |ü(y)|dy.

x

Theorem 2.3- Let the potential U satisfy a 'jrowlh condition of order 0.

Then the matrix functions R, R', R", L, L' and L" are

( i) continuous in x and k on R*(IE \{0})

(ii) analytic in k on (C for each x £ E.

Theorem 2.4. Assume [[0]]. Then the limits prescribed in (2.5) and (2.6)

for R, R', L and L' are. unifox-.., in k on compacta t= C \{0}. For the limits

in the non-prescribed directions we find if k € f :

X-*k)"1 f(2.13a) lim R(x,k) « I - (2ik) U(y)R(y,k)dy, lim R f(x,k) = 0,

(2.13b) lim L(x,k) = I - U i k ) " 1 U(y)L(y,k)dy, lim L'(x,k) = 0 .

rte limits in (2.1,') are uniform in k on compaata'zZ .

Theorem 2 . 5 . l,et U satisfy a growth condition oe order 1. Then the problems

(2.7), (2.8) for R, L with k = 0 have a unique solution that is continuous

in x and bounded for x • -•», x -• •» respectively. R, L satisfy (V..b), (".(*>

in classical sense for all ( x , k ) £ R.*£ .

The functions R , R ' , R " , L , L ' -jnd L" arc c-.mt i.nu,<ut' in ( x , k ) ->n 'B^C+. ',';:^

Units ppcjsi'ibi'd in (',',.;>) and (2.6) for R, ! \ ' , L m,l h' are :viiforr> in k ••/

177

Theorem 2 .6 . Suppose [ [ 1 ] ] . Then the matrix L has the Fourier representation

(2.14) L(x,k) = I + e N(x,s)ds, x € R, k £ <C+,

wit/2 H(x,s) £ L"Xn 0 L"X" (0 < s < +») / o r e-rc/z x.

The kernel N i s an element of C™*n(R^) = {W e Cn*n (]Rx[0,»))| Vx € K :

lim W(x,s) = 0 } , which is diffei'entijble with vespt.ot to x as •jell is>

to s and N , N £ c"Xn(E2) .X S U T

r, 'A has the important property

(2.15a) N(x,0+) = J j U(y)dy, ..7u :ftat

(2.15b) U(x) = -2 II ( x , 0 + ) , x 6 H.

2.2. Scattering coefficients.

Let us assume [[0]]. For k e ]R\{0} the pairs '?„ (x,k) .^(x.-k) and

¥ (x,k),ï (x,-k) constitute fundamental systems of solutions of

equation (2.3).

In particular we have

(2.16a) 'i'r(x,k) = H'£(x,k)F+(k) + 'l'£(x,-k)IMk)

(2.16b) V£(x,k) = ?r(x,k)TJ_(k) + '?r(x,-k)L+(k)

where the scattering cot1 cients R+, R_, L_ and L are n«n matrices

depending on k E R\{0}.

Substituting (2.1óa) into (2.16b) and vice versa, we find

( R (k)L (k) + R (-k)L (k) = I, L (k)r. (k) + L (-k)R (k) = 1,(2.17) \

1 R_(k)L_(k) + :l+(-k)L+(k) = 0, L+(k)R+(k) + L_(-k)R_(k) = 0.

From (2.5-6) and (2.16) we obtain the asymptotic behaviour for ':;' * "

of f• ., "/\, i and ;' with k 6 R\fO} fixed:L t. r r

173

(2.18a) f (x,k) « e~ l k xi for x ->• -«

« e ikxR+(k) + e~ikxR_(k) f o r x - +<*

— 1 —i Vv(2.13b) (ik) V ( x , k ) ra -e 1KXI for x * -«

«e l k x R + (k ) - e"lkxR_(k) for x > +"

i if x(2.18c) fe(x,k) « e I for x - +«

« e lkxL+(k) + e"l'<XL_(k) for x - -«

— 1 il/v

(2.18d) ( ik ) ' ^ ( x , k ) « e I for x

p» e 'L (k) - e L (k) for x * -*

Here t h e f» s i g n d e n o t e s t h a t t h e d i f f e r e n c e b e t w e e n l e f t and r i ^ h t hand

s i d e t e n d s t o 0 .

U s i n g ( 2 . 1 3 ) , ( 2 . 5 - 6 ) and ( 2 . 7 - 3 ) we f i n d f o r k C TR\\O}:

( 2 . 1 9 a ) R_(k) = l i m ( 2 i k ) ~ 1 e 1 K X { i k r ( x , k ) - r ' ( x , k ) ;

= l i m R ( x , k ) - j j - k R ' ( x . k ) = 1 im I - - ^ U ( y ) R ( y , k ) J y

(2.1%) R + ( k ) = 7 T k I e~

( 2 . 1 S c ) L + ( k ) = I - - ^ U ( y ) L ( y , k)Jy

( 2 . 1 9 d ) L _ ( k ) = ^ i j r j e 2 l k y l i ( y ) L ( y , k ) J y .

We now o b s e r v e t h a t t l i e r i g h t - h a n d s i d e s of ( 2 . 1 9 a ) and ( 2 . 1 9 c ) a r e w e l l

d e f i n e d f o r a l l k £ C M O J . Thus we e x t e n d t h e domain o f R_ and L from

K. \ i01 t o C \ i O t by d e f i n i n g :

( 2 . 2 0 a ) K_(k) s l - T7jj7 U ( y ) R ( y , k ) d y k C C + V O l

(2 .20K1 L + ( k ) s i - ji-^ ! J ( y ) L ( y , k ) d v !< £ C + ' - . O t .

17')

It follows from theorem 2.3, that ll_ and L are analytic on C+ and

continuous on C \{0}. By (2.13) one has for k £ C+:

(2.21) lim R(x,k) = R_(k), lim L(x,k) = L+(k),

where the limits are uniform in k on compacta c: C .

rurtheriiiore, the determinants of t -i matrices in (2.20) satisfy

/!< (x,k)?„(x,k)(2.22) det R_(k) = det L+(k) = (2ik) det/

x £ 1R, k € £+\{0}.

To prove this we first note that any pair of solutions 'i',,':'o of (2.3)

satisfies

-j— det' ) = 0, i.e. tlie determinant or the 2n-2n njtcixI - VA,KJ TrjVXitw

I ) is constant for x £ K. By continuity it suffices to prove

1 2(2.22) for k £ C+, in which case:

det

v ,tr £ J = det[

¥l ÏÓ ^R'-ikR L'+i

,h R v ,1 R_ xim detl ) = detf ) = (2 ik) n de t R

++- ^-L' -!l'+2ikR ' ^0 2ikR ;

I L= lim det( ) = det( + ) = (2ik)"det 1,

V ^ 2ikL

/I L N

( + ) = (2ik)"det 1,

: 1 imxi

2.3. Bound states.

We now study the bound states of p^uation (2.1) in C , i.e. values

of k with lm k > 0 for which (2.1) possesses a nontrivial quadratically

130

integrable solution ij). They are characterized by:

Lemma 2.1. If k» £ C+ and [[0]], then the following statements are

equivalent

( i) det R_(kQ) = 0

(it) There exists a nontrivial y £ L„(E) such that

(2.23) v" + (k* - 'u(x))g' = 0.

Proof: Suppose det R_(kQ) = 0. Then by (2.22) we can find a,b £ Cn\{0}

with

Putting v(x) s * (x,ko)a = -'1'„(x,kQ)b we obtain from (2.4-5-6) that

Hi decays exponentially for x •* ±°° and therefore belongs to L„(]R) .

Furthermore it is evident that % satisfies (2.23) and Q £ 3.

Conversely, suppose (ii) holds. If det R_(kQ) * 0, then by (2.22) the

columns of "/ (x,k„) , V -.(x,k„) form 2n linearly independent solutions ofr U i- 0

(2.23). Thus there are a,b £ c", (a,b) * (0,0), such that

Using (2.4-5-6) and (2.21-22) we conclude that for x * ±« the function

,' grows exponentially in at least one direction. This contradicts the

fact that *• is in L?(lR)n. a

We nexc present a property of bound states that will play an important

role in subsection 2.4.

T h e o r e m 2 . 3 . . V : !••!•' ; . • _ • < : , " ; ; . i , : L U <; y u ; f j , i , : > ' • : . • : • ' • - . > ; ; ï : .••••: . • ' . • • • ? , • } ' 0 ,

.'•.'• k „ c C + : ) , : ; : . \ o n j . - t r . ? . ' . , • . j f ( S . I ) . A . u ' . i - i , - f < j > - •' • .•••••;.•.•'.• ••',.;• - ; : . - ••;;•.'•.•'

K~ ( k ) ,;.:.: ,1 ,u'W;•••'.? :\>,\? 71 k = k Q -jith iV.^;:.:;-.

(2.24) r>r(k

0) = Hra (k-ko)R~'(k).k-kQ

181

'Thin there in a unique matrix C (k,,) nuc-h thai for x £ K

(2.25) 4-r(x,k0)Dr(k0) = H£(x,k0)C r(k0).

froot: Since for potentials with compact support the proof is immediate,

we shal' apply a truncation procedure.

For U • • I we define u' (x) = i..(x)U(x) with *.,(x) = 1 for 'x' - 'i-1,

a (x) = N-1x| for N-1 • |xj • ;•}, u (x) = 0 for jxj • N.

Plainly, U'* satisfies (2.2) with i' = 1. From (2.4-9-11) and (2.19a-b) itN N '•) T>) -M IJ

is clear that the functions V ,K ,'i'\,V ,TC_ and R associated with V" can

be extended to analytic functions with respect to k on C M O } . Moreover,

the relation (2.16a), which we obtained for k € E•i0', remains valid

for k £ <C\IOl by analytic continuation. Thus for :•: t E and k € C\<0!

one has:

(2.2o) 4'"(x,k) = li'il(x,k)K J(k) + i'!(x,-k)R"(k) .

Concerning the behaviour as N > •" we first note that

(2.27a) j |UH(y) - U(y)idy -'' 'L'(y)'ily • I) for ". • • .

F u r t h e r r a o r e , f o r x € R and lm k 0 t h e f o l l o w i n g e s t i m a t e l i o l d s

( 2 . 2 7 b ) maxi ] R A ' ( x , k ) - R ( x , k ) [ , ! L N ( x , k ) - U x , k ) ! , R ^ ( k ) - K _ ( k ) i

-r~j exp(-^A-) j '"''(yJ-L y) ,dy with A = 'U(y)idy.

Indeed, consider the integral, equation

R""(x5k)-i;(x,k) = j (2ik)~l(e2lk(x~y)-l)(r*(y)-U(y))R(y,k)dy

T -1 -'ik(x-v)+ (2ik) (e~ ' -1)1-' (y)(R (y,k)-R(y,k

liy ( 2 . 9 - 1 J ) we have t he e s t i m a t e j R ( y , k ) | • cxpf-rrr]"}, which y i e l d s

132

(2.28) |RN(x,k) - R(x,k)| < -rij- exp^-yly^ | |UN(y) - U(y)jdy

fIterating (2.28) we find

(2.29) |Ax,k, - R(x,k) | : - e x p ^ j j |u"(y) - U(y) |dy.

Similarly we derive (2.29) with left hand side |LN(x,k) - L(x,k)|. The

proof of (2.27b) is completed by taking x -» «° in (2.29) and using (2.21).

We shall apply the fundamental relation (2.26) for suitably restricted

values of N, x and k.

First we choose ;; = (k||k - k.| '- t }, > •• 0, with 17 <= « and det R_(k) * 0

for k f ÏÏ\{k0}. By (2.27) there is an N , such that for N • N

( i) det R^(k) * 0 for k Ê 3.'

(ii) |LN(x,k) - L(x,k)| • j for x € E and k 6 ÏÏ.

Finally, by virtue of theorem 2.4 we can pick xQ, such that for x - x»

and k £ :L one has |L(x,k) - lj • |. Combination with (ii) yields that

for N "• 11,, x > x^ and k £ '.'. it holi i) that JL' (x,k) - l| •• j, so thatNF.,(x,k) is invertible.

Now let us take N • NQ and x • x . For k t .)•. we can rewrite (2.26) as

follows:

i'(xk)] i'Since t'i'„(x,k)] i',,(x,-k) is continuous on and analytic on , , we obtain

where •)•- is traversed counterclockwise.

133

Next we let N -* °°. By (2.27) the integrand on the left converges uniformly

in k on 3ft, yielding

,nf ^'(x.kH (x,k)R~'(k)dk - Urn ƒ R^(k)tRN(k)]"1dk.9 " l r

N^oo 3 " +

Applying Cauchy's residue theorem we find

3flJ f~1(x)k)>Ci.(x>k)R"

1(k)dk = ~1

Hence

C (kn) = (2Tii)"1 lim ƒ R"(k)[RNCk)]"1dk.

fJ-K»

Thus (2.25) holds for x > x„ and therefore necessarily for all x € E.

2.4. The Gel'fand-Levitan equation.

In this subsection we shall derive a linear integral equation for

N(x,s), the Fourier kernel introduced in (2.14).

Let the potential U satisfy a growth condition of order 1. We shall make

the following extra assumptions:

(2.30a) det R_(k) * 0 for k £ R M O } and for |k| • .-, lm k 0 with

e > 0 small enough.

(2.30b) lira R~'(k) exists.k-i-0lm kïO

(2.30c) All poles of R_ (k) in C+ are simple.

Since det R_(k) is continuous on C M O } and analytic on C , whereas

lim det R_(k) = 1, it follows from (2.30a) that det R_(k) has at most

initely many zeros in C • Let us denote these by k., j = 1,2,...,N.

Because of (2.22) and lemma 2.7 we can characterize k.,k7,...,k, as the

184

bound states of (2.1) in <£+\{0}.

We now introduce

A (k) = R (k), the right transmission coefficient

B (k) = R+(k)R~ (k), the right reflection coefficient.

By (2.30) the matrix A is continuous on C+\{k ,k_,—,k } and analytic

on <C+\{k. ,k„,.. .,kN> with simple poles at k. , k„, •••, k^. Let us

examine B . A priori B is defined on E\{0}. However, theorem 2.5 tells

us that lim L(x,k) = I uniformly in k on C and thus for x„ sufficientlyX-t-oo + 0

large we obtain from (2.16a)

Br(k) = 4'^1(x0,k)4'r(x0,k)Ar(k) - ï"

1 (xo,k)fc(xo>-k), k € K\(0},

where the right hand side is continuous in k on all of E. Therefore, in

a natural way we extend B to ;

and (2.19a-b) it is clear that

a natural way we extend B to a continuous function on R. From (2.9-10)

Br ( k ) =2lk J e~2lkyU(y)dy + 0(p-), k * ±».

Since U £ L" X" fi L" X n(E), Fourier theory implies that kB (k) is an

element of L " X " ( K ) with the property lim kB (k) = 0. Note that both1 *• I k I +» r

— and kB (k) are quadratically integrable on |k| ? 1. Hence by

Cauchy-Schwarz B is absolutely integrable on |k| Ï 1. Thus we have found

Br 6 L"X" n h^n n C n X n(K) and Br(k) = o(-rjU) for k - ±».

As a consequence the function B defined by

cont( 2 . 3 1 ) B

c o n t ( z > = 2 T j e l k Z Br ( k ) d k , z € R

belongs to L n X n n cj"n(ï).

we introduce the right

1,2 N, determined as in theorem 2.3, i.e.

Next we introduce the right normalization coefficients C (k.),

135

(2.32) Vr(x,kjDr(k.) = ï£(x,k.)Cr(k.),

D (k.) = lim (k - k.)A (k).r J J r' J k+k. J *•

In terms of these we form the discrete counterpart of (2.31), namely

N ik. z

(2.33) B,. (z) = -i .£ e J C (k.), z € K.

discr j=1 r j '

Finally, we define

B = B.. + B

discr cont

We shall call the aggregate of quantities IB (k),k.,C (k.)} the (right)

scattering data of the potential U.

Our main result is now given by:

T h e o r e m 2 . 9 . / ƒ t •:,• , :>ni 11: ."-•->.•;<-; ('.'.'. ".' ;jftu (' = 1 ,!>t,; ("..•<:)> a--- fu'j'i' 'r :,

ik.-n the b\».ifU:>' kcrnc * N ( x , s ) i.n t>',jdn:-'i\: '». (D.I!) .i.;r ;.•;,•'/•'.• '•'..•

/..i/ / .'Ui'.nj ffKï.'jr*;/ ,\):i-ii io'i .•".'.•• x 6 E , s : 0 :

(2.34) N(x,s) + B(2x + s) + N(x,t)B(2x + s + t)dt = 0.

0J

//.-•i1.' i.hi' oa>'!ui>l-c x .ii>:n\ii't' •!''• ,i y a i w i - ' ci'.

r'tK' m a t r i x B / / / , ; / •jnOi't'ti:' ; • • ) ; ' ; ' C n ! - \ i > ' . i ' . ' . / ; / . J f i n n ' t t .",*'•;; . ' \", , j ; , - • , ' . > " • . ' . • ' • . ' . • ;

b j t . h i : j ' o ! f o u i n j s ' r ' a ' i c i ' i n j < L I : J : ;•';,• r ' ^ j : ' . : i'< 'j\'< \*! ?'• ".' •'•1-'.•'.''•"'''"' " • ' ^ ( ' O ,

ih,,: bound urai-rn k . . ' ƒ C". /^ ' 'z C + \ ) 0 } .,--,'J .'•'/.• .•','. J-':,- i n-;. .-. ' . ' : : ; • , ' , ' • ;

- - - - - - - ^ cr(k.).

Proof: For a matrix ',s(x,.), depending parametrical ly on x, we shall use

the following notation of the Fourier transform

(F>)(x,k) = j e .:.(x,s)ds, k e TR

(F"'*)(x,s) = - j e l k s:(x,k)dk, s e n .

The proof is based on the fundamental identity (2.16a), which we rewrite

as

186

¥ (x,k)A (k) = Y„(x,k)B (k) + lf (x , -k) , x £ K, k £ E.L ie -C. r -c

H*| lf y 1 If V

S u b s t i t u t i o n of t h e fo rmulas Y = e R, 'K„ = e L ( s e e ( 2 . 4 ) ) y i e l d s

r r + ï,

where L(x,k) a L(x,-k). According to theorem 2.6 we can represent L as

t = I + 2TTF J with J(x,s) = 0 for s < 0 and J(x,s) = N(x,s) for s ? 0.

Inserting this we obtain:

(2.35) RAr - I = e 2 l k X B r + 27<e2xkx(F~1 J)B r + FJ.

Mote that each term in (2.35) belongs to L_ (-«> •• k • +«>). Applying

F to both sides one gets

(2.36) F"'(RAr - I) = F"'(e2lkx13 r) + 2HF"1 (e2 l k x(F"'J)B r> + J .

We shall show that for s ; ' 0

(2.37a) {F"1(e2 l k x i i r)}(x,s) = \Qnt.(2x + s)

( 2 . 3 7 b ) {2TIF ' ( e ^ K X ( F ' j ) B ) } ( x , s ) = N ( x , t ) B ( 2 x + s + t ) d tr QJ cont

(2.37c) fF~1(RAr-I)}(x,s) = -Bdiscr(2x+s) - j N(x,t)Brf.gcr(2x+s+t)dt.

Clearly the integral equation (2. 34) is an immediate consequence ot (2.36-37).

ad (2.37a) Trivial, since the integral in (2.31) converges absolutely.

ad (2.37b) Because the integrals in (2.14-31) are absolutely convergent

we may interchange the order of integration.

ad (2.37c) Observe, that RA - I is continuous on C +\ik ),k 2 >—,k N} and

analytic on C+\(k .t^, • • • ,^) with simple poles at k ,k?,... ,k^,

whereas RA - I = (KTT T ) f° r |k| "* "» lm k 0. Thus, using

standard limit procedures, we find by Cauchy's residue theorem:

N{F~'(RA - I ) J (x , s ) = -j- 2ni £ Urn (k-k. )e l k s(R(x,k)A (k) - I)

r ^ N J"1 k^kj J

= i . ^ e l k J S R(x,k j )D r (k . )

N ik.(2x+s)= i ^ e J L(x,k j)C r(k j)

187

= -B,. (2x + s) - N(x,t)B,. (2x + s + t)dt. •discr 0J ' discr

We shall refer to (2.34) as the Gel'fand-Levitan equation. Though it

remains to be proven, we expect that under mild conditions equation (2.34)

is uniquely solvable for all x € TR. In that case one can reconstruct the

potential from the scattering data by the following procedure:

Given the scattering data {B (k),k.,C (k.)} one calculates the function

B(z) = -i e J ^(kj) + 27 { eikzBr(k)dk, z € E,

and then solves the Gel'fand-Levitan equation (2.34). The potential U is

found from N by the formula U(x) = -2N (x,0 ) as we saw in (2.15b). In

the literature the above procedure is known as inverse scattering.

Finally, let us remark that the notation used in this paper has been

chosen so as to exploit analogies with the reasoning performed in [3].

However, the form (2.34) does not appear in the literature, except for

the reference [3] itself. What does occur in the literature is a more

symmetric version of (2.34), which is readily obtained via the

transcription.

(2 .

(2.

38)

39)

0(y

ÜI(£

;x)

) =

= 2N(x

2B(25)

,2y)

For future reference let us reformulate the aforementioned inverse

scattering procedure in this symmetrized version:

Given the scattering data {B (k),k.,C (k.)}, calculate the function

N 2ik.£ (•«>(2.40) u(c) = -2i X e J Cr(k.) + \ e Z l kS (k)dk, U I.

Next, solve the Gel'fand-Levitan equation

r(2.41) 6(y;x) + u(x+y) + 6(z;x)u(x+y+z)dz = 0

with y > 0, x € E.

The potential U is then found from S by the formula

188

(2.42) U(x) = -Px(O+;x).

References

[1] F. Calogero and A. Degasperis, Nonlinear evolution equations solvableby the inverse spectral transform associated with the multichannelSchroedinger problem, and properties of their solutions, Lett.Nuovo Cimento 15 (1976), 65-69.

[2] P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. PureAppl. xiath. .32 (1979), 121-251.

[3] W. Eckhaus and A. van Harten, The Inverse Scattering Transformationand the Theory of Solitons, North-Holland Mathematics Studies 50,1981 (2nd ed. 1933).

[4] M. Wadati, Generalized matrix form of the inverse scattering method,Topics in Current Physics 17: Solitons (R.K. Bullough and P.J. Caudrey,ed.), Springer, 1930.

[5] M. Wadati and T. Kamijo, On the extension of inverse scattering method,Prog. Theor. Phys. 52 (1974), 397-414.

189

CHAPTER TEN

UNIFICATION OF THE UNDERLYING INVERSE SCATTERING PROBLEMS

1. Introduction.

In this chapter we show that the inverse scattering problems occurring

in the present thesis can all be viewed as special cases of the inverse

scattering problem for the matrix Schrödinger equation treated in

Chapter 9.

For the first three chapters there is nothing to prove since the scalar

Schrödinger problem occurred. In the Chapters 4 to 3 the Zakharov-Shabat

problem was the underlying scattering problem. Hence, it suffices to

concentrate on that.

Let us consider a slightly generalized version of the Zakharov-Shabat

system, as was studied in [i], [2], namely

,-ik

<•••> C M : i nwhere p(x) and q(x) are complex functions and k is a complex parameter.

It is understood that the potentials p and q have the regularity and decay

properties

(1.2a) p,q £ c'(lR)

(1.2b) lira p(x) = lira q(x) = lim p'(x) = lim q'(x) = 0

i X I -x» I X I -*» I X I -••» ' X ] -x»

190

(1.2c) I (|p(x)| + |q(x)|)dx < +0C

(1.2d) j (1 + |x|)(|p(x)q(x)| + [p'CxJ i + |q'(x)|)dx < +».— 00*"

In addition we shall require some conditions on the zeros of the Wronskian

of the right and left Jost solutions, to be specified presently in (4.1).

Clearly, the case p = -q was treated in Chapter 3 under the same

conditions except for a slight relaxation of (1.2d).

The basis of the analysis presented here is the remarkable fact -

already pointed out in [3] - that the system (1.1) is connected with a

special 2x2 matrix Schro'dinger equation. Namely, let , = (i^.^^) satisfy

(1.1). Differentiating with respect to x we obtain

(1.3a) v" + (k2 - U(x))* = 0 with

/PI q'(1.3b) U(x) = (

V pq

By (1.2) the potential U(x) satisfies Ch. 9, (2.2) with ? = 1, so that the

inverse scattering problem for (1.3) was already considered in Chapter 9.

We shall apply the results obtained there to derive an inverse scattering

formalism for (1.1).

This derivation forms an alternative for the one «iven in [2], Ch. 5. In

the sequel all entities introduced in Chapter 9 for general U(x) will -

without any change in notation — be associated with the special

potential U(x) in (1.3) exclusively.

2. Jost functions.

Following [2], we first introduce the Jost functions , (x,k),

j^(x,k) for Im k S 0 and J'r(x,k), ^(x.k) for Im k •: 0 as the special

solutions of (1.1) uniquely determined [2] by

(2.1a) •; (x,k) = e~lkxr(x,k), lim r(x,k) = (hr \j

x->— "•

(2.1b) Cr(x,k) =el k x?(x,k), Jim f(x,k) = (°)

191

(2.1c) ifv(x,k) = elkx£(x,k), H m £(x,k) = (°)

(2.ld) i|^(x,k) = e X£(x,k), H m ü(x,k) = (g).

x-*+°°

liere ^ , r, etc. denote vector functions with two components.

Though easily verified, the following fact is crucial for our analysis:

The Jost functions 1' (x,k) and ll'„(x,k), associated with the 2*2 matrix

Schrödinger problem (1.3) and first introduced in Chapter 9 in a more

general context, as well as the corresponding 2*2 matrix functions R(x,k)

and L(x,k), can all be expressed in terms of the vector functions introduced

in (2.1). Specifically one has for lm k ï 0

(2.2a) H'r(x,k) = (i))r(x,k)Jr(x,-k)).

(2.2b) fc(x,k) = (^(x,-k)i^£(x,k)),

(2.2c) R(x,k) = (r(x,k)r(x,-k)).

(2.2d) L(x,k) = (?(x,-k)£(x,k)),

i.e. ty is the first column of the 2*2 matrix f , etc.

Combining the representations (2.2) with the results derived i'i Chapter 9

we immediately learn new facts about the Zakharov-Shabat Jost functions

that somehow were overlooked in the literature. For instance, Ch. 9,

lemma 2.1 gives us a reformulation of the problems (1.1), (2.1) as

integral equations different from the wellknown form [2] (5.2.1). Further-

more, Ch. 9, theorem 2.2 tells us that r(x,k), r(x,k), C(x,k), f(x,k) can

be represented in the form of a Neumann series in which the n term is

of order |k| as |k] •+ °°. To illustrate why this is pleasant let us

quote from [2], p. 151, where the asymptotic behaviour for |k| -»• « of the

Zakharov-Shabat Jost functions is discussed:

"In the case of the Schrödinger equation the n-th term in the Neumann

series of the solution is of the order |k| , whereas this is no longer

automatically the case here. Though this fact was very pleasant before

(no singularity at k = 0), it works now in the wrong direction: we have

to work harder to get the asymptotics for |k| -*•"•".

This quotation is followed by a formal expansion procedure to find the

192

first few terms in the asymptotic expansion of r(x,k) for |k| ->• <».

Including the correctness-proof the derivation takes two pages whereas

the asymptotics for r, t, t is only mentioned but not proven. It is

therefore surprising that Ch. 9, theorem 2.2 gives us the full asymptotic

expansion of r, r, t, t for |k| -»• •» without any pain. Next, let us

mention some consequences of Chapter 9 and (2.2) that are also occurring

in [2], Ch. 5, albeit after a different reasoning.

By Ch. 9, theorem 2.5, the vector functions r and t are continuous

in (x,k) on RxC and analytic in k on C for each x £ R. Similarly, f

ind I are continuous in (x,k) on K*C_ and analytic in k on C_ for each

x £ K. Another consequence of Ch. 9, theorem 2.5, is that the limits

prescribed in (2.1a-c) and (2.1b-d) for r,£ and r,£ are uniform in k on

C and <C respectively.

Finally, it follows from Ch. 9, theorem 2.6 that the vector functions

t and I have the Fourier representation

(2.3a) £(x,k) = C ) + elkSn(x,s)ds, x C E, lm k -• 00J

(2.3b) £(x,k) = (') + e n(x,s)ds, x € K, Ink •: a,

0j

where the kernels n(x,s) and ?.(x,s) belong to L2 fi \A (0 ••' s ' +»') for

each x. Furthermore they are elements of

C*0(1:p = {w E C' (Kx [0 ,«» ) ) |Vx t K : l i m w(x ,s ) = ( p ) } .

By Ch. 9, (2.15a) and (1.3b) their values at s = 0 are related to the

potentials p and q in the following way

(2.4a) -2n(x,0 ) = ( )-x

fC° p(y)q(y)dy/

,- I™ p(y)q(y)dyv(2.4b) -2n(x,0 ) = (

V p(x) '

3. Scattering coefficients and bound states.

As a next step let us introduce a convenient set of scattering

coefficients for the problem (1.1) and see how these are related to the

Itt

scattering coefficients R+, R_, L_ and L associated with the potential

J(x) in (1.3b).

Mote first that for k G E either of the pairs i/; , ty and ij;., q>„ forms

a basis of solutions of the system (1.1). Thus we have

(3.1a) Yr = r+i|j£ + r j ^ (3.1b) ^ = tji^ + £+^r

where the scattering coefficients r , r etc. depend on k 6 E. They

satisfy a number of relations, of which we mention

(3.2a) r_f+ - f_r+ = 1

r r -1 I I r_ = I , f = l_(3.2b) ( ~\ = ( ~ \ i.e.

The relation (3.2a) is easily derived from the fact that the Hronskian

of ty and ip , i.e. W(ij; fty ) = (ip ) ( ) o — ((|J )O('^ ),» is independent o

x. Further (3.2b) holds by compatibility.

Using (2.2a-b) we can rewrite (3.1) as:

/ 0 r (-k)N ,r (k) 0(3.3a) ¥ (x,k) = ï^(x,k)( ~ ) + V„(x,-k)(

r •" V +(k) 0 ' \ 0 f+(-k)

/ 0 £ (k)x 7l (-k) 0 N

(3.3b) n(x,k) = H' (x,k)( + f (x,-k)(V (-k) 0 J r ^ 0 C (kr

for x,k £ E. From (3.2-3) we conclude that the matrices R+, R_, L_, L+,

defined in Ch. 9, (2.16) for k 6 K\{0), can be extended to k £ 1R and

are given by

(3.A) K (k) = 1 R_(k) = ( "V +(k) 0 ' X 0 i+(-k)'

, 0 -?_(k)N /^(-'O 0 vL (k) = ( ) L (k) = ( ) .

^-r.(-k) 0 ' ^ 0 r (kV

194

The domain of R and L is extended to the upper half plane by Ch. 9,

(2.20). However, owing to special features of the system (1.1) there is

another way to perform the extension. Hereto we note that the Wronskian

W(I)J ,i))/>) = W(r,£) is well-defined for k € C , independent of x and for

k € E. equal to r_. Similarly W<$r ,ijy„) = W(f,•£) is well-defined for

k £ C , independent of x and for k £ TR equal to -r . Hence, we can

extend the domain of R_ and L from R to C. by the following definitions

(3.5) r_ = r ^ 2 - r ^ on C+

On C \{0} the extension given in (3.4-5) coincides with the one in Ch. 9,

(2.20). Indeed, if p and q have compact support this follows from the

principle of analytic continuation. For general p and q it is

demonstrated by applying a truncation procedure.

In the following, when writing R_, t, , we shall mean the extension by

(3.4-5). From Ch. 9, theorem 2.5 it is clear that R_ and L are analytic

on C and continuous on C .

We are now able to determine the limits of r, f, I, C. in the

directions that are not prescribed in (2.1). From Ch. 9, (2.21) and (2.2),

(3.4) we obtain

/ r - ( ! ° \(3.6) lim r(x,k) = I ) uniformly in k on compacta c C

x+co \ 0 J +

/ 0 >lim £(x,k) = i E uniformly in k on compacta cr C+

x->~c» ^ r (!c)'

/ "0 \lini r(x,k) = ( I uniformly in k on compacta c: Cx-«> s f + ( k ) '

/f+(k)\lim C(x,k) = ! ! uniformlv in k on compacta cr C .x-*-» \ 0 '

Evidently we can reformulate (3.5) as follows

(3.7) r_ = det( + ].ij;j,) on ~C+, f+ = detC^J^) on C_.

As an immediate consequence we have

195

Lernna 3 . 1 . If Ira k > O and r_(kg) = 0 then there ia an u (k Q ) £ C\{0}

such that

(3.3a) ijjr(x,ko) = a (k 0 H £ (x ,k 0 ) .

If lm ie, < O and f+(kQ) = O then thar-e is an 5(k0) £ CMO} <JUCT7 r.foi

(3.3b) ^ ( X . Ü Q ) = a(ko)+^(x,ko) .

Furthermore, using (2.1), (3.6-7) and reasoning as in the proof of Ch. 9,

lemma 2.7 we arrive at

Lemma 3.2. If k1 6 C\1R, than the. foilouinj statements ajv equivalent

( i) k0 c (k Ê C+|r_(k) = 0} U {k'e C_|r+(k) = 0}

(ii) k^ is a bound at.ite of (1.1), i.e. t-he.ps wrists a \.»i :!•;''.'•' r'

<p 6 L 2 ( R )2 such thai

4. Inverse scattering.

We shall now show how the results from Ch. 9, subsection 2.4 lead to an

inverse scattering theory for the generalized Zakharov-Shabat system

(1.1). In addition to (1.2) we make the following assumptions:

(4.1a) r_(k) * 0 and f+(k) * 0 for k £ E

(4.1b) All zeros of r_ in C+ and f in C_ are simple.

Plainly, (4.1) guarantees that R_(k) as defined in (3.4) fulfills the

requirements made in Ch. 9, (2.30). Consequently, r_ am) f have at riost

finitely aiany zeros. We shall write k.,k?,...,k, for the zeros of r_ in C+

and k.,k. , ...,k-j for thi zeros of f in C_. Because of (3.7) and lemma 3.2

•jt' can characterize k. ,k_ ,... ,k .,k. ,k9 ,. .. ,k", as the bound states of (1.1)

1'Jf)

in C.

Let us calculate the matrix functions introduced in Ch. '), subsection

2.4. tor the right reflection coefficient we obtain from (3.4)

0 b (-k) r (k) f (k)Br(k) = I ), b (k) 3 — m , hr(k) - -r^r , k H R.

\(k)

Hence, by Ch. ), (2.31)

, 0 b (zKli (.) = { < O n L ), z C K, with

^b (Z) 0 /cont

b (z) * 4~ I elkzb (k)dk, b (z) s -L o~lkZb (k)dk.cont 2'; _ J r com 2 } r

We next compute U,. . Renumbering the hound states !;. , k. if ncrtssarv,discr J J

we may represent the zeros of det R_(k) in (.' as k .,!<.,,. ...k,,, ,k ; ,

*••'kd'kd+l'""'kN' w h e r e k:j-d + i £ ~ ki' ! = l,2,...,d. Thus one has

Dy le inraa 3 . 1 t h e r e a r e . ( k . ) , t ( k . ) £ CWO) s u c h t h a t

( 4 . 2 a ) , r ( x , k . ) = . ( k . ) , f ( x , k . ) , j = 1 , 2

( 4 . 2 b ) , ( x , k . ) = ~ i ( k . ) , . , ( x , k . ) , j = 1 , 2 , . . . , d .

u s i n g C h . i , ( 2 . 3 2 ) a n d ( 2 . 2 ) , ( 3 . 4 ) , ( 4 . 2 ) w e f i n d

for j = l,2,...,N-d

for j = N-d+1,...,d

for j = d+1,...,N

and so by Ch. 9, (2.33)

13.- (z) =discr \b.. (z)discr

b,. (z)discr

discr

ïk.za J

discr

z £ IR, with

- i k .zJ

(k.)

cj ~

r S(k.)

cj ~ f'(E.)J + j

Setting b s D + b.. , b s b + E.. , we finally have° cont discr cont discr

InserLion of these results into Ch. 9, theorem 2.9 gives us:

T h e o r e m 4 . 1 . i f ' ' V ' a o n d L r l o n . - ï { 1 . 2 ) a n d ( 4 . 1 ! < ; ? v ' V : . ' V . ' l e I , .'•'.••";

:•'y.u'Lr'i1 Sijt'ncl N ( x , s ) = ( n ( x , s ) n ( x , s ) ) dcfl'iel in !:\.-'-> .; n /<•;'?• \; •

•'•): '.oiöinj {.ntcjpj; ciua'ion foi» x 6 E , s • 0 :

(A.3) W(x,s) + B(2x + s ) + :- i (x,t)B(2x + s + t )d t = 0.

198

the scattering data associated with the potentials p and q, namely: the

rijhv reflection coefficients b , b , the bound states k., k. of (1.1) in

(D and the right normalization coefficients c , c..

Theorem 4.1 paves the way to an inverse scattering theory:~ ~ r ~ r

Given the scattering data {b ,b ,k.,k.,c.,c. }, one computes B and solves

(4.3). The potentials p and q are then found from !J by (2.4). It should be

noted, however, that the problem of finding sufficient conditions for

unique solvability of (4.3) covering a wide class of potentials is still

open.

It follows from the last remark of Chapter 9 that we can cast the

above inverse scattering procedure in a more symmetric form, namely:

Given the scattering data {b ,b ,k.,k.,c.,c. }, compute the function

(4

(4

(4

.4a) u)

.4b) .2

.4c) :~

Next, solve

(O

(O

the

_ /

= -2

= 2i

Gel

0 ,«,

(O 0 '

d 2ik.

d -2ik.

'fand-Levitan

• ! f™ _11 -J

equation

ik£:br(k)dk

2 l K b (k)dk.

(4.5) ri(y;x) + '..(x+y) + fi(z;x)^(x+y+z)dz = 0

with y • 0, x G E.

The potentials p and q are then found from fi = (b b) by

(4.oa) ;>(x) = -b,(0 ;x)

(4.ób) q(x) = -bj(0 ;x).

References

[I] .-I.J. Ablowitz, Lectures on the inverse scattering transform, Stud.Appl. Math. 53 (1978), 17-94.

[2] W. Kckhaus and A. van Harten, The Inverse Scattering Transformationand the Theory of Solitons, North-'.loLland Mathematics Studies 50,1981 .

[3] M. Hadati and T. Kamijo, On the extension of inverse scatteringmethod, Prog. Theor. Phys. 52 (1074), 397-414.

.'00

CONCLUDING REMARKS

In what follows we have collected a number of additional results,

each with some interest of its own. We present these results without proof.

(i) Let u(x,t) be a real solution of the KdV obtained via 1ST

as described in Chapter 2. Suppose there are x.,x E TR, x. * x~

and t, > t. > 0, such that for all x £ K

uCx^+x,^) = u(xi-x,ti), i = 1,2.

Then one has:

(a) If x. > x2, then u = 0.

(b) If x < x„, then either u = 0 or u(x,t) = -2K2sech2[x(x-x+-4K2t)]

with

K = èV—— and x = — — — .H 2 ll 2 C1

In particular this shows that, apart from the 1-soliton solution,

any nontrivial solution of the KdV obtained via 1ST can have

spatial symmetry for at most one value of the time t.

(ii) By combining the remark made in Chapter 2 after the proof of

lemma 5.1 with the other results from that chapter one easily

arrives at the following theorem:

•••...•midci' JL t-;>ai>am,-.ici' family u(x,t), t s t Q, tQ £ K ,;ƒ JV U ?

;\ t^atiala in the S^hroding^r r.^.ii.teving ;i>jb!cr;, r,at-iüf:i in.j

j'of fi.:-ud t > t the. oMdiiicnc. .::-v;.ed in Ch. 2, hhj.-.jt:i-rn 2.1.

u ( x , t ) . vh'iz-; u d(x,t) f-y th,. >'e f.-;-'.. i •»'..•:.• .-.;,-• •ƒ u ( x , t ) , .'.. .

••':,• : •. lari.;' :^i!': J<J,I: Uyi'-ng .iar.i { 0 , K . ( t ) , c ^ ( t ) } . /... S2_ .•••

•ji.se>! /•;, •••'•. 2 , ( 2 . 2 6 c ) ,:>id .'.. ; V :•• /.• •: :. 4 5 .

A . : : , - , -I, ,•• ••>.'• - . : . • • . : • : • . • ^ . c , , . . . , c N , . • , " . • ; ; . . • •

201

Assume furthermore, that the function y •*• Si (x+y;t) is strong-

l.y diffcpcntiablc in V with respect to x at every point ( x , t ) ,

x 6 E , . t i t... Let there exist a function a:[t_,<») -*-K, uuoh that

in the parameter' region t £ t~., x § a(t),the functions Q and Si',

with ft' the utronij x-dcrivatiw; of SI , satisfy:

(a) max | |s?c(x+y;t) | , |sV (x+y;t) | j S H ( y , t ) , y > 0,

with H(y, t ) a monotonitnil hj decreasing function of y

fur fixed t , such that a ( t ) = sup H(y, t) ->• 0 as t ->• <°.0<y<+«>

(b) 0/°O|ftc(x+y;t) |dy < oQ < 1 and

0/"|i^(x+y;t) |dy £ a, < +», with aQ and a^ constants.

Then we haoc:

sup |u(x,t)-u,(x,t) | = 0(a(t)) as t -»• m.xga(t) a

(iii) The inverse scattering formalism associated with the self-

adjoint Eakharov—Shabat system

where q = q(x) is a real function and C, a complex parameter,

can be simplified. As in Chapter 4 the Gel'fand-Levitan equation

appearing in the literature can be reduced to a scalar integral

equation containing only a single integral. With the help of this

simplification one can analyse in a way similar to Ch. 2, section

the asymptotic behaviour corresponding with the various nonlinear

evolution equations (all solitonless) solvable via the associated

inverse scattering method, such as the solitonless mKdV, sinh-

Gordon, etc.

(iv) The explicit structure of the constants in the niKdV analysis

presented in Chapter 5, shows that related results are valid

in coordinate regions t > 0, x s -u -vt , •• = — + e, with

u,v,e nonnegative constants and e > 0 sufficiently small.

202

AWE SDU

AN OPEN PROBLEM

In this thesis we have succeeded, by a more or less uniform method,

to reveal the asymptotic structure for large time for solutions of a

number of interesting nonlinear evolution equations solvable by the

inverse scattering method. However, any reader familiar with the soliton

field no doubt has noticed the absence of the wellknown nonlinear

Schrödinger equation (NLS)

(1a) iqt = qxx + 2q2qX, -«> < x -• +», t •> 0

(lb) q(x,0) = qQ(x),

where the initial function ^Q(X) is an arbitrary complex-valued function

on R, sufficiently smooth and rapidly decaying for |x[ •* °° and

satisfying conditions similar to Ch. 3, (7.2) so as to make the

Zakharov-Shabat inverse scattering method associated with Ch. 8, (2.1)

work.

Note that the dispersion relation w(O = -i,2 associated with the

linearized version of (la) is an even function, so that the group

velocity dw/di; = -2c, is not of one sign for all real £. This forms a

significant difference with the other problems treated and has direct

consequences for the method employed in this thesis.

Let us nevertheless see what our method produces.

203

To start with let us point out that for the problem (1) the Zakharov-

Shabat inverse scattering method runs as follows (see [1], [2], [4], [6]

for details).

Having computed the (right) scattering data [b^(0 , C • ,CL. } associated

with q o(x), when introduced as a potential into Ch. 8, (2.1), one puts

(2a) c5(t) = cyexp{-4i£2. t }, j = 1,2,...,NJ J J

(2b) b U,t) = b (c)exp{-4i;2t}, -» < ,; • +».

Then by the solvability of the inverse scattering problem, there exists

for each t > 0 a smooth potential q(x,t) having {b (<;,t), c,. ,C. (t) } as

its scattering data. The function q(x,t) is the unique solution to the MLS

initial value problem (1).

Now, let q,(x,t) denote the reflectionless part of the solution

q(x,t) of (1). Then, since the group velocity is not of constant sign,

tnere is no particular region of the x-axis singled out on which

q(x,t) - q,(x,t) might be concentrated as t -+ «. However, we may expect

an overall decay of q(x,t) - q,(x,t) for t -• «>. In fact, some authors

[3], [5] claim the existence of special solutions to (1) such that

q - q, decays in time as t more or less uniformly in x on R.

Let us see what the techniques of Chapter 3 yield in this case. By

theorem 4.t of that chapter we can estimate q — q, in terms of the

scattering data in the following way:

For each x t E and t "> 0 one has

(3) |q(x,t) - qd(x,t)| S u?Q[S2 j |:'c(x+y;t)

+ sup !;:

with

(4) !.c(s;t) = -i

and a 0 the constant given by Ch. 3, (4.2).

Now, a detailed examination of the right hand side of (3) shows that in

all relevant coordinate regions the second term behaves as t " as t •• ™,

but the first term tends to a constant!

Note the difference with the cmKdV problem Ch. 8, (7.1) where theorem 4.1

204

revealed neatly the asymptotic structure (see Ch. 3, (7.7-8) and the sub-

sequent discussion).

As an explicit prototype problem, let us mention the special case

that the initial function qn(x) in (1b) has the scattering data

{b (c) = e ,0,0}. Then the integral (4) can be evaluated in closed

form, yielding

(5) fic(s;t) = it (1 + 4it)~iexp{-s2/(1+4it)}.

Let us write out in full glory the Gel'fand-Levitan equation Ch. 8, (3.2-3)

in the special case (5). For x E R, y > 0 and t > 0 one then has

r(6a) g(y;x,t) + w(x+y;t) + i3(z;x,t)io(x+y+z;t)dz = 0

0J

0 -lr"i(1-4it)~^exp{-s2/(1-4it)}v

(6b) ü>(s;t) = [ i . )V ( 1 + 4it) iexp{-s2/(1+4it)} 0 '

with s £ E.

Here the unknown B(y;x,t) is a 2x2 matrix function of the variable y,

whereas x and t are parameters. Of crucial importance is the behaviour of

8(0 ;x,t) as t ->• +<*> in appropriate regions x S a(t). This gives the

behaviour of q(x,t) through the relations Ch. 3, (3.4-5).

Since the integral equation (6) is an explicit integral equation, no

particular knowledge of inverse scattering is required to attack it.

References

[l] M.J. Ablowitz, Lectures on the inverse scattering transform, Stud.Appl. Math. 58 (1973), 17-94.

[2] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inversescattering transform - Fourier analysis for nonlinear problems, Stud.AppL. Math. 53 (1974), 249-315.

[3] M.J. Ablowitz and U. Segur, Asymptotic solutions and conservationlaws for the nonlinear Schrcidinger equation I, J. Math. Phys., 17(1976), 710-713.

205

[4] W. Eckhaus and A. van Harten, The Inverse Scattering Transformation andthe Theory of Solitons, Uorth-Holland Mathematics Studies 50, 1981(2nd ed. 1983).

[5] H. Segur, Asymptotic solutions and conservation laws for the nonlinearSchrodinger equation II, J. Math. Phys., 17 (1976), 714-716.

[6] S. Tanaka, Non-linear Schrodinger equation and modified Korteweg-deVries equation, construction of solutions in terms of scattering data,Publ. R.I.M.S. Kyoto Univ. 10 (1975), 329-357.

206

SAMENVATTING

In 1967 ontdekten Gardner, Greene, Kruskal en Miura een methodeom het beginwaardeprobleem voor de Korteweg-de Vries vergelijking (KdV)in principe op te lossen. Niet lang daarna bleek dat deze methode nietop zichzelf stond, maar met succes gebruikt kon worden otn andere belang-rijke niet-lineaire partiële differentiaalvergelijkingen te bestuderen.De methode kreeg de naam: inverse scattering technique (1ST).

Aan de hand van de KdV laat zich de 1ST als volgt omschrijven:Het beginwaardeprobleem voor de (niet-lineaire) KdV wordt gereduceerdtot het oplossen van de (lineaire) Gel'fand-Levitan integraalvergelijking.De beginwaarde voorgeschreven voor de oplossing van de KdV wordt alspotentiaal geïntroduceerd in de Schrödinger vergelijking en levert dande zogenaamde scattering data met behulp waarvan de coëfficiënten van deGel'fand-Levitan vergelijking worden gedefinieerd.

Echter, expliciet oplossen van deze integraalvergelijking is slechtsmogelijk als de reflectiecoëfficiënt geassocieerd met de beginwaardenul is. Dan ontstaat de vermaarde "pure N-soliton oplossing" met Nhet aantal discrete eigenwaarden in de Schrödinger vergelijking.Het asymptotisch gedrag van deze oplossing is uitgebreid bestudeerden blijkt te corresponderen met een decompositie in N solitonen, d.w.z.gelokaliseerde golven die na onderlinge interactie hun oorspronkelijkevorm en snelheid behouden en alleen een faseverschuiving (phase shift)aan deze interactie overhouden.

In het algemene geval dat de reflectiecoëfficiënt niet overal nul is,is de asymptotiek aanmerkelijk gecompliceerder. Een eenvoudige heuristi-sche redenering leidt al gauw tot het vermoeden dat naarmate de tijd ver-strijkt de oplossing uiteen zal vallen in twee componenten: een soliton-component bestaande uit N naar rechts lopende solitonen en een nonsoliton-component bestaande uit dispersieve golven die naar links lopen.Het feit dat destijds geen enkel bewijs van dit vermoeden bekend was,vormde de aanleiding tot het onderzoek geboekstaafd in dit proefschrift.

Meer algemeen stellen we ons in dit proefschrift ten doel een volledi-ge, door bewijzen gestaafde beschrijving te geven van de manier waaropsolitonen te voorschijn komen uit verschillende klassen van niet-lineairepartiële differentiaalvergelijkingen oplosbaar via 1ST. Hierbij werken wein die coordinaatgebieden waar de nonsolitoncomponent is op te vatten alseen storing op de solitoncomponent. Een steeds weerkerend element in deanalyse van de verschillende niet-lineaire problemen op de voorafgaandebladzijden is dan ook de aanpak van de Gel'fand-Levitan vergelijkingvia een storingsargument.

De inhoud van dit proefschrift kan als volgt worden geresumeerd:

207

In Hoofdstuk 1 analyseren we de oplossing van het beginwaardeprobleemvoor de KdV in alle naar rechts meelopende coördinaten en geven eenvolledige en nauwkeurige beschrijving van het te voorschijn komen vansolitonen. Deze asymptotische analyse wordt in Hoofdstuk 2 uitgebreidmet zeer expliciete schattingen van de nonsolitoncomponent op rechter-halflijnen die langzaam naar links lopen. In Hoofdstuk 3 berekenen we dephase shifts van de KdV-solitonen wanneer niet alleen de interactie metde andere solitonen maar ook die met de nonsolitoncomponent in aanmerkingwordt genomen. De interactie met de nonsolitoncomponent blijkt te resul-teren in een extra phase shift naar links. In Hoofdstuk 4 beschouwen wede vraag hoe goed een oplossing van een niet-lineaire, via 1ST oplosbaregolfvergelijking wordt benaderd door zijn solitoncomponent in een ruimerkader door schattingen af te leiden voor het verschil van een reëlepotentiaal in het Zakharov-Shabat systeem en zijn reflectieloze component.Verder wordt het bijbehorende inverse scattering formalisme aanzienlijkvereenvoudigd. Gebruikmakend van de schattingen uit Hoofdstuk 4 wordenin Hoofdstuk 5 asymptotische schattingen voor de oplossing van hetmodified KdV (mKdV) beginwaardeprobleem verkregen, welke op hun beurtbenut worden in Hoofdstuk 6 voor de berekening van de phase shifts van demKdV-solitonen. In tegenstelling tot het KdV geval beschouwd in Hoofdstuk3 blijkt bij de mKdV de interactie met de nonsolitoncomponent te resulte-ren in een extra phase shift naar rechts. In Hoofdstuk 7 passen we deresultaten uit Hoofdstuk 4 toe voor een asymptotische analyse van deoplossing van het sine-Gordon beginwaardeprobleem. De schattingen uitHoofdstuk 4 worden in Hoofdstuk 8 gegeneraliseerd tot het geval vancomplexe potentialen. Als toepassing analyseren we het complexe mKdV(cmKdV) beginwaardeprobleem. In Hoofdstuk 9 ontwikkelen we een inversescattering formalisme voor de matrix Schrödinger vergelijking met niet-Hermitische potentiaal. Dit matrix Schrödinger probleem vormt, zoals inHoofdstuk 10 wordt aangetoond, een natuurlijke synthese van de scatteringproblemen optredend in dit proefschrift. In een appendix illustreren weaan de niet-lineaire Schrödinger vergelijking, dat de in dit proefschriftontwikkelde technieken falen voor vergelijkingen waarvan de geassocieerdegroepssnelheid niet tekenvast is en stellen vervolgens een interessantopen probleem.

208

CURRICULUM VITAE

Geboortedatum: 31 januari 1950Geboorteplaats: ZwolleEindexamen gymnasium 3: 1969 (cl.)Doctoraalexamen wiskunde aan de R.U.Groningen: J978 (cl.)Specialisatie: FunctionaalanalyseAfstudeerdocent: Prof. dr. G.E.F. ThomasAfstudeeronderwerp: De Stieltjes integraalvergelijkingPromotieplaats aan het Mathematisch Instituut van deR.U.Utrecht: 1978-1984Huidige functie: medewerker aan de afdeling ToegepasteWiskunde (vakgroep ADAM) van de T.H.Twente

209

SOLITON SOLUTIONS

OTTlti OFFER

TVPING TWISTS

INSUMMABLE INSÖMNTA

STELLINGEN

behorende bij het proefschrift Studies in Soliton Behaviour

van P.C. Sehuur

1. Beschouw in de complexe Hilbertruitnte JC = L2[O,1] met standaard-

inprodukt de twee t-parameter families van operatoren F.(t)

en F„(t), beide met domein C[O,1] , gegeven door

Ft(t)v = v(0), t ï 0,

[F2(t)v](x) =|lév(0)' t > 0

(o t = o.

Dan zijn F,>F2 e n F]F2 s t e r k differentieerbaar in 3C op CfO,1]

met betrekking tot t s 0, terwijl niet geldt

2. Zij R > 0 en f:[O,R)—*C continu met f i 0.

Dan is er een 6 > 0 zodanig dat voor 1 < x < 1 + 6 geldt

( x - 1 ) 20 / R | k | 0 x" k f (yx" k ) | 2 dy < 4 0 ; R | f ( Z ) | 2 dZ.

3. De bewering in W. Rudin, Functional Analysis, p. 341,

dat iedere symmetrische operator in een Hilbertruimte

een gesloten symmetrische voortzetting bezit, kan door

een eenvoudig tegenvoorbeeld weerlegd worden.

4. Zij E de klasse van alle complexwaardige functies g(x) ,

gedefinieerd op (0,»), zodanig dat (1+x) g(x) tot L1(0 < x < +»)

behoort. Zij A 6 C, Re X * 0.

Dan heeft de Stieltjes integraalvergelijking

f(x) = g(x) + Xj £2± dy, x > 0

voor iedere g E E een oplossing f € E.

(P.C. Schuur, De Stieltjes vergelijking, h'.IJ. Gt\niinjcn,

inu-rn rapport, IH?8)

5. Zij S een (niet noodzakelijk dichtgedefinieerde) isometrische

gesloten operator in een complexe Hilbertruimte.

Dan is het spectrum van S een van de volgende verzamelingen:

(i) n e c| |x| i 1} (Ü) {A e c| |A| % 1}

(iii) (C (iv) een deelverzameling van

{x e c| |x| = 1}.

6. In het boek W. Magnus, F. Oberhettinger, R.P. Soni, Formulas and

Theorems for the Special Functions of Mathematical Physics

vertonen twee afzonderlijk vermelde formules bovenaan p. 9

een frappante gelijkenis:

ƒ (cos<)*cos (fit) At — -

Rex> — 1,

{ {cos l)' cos (yt) dt = ~ r(x + J)

» 2 ^(-p+^r-f-'+i)Rex> - I .

7. De buitenkant van het blad Mededelingen van het Wiskundig

Genootschap is voorzien van de zinsnede "verschijnt maandelijks

(negen maal per jaar)". Deze zinsnede is onprecies en inwendig

tegenstrijdig en dus niet passend voor een wiskundig tijdschrift.

8. De rits is een manonvriendelijke uitvinding.

9. Of iemand een baan al dan niet krijgt, wordt te dikwijls bepaald

door het misverstand in psychologische tests, dat een oneindige

rij bepaald zou zijn door specificatie van een eindig aantal

termen.

10. Veel cartoons suggereren dat sokken breien begint bij de teen

en eindigt bij de boord. In werkelijkheid gaat het andersom.

11. Wie blij is met een dode mus is een dierenbeul.