Physica D 217 (2006) 77–87www.elsevier.com/locate/physd

Freedom in the expansion and obstacles to integrability in multiple-solitonsolutions of the perturbed KdV equation

Alex Vekslera, Yair Zarmia,b,∗

a Department of Physics, Beer-Sheva, 84105, Ben-Gurion University of the Negev, Israelb Department of Solar Energy and Environmental Physics, Jacob Blaustein Institute for Desert Research,

Sede-Boqer Campus, 84990, Ben-Gurion University of the Negev, Israel

Received 29 June 2005; received in revised form 26 February 2006; accepted 21 March 2006

Communicated by B. Sandstede

Abstract

The construction of solutions of the perturbed KdV equation encounters obstacles to asymptotic integrability beyond the first order, when thezero-order approximation is not a single-soliton wave. In the standard analysis, the obstacles lead to the loss of integrability of the Normal Form,resulting in a zero-order term, which does not have the simple structure of the solution of the unperturbed equation. Exploiting the freedom inthe perturbative expansion, an algorithm is proposed that shifts the effect of the obstacles from the Normal Form to the higher-order terms. Theexpansion has been carried out through third order. Through this order, the Normal Form remains integrable, and the zero-order approximationretains the structure of the unperturbed solution. For multiple-soliton solutions, the resulting obstacles decay exponentially away from the soliton-interaction region, which is a finite domain around the origin in the x–t plane. The effect is demonstrated in detail through second order for thetwo-soliton case, where the obstacles generate a second-order radiative tail that emanates from the origin, and decays exponentially away fromthe origin. These results suggest a new meaning to “asymptotic” integrability: The zero-order term is determined by an integrable Normal Form,and the higher-order terms in the expansion of the full solution tend to those of the integrable case (when no obstacles exist) asymptotically awayfrom the origin.c© 2006 Elsevier B.V. All rights reserved.

Keywords: Perturbed KdV equation; Normal Form expansion; Obstacles to integrability

1. Introduction

The KdV equation [1], to which the terms that have beenneglected in its derivation are added as a small perturbation,is often analyzed by the method of Normal Forms (NF) [2–8], where the NF is the equation that governs the evolution ofthe zero-order approximation to the solution of the perturbedequation. The motivation is the expectation that like the KdVequation, the NF will be integrable and preserve the wave natureof the solution of the unperturbed equation. When the zero-order approximation is a single-soliton state, this expectationis borne out [4,8]; the NF then merely updates the dispersion

∗ Corresponding author at: Department of Solar Energy and EnvironmentalPhysics, Jacob Blaustein Institute for Desert Research, Sede-Boqer Campus,84990, Ben-Gurion University of the Negev, Israel. Tel.: +972 8 659 6920.

E-mail address: zarmi@bgu.ac.il (Y. Zarmi).

0167-2789/$ - see front matter c© 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2006.03.011

relation obeyed by the wave velocity [9–11]. The situation isdifferent when one seeks a solution for which the zero-orderterm is a multiple-soliton state. Except for specific forms ofthe perturbation, from second order and onwards, the standardformalism generates terms, which cannot be accounted for bythe perturbative expansion of the solution (the Near IdentityTransformation-NIT) [3–8]. One is forced to include them inthe NF [4,7,8], rendering the latter non-integrable (hence theterm “obstacles to integrability”), and spoiling the simplicity ofthe zero-order solution.

The loss of integrability of the NF is a consequence of anassumption made in the standard NF expansion, that, startingfrom the first order, the terms in the NIT are differentialpolynomials in the zero-order approximation.1 In this paper,

1 The problem arises already in the NF analysis of perturbed ODE’s, where,customarily, higher-order corrections in the NIT are assumed to be polynomials

78 A. Veksler, Y. Zarmi / Physica D 217 (2006) 77–87

an alternative algorithm is proposed, which allows for anadditional t- and x-dependence in the higher-order terms,a dependence that cannot be accounted for by differentialpolynomials in the zero-order approximation. The addedfreedom enables one to shift the effect of the obstacles tointegrability from the NF to the NIT. The NF then remainsintegrable. Its solution (the zero-order approximation) retainsthe multiple-soliton character of the solution of the unperturbedequation. Moreover, just as in the single-soliton case, the NFmerely updates the dispersion relation obeyed by the velocityof each soliton. Our algorithm yields obstacles that vanishidentically when computed for a single soliton. (Obstacles thatare not computed according to this algorithm do not vanishin the single-soliton case. The fact that they do not emergein that case is then discovered through explicit calculation.)In the multiple-soliton case, our obstacles do not vanishidentically. However, away from the soliton-interaction region(a finite domain around the origin in the x–t plane) theydecay exponentially. As a result, they generate in the NIT abounded, exponentially decaying tail, so that, away from theorigin in the x–t plane, the effect of the loss of integrabilityon the approximate solution disappears. The general analysishas been carried out through third order. The second-orderobstacle is a differential polynomial in the zero-order solution,and is expressible in terms of symmetries of the KdV equation.(We call this a “canonical” obstacle.) The third-order analysisprovides a picture for what happens in orders higher thansecond: The obstacles are, again, localized around the origin,but contain contributions which cannot be written as differentialpolynomials. They are functionals of the canonical obstacles. Adetailed exposition is presented for the case of the two-solitonsolution, through second order.

Most of the paper focuses on the analysis through secondorder. The general aspects of the NF analysis of the perturbedKdV equation are reviewed in Section 2. The emergence ofobstacles to integrability in the NF in the standard analysisis outlined in Section 3. Section 4 reviews the NF analysisof the single-soliton case, where no obstacles to integrabilityemerge. In Section 5, the first-order analysis is reviewed for thecase of a general zero-order solution (i.e., not a single soliton).This leads to the introduction in Section 6 of the concept of“canonical” obstacles, which are differential polynomials inthe zero-order approximation. They can be expressed in termsof symmetries of the unperturbed equation, vanish explicitlyin the single-soliton case, and are localized in the multiple-soliton case. In Section 7, it is shown how to shift the effectof the obstacles to integrability from the NF (i.e., from thezero-order term) to the second-order correction in the NIT. Allthat is required is that the second-order correction containsterms that cannot be expressed as differential polynomials inthe zero-order solution. In Section 8, the way to ensure that theobstacles to integrability are localized around the origin when

in the zero-order term. This assumption works for systems of autonomousequations with a linear unperturbed part. Inconsistencies may emerge in allother cases, and are resolved by allowing for explicit time dependence in theNIT. The issue is discussed in part in [25].

the zero-order solution is not a single soliton is presented. Allthat is required is that the differential polynomial part of thesecond-order correction in the NIT has the structure found inthe single-soliton case. Section 9 presents the detailed analysisof the two-soliton case through second order. It is shown thatthe canonical obstacle to integrability is not a resonant drivingterm, so that all it can generate in the second-order solution isa decaying radiation tail. Section 10 presents arguments as towhy the approach delineated in the previous sections applies tohigher orders as well as to multiple-soliton solutions, with morethan two solitons. The main results of the third-order analysisare presented, from which it becomes clear how to extend themethod to higher orders. Concluding comments are presentedin Section 11.

2. NF analysis of the perturbed KdV equation

In this section, general aspects of the NF analysis of theperturbed KdV equation are reviewed2:

wt = 6wwx + wxxx + ε(30α1w2wx + 10α2wwxxx

+ 20α3wxwxx + α4w5x )

+ ε2(140β1w3wx + 70β2w

2wxxx + 280β3wwxwxx

+ 14β4ww5x + 70β5w3x + 42β6wxw4x + 70β7wxxwxxx

+ β8w7x ) + O(ε3) (|ε| � 1). (2.1)

The unperturbed equation,

ut = 6uux + uxxx ≡ S2[u], (2.2)

is integrable [12–24]. Its pure soliton eigen-solutions are thesingle soliton,

u(t, x) = 2k2/ cosh2(k{x + v0t + x0}), (2.3)

where

v0 = 4k2, (2.4)

as well as multiple solitons, which may be expressed by theHirota formula [12]. For example, the two-soliton solution isgiven by

u(t, x) = 2∂2x ln

{1 + g1(t, x) + g2(t, x) +

(k1 − k2

k1 + k2

)2

× g1(t, x)g2(t, x)

}(gi (t, x) = exp[2ki (x + v0,i t + x0,i )]). (2.5)

In Eq. (2.5), each v0,i is related to ki by Eq. (2.4). If theinitial conditions employed do not correspond to pure solitoneigen-solutions, then the solution develops dispersive tails [22–24]. This paper focuses on the pure soliton case.

Away from the interaction region of the solitons, Eq. (2.5)asymptotically reduces to a sum of two single-soliton solutions(see Fig. 1):

2 The numerical coefficients in Eq. (2.1) have been chosen so as to conformto the structure of the symmetries of the unperturbed equation, discussed in thefollowing

A. Veksler, Y. Zarmi / Physica D 217 (2006) 77–87 79

Fig. 1. Two-soliton solution (Eq. (2.5); k1 = 0.3, k2 = 0.4.

u(t, x) → 2k21/ cosh2(k1{x + v0,1t + ξ0,1})

+ 2k22/ cosh2(k2{x + v0,2t + ξ0,2}). (2.6)

The deviation of the exact solution from the asymptoticform falls off exponentially as the distance from the interactionregion grows.

Returning to Eq. (2.1), we assume a Near IdentityTransformation (NIT) for w:

w = u + εu(1)+ ε2u(2)

+ O(ε3). (2.7)

The evolution of the zero-order term, u(t, x), is governedby the Normal Form (NF), which is constructed from Sn , thesymmetries of the unperturbed equation [2–8,13–24]:

ut = S2[u] + εα4S3[u] + ε2β8S4[u] + O(ε3). (2.8)

The Sn obey the recursion relation [15–21]

Sn+1[u] = ∂2x Sn[u] + 4uSn[u] + 2ux Gn[u], (2.9)

with

Gn[u] = ∂−1x Sn[u]. (2.10)

For the present analysis, we shall need

S1 = ux

S2 = 6uux + uxxx

S3 = 30u2ux + 10uuxxx + 20ux uxx + u5x

S4 = 140u3ux + 70uuxxx + 280uux uxx + 14uu5x

+ 70u3x + 42ux u4x + 70uxx uxxx + u7x .

(2.11)

Eq. (2.8) is integrable; the single- and multiple-solitonsolutions of Eq. (2.2) are also solutions of Eq. (2.8), with eachvelocity updated according to a dispersion relation [9–11]:

v = 4k2+ εα4(4k2)2

+ ε2β8(4k)3+ O(ε3). (2.12)

Using Eqs. (2.7) and (2.8) in Eq. (2.1), one obtains in O(εn)

the n′th-order homological equation, which determines u(n),the n′th-order term in the NIT. The first-order equation is:

u(1)t [u; t, x] − 6∂x (uu(1)) − ∂3

x u(1)+ α4S3[u]

= 30α1u2ux + 10α2uuxxx + 20α3ux uxx + α4u5x . (2.13)

The second-order homological equation has the followingstructure

u(2)t [u; t, x] − 6∂x (uu(2)) − ∂3

x u(2)+ β8S4[u]

= +140β1u3ux + 70β2u2uxxx + 280β3uux uxx

+ 14β4uu5x + 70β5u3x + 42β6ux u4x + 70β7uxx uxxx

+ β8u7x + Z (2). (2.14)

In Eq. (2.14), Z (2) is the contribution of the first-order partof the solution, u(1), to the second-order homological equation.It is given in Appendix A (Eq. (A.1)). In Eqs. (2.13) and (2.14)and in the following sections, partial derivatives with respectto t and x are applied to the dependence of u(n) (n = 1, 2) onthese variables both through u(t, x) and through the explicit t-and x-dependence.

The rational behind the construction of the NF, Eq. (2.8),as the equation that governs the evolution of the zero-orderterm is well known [2–8,13–24]. The symmetries, if left inthe homological equations (e.g., Eqs. (2.13) and (2.14)), wouldgenerate secular terms in the higher-order corrections. Shiftingthe contribution of symmetries from the homological equationsto the NF eliminates the primary and most obvious cause forthe emergence of unbounded higher-order corrections. Still,as there may be other sources of unbounded solutions, withany specific expansion algorithm, one ought to check whetherthe higher-order corrections constitute valid approximationsat least as far as |t |, |x | = O(1/ε), which is the standardexpectation of the method of NF.

3. Review of standard NF analysis [2–8]

In the standard analysis, the higher-order terms in Eq. (2.7)are assumed to be differential polynomials in u, with no explicitdependence on t and x . No obstacle is encountered in the first-order equation, Eq. (2.13) [2–8]. The structure of u(1) is

u(1)= au2

+ bq(1)ux + cuxx(q(1)

≡

∫ x

−∞

u(x, t)dx

). (3.1)

(All terms in u(1), must have a total weight of 4, whenthe weights assigned to u, ∂t and ∂x , are 2, 3 and 1,respectively [13–19].) Eq. (2.13) is solved for u(1), yielding:

a = −5α1 +53α2 +

103

α4, b = −103

α2 +103

α4,

c = −52α1 +

53α3 +

56α4. (3.2)

In second order, assuming that u(2) is a differentialpolynomial in u, with no explicit dependence on t and x , itmust have weight 6. The formalism allows for the followingdifferential polynomial:

u(2)= Au3

+ Bu2(q(1))2+ Cuux q(1)

+ Duuxx

+ Eu(q(1))4+ Fuq(1)q(2)

+ Gu2x + Hux (q

(1))3+ I ux q(2)

+ Juxx (q(1))2

+ K uxxx q(1)+ Luxxxx (q(l)

= ∂−1x (u)l). (3.3)

80 A. Veksler, Y. Zarmi / Physica D 217 (2006) 77–87

Fig. 2. Obstacle for the two-soliton solution in standard NF analysis (Eq. (3.4));parameters as in Fig. 1.

Despite the wealth of coefficients in Eq. (3.3), not all theterms on the r.h.s. of Eq. (2.14) can be accounted for. Theunaccounted-for terms constitute the second-order obstacle,R(2). The structure of the latter depends on the choice ofcoefficients. For example, if one chooses to account for as manyterms on the r.h.s. of Eq. (2.14) as possible, the coefficientsin Eq. (3.3) obtain the values given in Appendix A (Eqs.(A.2)–(A.10)), and the unaccounted-for term is:

R(2)St = µu3ux . (3.4)

The coefficient µ is given in Eq. (A.11). The subscript Ststands for the standard analysis. The structure of this term(with µ omitted) is shown in Fig. 2 for the two-soliton case.In general, the structure of other forms of the obstacle is similarto that of R(2)

St .As the obstacle, R(2) cannot be accounted for by the NIT in

Eq. (2.14), one is forced to include it in the NF. Thus, againstone’s expectation, Eq. (2.8) has to be modified into

ut = S2[u] + εα4S3[u] + ε2{β8S4[u] + R(2)

}. (3.5)

Obstacles spoil the integrability of the NF, because theycannot be written as linear combinations (with constantcoefficients) of symmetries. Whereas Eq. (2.8) is solved by thesame single- or multiple-soliton solutions of the unperturbedequation (with updated velocities), Eq. (3.5) is not. Its solutionloses the KdV multiple-soliton structure, and may have to becomputed numerically. For example, the obstacle in the analysisof [4,8] for a two-soliton case leads to a zero-order solutionthat contains an O(ε2) radiation term, an O(ε4) time-dependentupdate of the wave numbers, and a new soliton with an O(ε4)

time-dependent wave number and an O(ε8) amplitude.The advantage of the standard analysis is that the higher-

order corrections in the NIT are differential polynomials inu, the zero-order approximation. If the latter is bounded andfalls off sufficiently rapidly for |x |, |t | → ∞ (this holds, inparticular for single- and multiple-soliton solutions, also withthe addition found in [4,8]), these polynomials are boundedthroughout the x–t plane. Namely, there are no secular terms.This is better than generally expected in the method of NF,

where the validity of higher-order approximations is expectedto hold for |t |, |x | = O(1/ε).

4. Review of NF analysis in the single-soliton case throughsecond order

For later use, we review the expansion of the solution ofEq. (2.1) for the case of the single-soliton solution of the NF,Eq. (2.8), given by Eq. (2.3). The only modification relativeto the unperturbed single-soliton solution is that the velocity isupdated according to Eq. (2.12). Only some of the coefficientsin u(1) and in u(2) of Eq. (2.7) are determined, becausedifferential monomials, which are considered independent inthe general analysis, become related. (See Eqs. (A.12) and(A.13) for examples.)

In first order, one expects there to be no obstacle in thesingle-soliton case, because, already in the general case, adifferential polynomial solution for u(1) exists, which accountsfor all the terms in Eq. (2.13) and no obstacle emerges, provideda, b and c take on the values given in Eq. (3.2). In the single-soliton case, inserting Eqs. (2.8) and (3.1) in Eq. (2.13), one ofthe coefficients, a, b or c, in Eq. (3.1) remains undetermined.This is found by direct computation [4,8]. Choosing c as thefree parameter, u(1) obtains the form

u(1)s [u] =

16(6c − 15α1 + 10α2 − 10α3 + 15α4)u

2

+16(6c + 15α1 − 20α2 − 10α3 + 15α4)

× q(1)ux + cuxx . (4.1)

(The subscript s indicates that this is the case of a single-soliton zero-order solution.)

The solution of the second-order homological equation,Eq. (2.14), is also a differential polynomial, u(2)

s [u], and noobstacles to integrability emerge [4,8]. The values of thosecoefficients in Eq. (3.3) which can be determined are given inAppendix A (Eqs. (A.14)–(A.16)); the remaining ones are free.

The first- and second-order corrections in the NIT remainbounded for both |t |, |x | < ∞ because they are differentialpolynomials in u, the latter being a single-soliton solution,which is bounded throughout the x–t plane. (Integration of Eq.(2.3) shows that the functions q(1) and q(2) are also boundedthroughout the plane.) Namely, there are no secular terms. Thisis better than generally expected in the method of NF, where thevalidity of higher-order approximations is expected to hold for|t |, |x | = O(1/ε).

5. Absence of a first-order obstacle for a general zero-ordersolution

In Section 3, we reviewed the known observation that noobstacles emerge in the first-order homological equation, Eq.(2.13), even when the zero-order solution, u, is not a singlesoliton. It is instructive to see how this comes about.

In the analysis of Eq. (2.13) in the general case, thecoefficients a, b and c of the expression for u(1) (see Eq. (3.1))receive the values given in Eq. (3.2). In the single-soliton case,

A. Veksler, Y. Zarmi / Physica D 217 (2006) 77–87 81

u(1) assumes the form given by Eq. (4.1), and there is nofirst-order obstacle, although one parameter, e.g., c, remainsfree. Let us, for the moment, adopt Eq. (4.1) for u(1) also forthe general case. Before c is assigned the value specified byEq. (3.2), some differential monomials in Eq. (2.13) are notaccounted for; an “obstacle” emerges, given by:

R(1)= γ

(1)21 R21

(γ

(1)21 =

12(6c + 15α1 − 10α3 − 5α4)

).

(5.1)

In Eq. (5.1), R21 has the form

R21 = 3u2ux + uuxxx − ux uxx = S2G1 − G2S1. (5.2)

(The notation for the subscript used for the obstacle is self-evident.) Not surprisingly, with c given in Eq. (3.2), this first-order obstacle is eliminated. One also sees why c remainsfree in the single-soliton case: As shown in the following, R21vanishes if computed for the single-soliton solution of Eq. (2.3).

6. “Canonical” obstacles

R21 is the first example of “canonical” obstacles, defined as

Rnm = SnGm − Gn Sm . (6.1)

Whenever the zero-order solution, u, is not a single soliton,the algorithm described in the following sections generates inthe higher-order analysis obstacles that are functionals of Rnm .

The importance of the canonical obstacles is that all Rnmvanish when computed for the single-soliton solution. Thisproperty is a consequence of the recursion relation, Eq. (2.9),obeyed by the symmetries, Sn . To see this, consider, first, thesingle-soliton solution of the unperturbed KdV equation, whichwe write as

u(t, x) = us(ξ0) ξ0 = x + v0t (v0 = 4k2), (6.2)

where the subscript s indicates that this is the single-solitonsolution. Consequently, the unperturbed equation, Eq. (2.2), canbe rewritten as

v0∂ξ0 us = S2[us]. (6.3)

Eq. (6.3) is a relation between the first two symmetries in theKdV hierarchy (see Eq. (2.11)):

v0S1[us] = S2[us]. (6.4)

For localized soliton solutions that vanish at ξ0 = ±∞, Gn ,defined in Eq. (2.10), obey a similar relation:

G2[us] = 3u2s + us,xx = v0G1[us] = v0us . (6.5)

Substituting Eqs. (6.4) and (6.5) in Eq. (2.9), one readilyobtains by induction

Sn[us] = (v0)n−1S1[us], Gn[us] = (v0)

n−1G1[u]. (6.6)

The single-soliton zero-order solution of Eq. (2.8) has thesame functional form as the unperturbed solution, Eq. (2.3),the only change being that the velocity is modified by the

dispersion relation, Eq. (2.12). As the derivation of Eq. (6.6)involves only u and its spatial derivatives, it remains valid forthe single-soliton solution in the perturbed case as well.

The proportionality of all symmetries to S1 implies that thecanonical obstacles, given by Eq. (6.1), vanish explicitly inthe single-soliton case. More important, in the multiple-solitoncase, the solution approaches a sum of well-separated singlesolitons at an exponential rate as the distance from the soliton-interaction region grows. Consequently, the canonical obstaclesare localized around the origin, and vanish asymptotically at thesame rate as the distance from the origin grows.

7. Shifting the second-order obstacle from the NF to theNIT

The effect of the obstacle can be shifted from the NF to theNIT if one includes in u(2) an additional term, u(2)

r (t, x), whichdepends explicitly on t and x :

u(2)= u(2)

d [u] + u(2)r (t, x). (7.1)

u(2)d is the differential polynomial of Eq. (3.3).

(Eqs. (A.2)–(A.10) provide an example of a possible choice ofthe coefficients in Eq. (3.3).) u(2)

r accounts for the second-orderobstacle, R(2), through Eq. (2.14), which is reduced to an equa-tion for u(2)

r :

∂t u(2)r (t, x) = 6∂x {uu(2)

r (t, x)} + ∂3x u(2)

r (t, x) + R(2). (7.2)

With the NF relieved of the burden of accounting for theobstacle, it retains its preferred form of Eq. (2.8) and remainsintegrable, through second order. Its solution, the zero-orderterm, is the same multiple soliton as for the unperturbedequation, with soliton parameter updating according toEq. (2.12) for each soliton. Consequently, u(2)

d , the differentialpolynomial part of u(2), is bounded. The penalty paid for theloss of integrability is that, in general, the solution of Eq. (7.2)may not be expressible as a differential polynomial in u, andmay have to be solved for numerically.

8. Localized obstacle — second-order analysis

Our last task is to ensure that the second-order term, u(2), isbounded. In the previous section, it was pointed out that thedifferential polynomial part, u(2)

d , is bounded. Thus, we still

need to ensure that u(2)r of Eq. (7.1) is also bounded. For a

general obstacle, Eq. (7.2) may generate an unbounded solutionfor u(2)

r . The obstacle R(2)St of Eq. (3.4) is representative of

this problem. This obstacle overlaps with the solution over aninfinite range in t and x , in both the single- and the multiple-soliton cases. The plot in Fig. 2 of the obstacle of R(2)

St forthe two-soliton case demonstrates this statement. This infiniteoverlap lends R(2)

St the capacity to generate an unboundedterm in the solution of Eq. (7.2). A minor problem (more ofan aesthetic value) is that this obstacle does not vanish evenwhen u is a single-soliton solution, while there ought to be noobstacles in that case [4,8]. (The reason is that some of the

82 A. Veksler, Y. Zarmi / Physica D 217 (2006) 77–87

computational steps leading, for example, to Eq. (3.4) are notpossible in the case of the single-soliton zero-order solution.)

These difficulties are resolved if u(2)d , the differential

polynomial part in Eq. (7.1), is chosen to have the structureof u(2)

s , the differential polynomial that solves the second-orderhomological equation in the case of a single-soliton solution.However, now it is computed for u, the solution of the NF inthe general case. Eq. (7.1) then becomes:

u(2)= u(2)

s [u] + u(2)r (t, x). (8.1)

With u(2) of Eq. (8.1), u(2)s accounts for all the differential

monomials in Eq. (2.14), that can be cancelled when u is asingle-soliton zero-order solution of the NF. Consequently, theterms that are left unaccounted for by u(2)

s must vanish inthe single-soliton case by construction. Namely, the resultingobstacle vanishes explicitly in the single-soliton case. As aresult, in the multiple-soliton case, the resulting obstacle isexpected to be sizable only in the soliton-interaction region, afinite domain around the origin, and to decay exponentially tozero away from the interaction region. The reason is that, awayfrom the soliton-interaction region, the zero-order solutionapproaches asymptotically a sum of distinct single solitonsat an exponential rate. Since the obstacle is now a localizeddriving term in Eq. (7.2), the solution of that equation is alsolocalized around the origin. These statements are demonstratedin Section 9, for the two-soliton case.

We use u(1)s of Eq. (3.1), with the coefficients given by

Eq. (3.2), u(2) of Eq. (8.1) and u(2)s of Eq. (3.3). The coefficients

of u(2)s are given in Eqs. (A.14)–(A.16). A number of them

remain free. The second-order homological equation, Eq.(2.14), determines some of these coefficients. The resultingcoefficients are given in the Eqs. (A.17)–(A.19). The structureof the second-order obstacle is found to be

R(2)= {Q1u + Q2∂

2x + Q3ux∂

−1x }R21 + Q4 R31. (8.2)

The numerical coefficients Qi , 1 ≤ i ≤ 4, of Eq. (8.2)are given in Eqs. (A.20)–(A.23). The construction of R(2)

from canonical obstacles is an explicit expression of the factthat the obstacle must vanish identically in the single-solitoncase. In the multiple-soliton case, it decays exponentially awayfrom the interaction region of the solitons. An appropriatechoice of the still free coefficients in Eq. (3.3) exists (given inEqs. (A.24)–(A.26)), which simplifies Eq. (8.2) to:

R(2)= −

103

µu R21. (8.3)

µ is given in Eq. (A.11). The obstacle R(2) is accounted forby u(2)

r (t, x), through Eq. (7.2).

9. Second-order NF analysis in the two-soliton case

We now present the NF analysis of Eq. (2.1) when the zero-order approximation (solution of Eq. (2.8)) is the two-solitonsolution, Eq. (2.5), in which the velocities are updated throughEq. (2.12). The two solitons, i = 1, 2, have wave numbers ki .

Fig. 3. Canonical obstacle u · R21 (Eq. (5.2)) for the two-soliton solution;parameters as in Fig. 1.

For |x |, |t | → ∞, the obstacle vanishes exponentially. (Fig. 3demonstrates this property.) Substituting the expression for thetwo-soliton solution, Eq. (2.5), in Eq. (8.3), one finds after adetailed inspection that the asymptotic behavior of the obstaclealong soliton no. 1 is:

R21 ∝ e−2|k2(x+4k22 t)|

|k1(x + 4k21 t)| → C,

|k2(x + 4k22 t)| → ∞, (9.1)

and there is a similar behavior along soliton no. 2. Far from bothsolitons, the behavior is

R21 ∝ max{e−2|k1(x+4k21 t)|−4|k2(x+4k2

2 t)|,

e−4|k1(x+4k21 t)|−2|k2(x+4k2

2 t)|}

|k1(x + 4k21 t)| → ∞, |k2(x + 4k2

2 t)| → ∞. (9.2)

The dispersion relation obeyed by these asymptotic termsdoes not resonate with the homogeneous part of Eq. (7.2).Hence, they generate for u(2)

r of Eq. (8.1) a bounded tail, whichemanates from the soliton-interaction region around the origin,and decays exponentially as the distance from the origin grows.As a result, the effect of the loss of integrability on the second-order correction of Eq. (8.1) decays asymptotically, so that theperturbed KdV equation becomes “integrable” asymptoticallyaway from the origin in the x–t plane.

10. Extension to higher orders, and to N solitons, N > 2

To see whether the new algorithm applies to higher orders,the analysis has been extended to the third order. In viewof the cumbersome nature of the computation, only the mainresults are presented. The most general third-order perturbationexpected in physical systems (see Eq. (A.27)) was addedto Eq. (2.1).

To begin with, no obstacles to integrability are encounteredwhen the zero-order solution, u, describes a single soliton.Eq. (2.12) for the eigenvalue update is modified to

v = 4k2+ εα4(4k2)2

+ ε2β8(4k2)3+ ε3γ14(4k2)4

+ O(ε4). (10.1)

A. Veksler, Y. Zarmi / Physica D 217 (2006) 77–87 83

Moreover, the solution for the third-order term, added tothe NIT, Eq. (2.7), is a differential polynomial, u(3)

s [u]. Thestructure of the most general differential polynomial allowedin this order is given in Eq. (A.28). Of the 29 coefficients, onlysix are determined in the single-soliton case.

We now turn to the multiple-soliton case. Following theprescription used in the second-order analysis (see Eq. (8.1)),we assume for the third-order term in the NIT, Eq. (2.7), theform

u(3)= u(3)

s [u] + u(3)r (t, x). (10.2)

Here u(3)s [u] is the differential polynomial obtained as the

solution of the third-order homological equation in the single-soliton case (now computed for the multiple-soliton solution).It accounts for all the terms generated by the formalism, whichexist also in the single-soliton case.

The third-order homological equation is reduced to anequation for u(3)

r . Its structure is similar to that of Eq. (7.2):

∂t u(3)r (t, x) = 6∂x {uu(3)

r (t, x)} + ∂3x u(3)

r (t, x) + R(3). (10.3)

R(3) is the third-order obstacle. It is the contribution thatis left after having accounted for that part of the driving termwhich can be eliminated in the single-soliton case. Therefore,by construction, R(3) must vanish explicitly in the single-solitoncase, and, hence, must be localized around the origin in the x–tplane in the multiple-soliton case.

However, the structure of R(3) is more complicated than thatof R(2), the driving term in the second-order equation, Eq. (7.2).R(2) is a canonical obstacle: it is a differential polynomial (seeEq. (8.2)) that vanishes explicitly in the single-soliton case, andis localized around the origin in the multiple-soliton case. R(3)

of Eq. (10.3) contains two terms:

R(3)= R(3)

d + R(3)nd . (10.4)

In Eq. (10.4), R(3)d is a differential polynomial, and R(3)

nd is

the contribution of u(2)r (t, x), the non-polynomial part of the

second-order term in the NIT (see Eq. (8.1)), to Eq. (10.3).The calculation yields that R(3)

d is a linear combination withcomputable, constant coefficients of any of the terms given inEq. (A.29). As some of the coefficients in Eq. (A.28) are stillfree, one can reduce the number of contributing terms, so that,for instance, only those involving R21 appear. The importantobservation is that, like in the second-order case (see Eq. (8.2))the differential polynomial part of the third-order obstacle isa linear functional of canonical obstacles, Rnm , operated uponby derivatives or integration with respect to x , or multiplied bydifferential monomials. All these operations guarantee that eachterm has the right weight (11) for the third-order obstacle. Thisrule also applies to the second-order obstacle of Eq. (8.2), wherethe overall weight is 9.

R(3)nd , the second component in the third-order obstacles, is

given by:

R(3)nd = 20(α4 − α2)q

(1)∂x (ux u(2)r )

+ 5{12α4uux − (3α1 − 2α2 − 2α3 − α4)uxxx }u(2)r

+ {10(α2 + 2α4)u2− 5(3α1 − 6α3 − α4)uxx }∂x u(2)

r

+ 20α3ux∂2x u(2)

r + 10α2u∂3x u(2)

r + α4∂5x u(2)

r . (10.5)

Thus, R(3)nd is a linear functional of u(2)

r , the non-polynomialpart of the second-order term in the NIT (see Eq. (8.1)).Consequently, it is a linear functional of the second-orderobstacle, R(2). Although R(3)

nd may not be expressible as

a differential polynomial in u (because u(2)r is not such a

polynomial), it is localized around the origin in the x–t plane,and falls off exponentially away from the origin, because u(2)

rhas this behavior. We have shown in the example of the two-soliton case that the non-resonant exponential fall-off of theobstacle, R(2), leads to a bounded, exponentially falling u(2)

r .Hence the effect of the driving term given by Eq. (10.5)on the third-order non-polynomial term, u(3)

r (the solution ofEq. (10.3)) will be similar.

In summary, the picture that emerges in second order recursin the third-order analysis. The localized obstacle (now morecomplicated than in second order) generates a radiative tail,which emanates from the origin, and falls off exponentiallyaway from the origin.

The picture that emerges is expected to recur in higher ordersas well. To begin with, one expects no obstacles to integrabilityto emerge in the single-soliton case. Namely, in that case, u(n),the n′th-order term in the NIT, can be a differential polynomial.This is seen by an induction argument as follows. Computingthe n′th-order perturbation for a single-soliton solution,Eq. (2.3) generates n + 2 terms of the form

sinh(k{x + v0t + x0})/ cosh2k+1(k{x + v0t + x0}),

k = 1, . . . , n + 2. (10.6)

If in all lower orders, u(k), k ≤ n − 1, have been solvedfor using differential polynomials, then these are also theonly terms contained in the driving term in the n′th-orderhomological equation. On the other hand, the number ofdifferential monomials that can contribute to u(n) grows muchmore rapidly (perhaps like n2 for high n: there are 3, 12, and29 terms in u(1), u(2), and u(3), respectively), each multipliedby a free parameter. Thus, there are ample opportunities foraccounting for the n + 2 terms of Eq. (10.6) with theseparameters.

We now turn to the multiple-soliton case. As in Eqs. (8.1)and (10.2), we write the n′th-order term in the NIT as

u(n)= u(n)

s [u] + u(n)r (t, x). (10.7)

In Eq. (10.7), u(n)s [u] is the differential polynomial that

solves the n′th-order equation in the single-soliton case, andu(n)

r (t, x) is a non-polynomial component. The homologicalequation becomes an equation for u(n)

r (t, x):

∂t u(n)r (t, x) = 6∂x {uu(n)

r (t, x)} + ∂3x u(n)

r (t, x) + R(n). (10.8)

By construction, the n′th-order obstacle, R(n), must vanishif computed for a single soliton. Hence, it must be localizedaround the origin when computed for a multiple-soliton

84 A. Veksler, Y. Zarmi / Physica D 217 (2006) 77–87

solution, and decay exponentially away from the origin. As inthird order, it will contain two contributions:

R(n)= R(n)

d + R(n)nd . (10.9)

R(n)d is the differential polynomial component, and R(n)

nd isthe non-polynomial part.

The structure of the second- and third-order obstacles leadsus to make a reasonable conjecture (for which we have noproof) regarding the structure of R(n)

d . It is expected to beexpressible as

R(n)d [u] =

∑l,p,m

γ(n)lpm f [u, q(l), ∂x , ∂

−1x ]Rpm[u]. (10.10)

The numerical coefficients, γ(n)lpm , are determined by the

coefficients that appear in the perturbation, and by the specificchoice of coefficients in the NIT. Eq. (3.3) defines q(l). Theoperator acting on the canonical obstacle, Rpm , is constructedso as to yield the correct overall weight (2n + 5) of this term.What is implied is that the canonical obstacles constitute abasis with the help of which one can expand any differentialpolynomial that vanishes explicitly in the single-soliton case,and is, hence, localized in the multiple-soliton case.

R(n)nd is expected to be a functional of all the non-polynomial

lower-order contributions to the NIT, u(k)r (t, x), with k < n. As

all u(k)r (t, x) are localized around the origin, their effect in the

n′th-order equation is concentrated around the origin.Finally, it has been demonstrated in detail that the algorithm

proposed in this paper avoids the difficulties encounteredin the standard analysis, through O(ε2) for the two-solitoncase. However, the N -soliton case, for N > 2, can beanalyzed in a similar manner, as the obstacles that are obtainedbehave in a similar manner: By construction, they must vanishexponentially away from the origin in the x–t plane, wherethe zero-order approximation asymptotes into a sum of wellseparated single solitons. Hence, the obstacles only contributein a finite domain around the origin also in the N -soliton casefor any N ≥ 2.

11. Concluding remarks

In this paper, its has been demonstrated that, order-by-order in the perturbative analysis, the effect of obstacles tointegrability in the perturbed KdV equation can be shifted fromthe NF to the NIT by allowing the higher-order corrections inthe NIT to depend explicitly on t and x . The penalty for the lossof integrability is the fact that the NIT ceases to be a sum ofdifferential polynomials in the zero-order approximation; someparts of the NIT may have to be found numerically. The gainis that the NF is constructed from symmetries only and, hence,remains integrable, and the zero-order term retains the multiple-soliton structure of the unperturbed solution. The computationwas carried through third order; the second-order analysis ofthe two-soliton case was presented in detail.

By construction, the obstacles that emerge through ouralgorithm in every order of the expansion must be localizedaround the origin, because they vanish explicitly in the case ofa single-soliton solution of the NF. The “canonical” part of theobstacles (see Eqs. (5.2), (6.1), (8.2) and (A.29)) is expressedin terms of symmetries of the unperturbed equation. The partwhich cannot be expressed as a differential polynomial in theobstacles that emerge from third order onwards (see Eqs. (10.4),(10.5) and (10.9)) also must be localized around the origin.Moreover, as the obstacles decay exponentially away fromthe interaction region in the multiple-soliton case, they cannotgenerate unbounded behavior in the higher-order corrections tothe solution, but only a decaying tail, which emanates from thesoliton-interaction region around the origin.

Our results are in agreement with the numerical studiesof [26]. That work shows that the numerical solution of theion acoustic plasma equations for a two-soliton collision isdescribed extremely well by the two-soliton solution of theKdV equation, which is derived as an approximation to theion acoustic plasma equations. The difference between the fullequations and the KdV equation can be viewed as a smallperturbation that is added to the latter. The result of [26] impliesthat the perturbation does not alter the nature of the solutionthrough second order. A dispersive wave found by [26] seemsto be consistent with a third-order effect.

The overall picture that emerges is that, at least throughthird order, an appropriate choice of the structure of the NIT(e.g., Eqs. (8.1), (10.2) and (10.7)) can be made, so that the NFremains integrable, and the effect of the resulting “canonical”obstacle to integrability can be shifted to the NIT, and confinedto a radiative tail that emanates from the origin, and decaysexponentially as the distance from the origin grows. Thismeans that, away from the origin, the effect of the loss ofintegrability is not felt in the order-by-order approximations tothe solution. Thus, “asymptotic” integrability acquires a newmeaning (which we have demonstrated in detail through secondorder): The approximate solution is given by a zero-order term,which is determined by an integrable NF (namely, constructedfrom symmetries only), and an NIT, the structure of whichtends to that of the integrable case (when no obstacles exist)asymptotically away from the origin.

Appendix A

A.1. Z (2) of Eq. (2.14)

Z (2)= 6u(1)u(1)

x + 30α1(2uux u(1)+ u2u(1)

x )

+ 10α2(uxxx u(1)+ uu(1)

xxx )

+ 20α3(uxx u(1)x + ux u(1)

xx ) + α4u(1)xxxxx

(u(1)= u(1)

[u; t, x]). (A.1)

Here derivatives of u(1)[u; t, x] with respect to x account for

its dependence on x through u as well as through the explicitdependence on x .

A. Veksler, Y. Zarmi / Physica D 217 (2006) 77–87 85

A.2. Coefficients in Eq. (3.3): Standard NF analysis

A =509

(9α21 − 3α1α2 + α2

2 − 6α1α3 + 20α2α3

− 8α23 − 21α2α4 + 8α2

4) −143

(5β2 + 30β3

− 29β4 − 20β5 + 30β6 − 20β7 + 4β8), (A.2)

C =100

3α1(α2 − α4) − 28(β4 − β8), (A.3)

D =259

(27α21 − 3α1α2 + 4α2

2 − 30α1α3

+ 58α2α3 − 24α23

+ 3α1α4 − 63α2α4 + 8α3α4 + 20α24)

−143

(10β2 + 40β3 − 41β4

− 30β5 + 45β6 − 30β7 + 6β8), (A.4)

G =259

(18α21 + 3α1α2 + 3α2

2 − 24α1α3

+ 40α2α3 − 16α23

+ 3α1α4 − 48α2α4 + 6α3α4 + 15α24)

−73(15β2 + 60β3 − 59β4 − 45β5 + 60β6

− 40β7 + 9β8), (A.5)

I =509

(3α1α2 + α22 − 7α2α4 + 3α2

4)

−143

(5β2 − 7β4 + 2β8), (A.6)

J =509

(α2 − α4)2, (A.7)

K =259

(α2 − α4)(3α1 − 2α3 + α4) −143

(β4 − β8), (A.8)

L =2572

(27α21 + 4α2

2 − 36α1α3 + 56α2α3 − 20α23 + 6α1α4

− 64α2α4 + 4α3α4 + 23α24)

−76(5β2 + 20β3 − 20β4 − 15β5 + 21β6

− 15β7 + 4β8), (A.9)

B = E = F = H = 0. (A.10)

A.3. Coefficient µ of the obstacle in standard analysis (Eqs.(3.4) and (3.5))

µ =100

9(3α1α2 + 4α2

2 − 18α1α3 + 60α2α3 − 24α23

+ 18α1α4 − 67α2α4 + 24α24)

+140

3(3β1 − 4β2 − 18β3 + 17β4 + 12β5

− 18β6 + 12β7 − 4β8). (A.11)

A.4. Examples of relations obeyed by the single-solitonsolution

∂x (uq(1)) = u2− u(q(1))2, (A.12)

uxx = −u2+ u(q(1))2. (A.13)

A.5. Coefficients in Eq. (3.3) in the single-soliton case

A =5

12(105α2

1 − 108α1α2 − 24α22 + 28α1α3

− 80α2α3 + 4α23

− 78α1α4 + 192α2α4 + 60α3α4 − 99α24)

−73(11β1 − 18β2 − 6β3 + 18β4 + 4β5 − 9β6

+ 9β7 − 9β8)

− B + C − E −43

F − G + H +43

I

− 3K + 6L , (A.14)

D =536

(225α21 − 252α1α2 − 56α2

2 + 12α1α3

− 200α2α3 − 44α23

− 162α1α4 + 528α2α4 + 300α3α4 − 351α24)

−73(8β1 − 14β2 − 8β3 + 17β4 + 2β5 − 3β6

+ 7β7 − 9β8)

− B + C − E −43

F − G + H +43

I

− 3K + 10L , (A.15)

J =572

(135α21 − 216α1α2 + 52α2

2 − 84α1α3

+ 40α2α3 + 28α23

+ 54α1α4 + 24α2α4 − 60α3α4 + 27α24)

−76(4β1 − 7β2 − 4β3 + 8β4 + β5 − 3β6 + β7)

− E −13

F + H +13

I + K − L . (A.16)

A.6. Constraints on the coefficients in Eq. (3.3) in the generalcase, with u(2)

s a differential polynomial in Eq. (8.1)

B = E = F = H = 0, (A.17)

K =259

(α1α2 − 2α2α3 − α1α4 + α2α4 + 2α3α4 − α24)

+16

C, (A.18)

L =5

72(135α2

1 − 176α1α2 − 28α22 − 84α1α3

− 40α2α3 + 28α23

+ 14α1α4 + 224α2α4 + 20α3α4 − 93α24)

+16

C +13

I. (A.19)

A.7. Coefficients of the obstacle in Eq. (8.2)

Q1 = −56(630α2

1 − 889α1α2 − 167α22 − 336α1α3

− 280α2α3 + 112α23

+ 31α1α4 + 1316α2α4 + 210α3α4 − 627α24)

+72(84β1 − 167β2 − 84β3 + 201β4 + 21β5

− 84β6 + 56β7 − 27β8)

−434

C +212

G − 24I, (A.20)

86 A. Veksler, Y. Zarmi / Physica D 217 (2006) 77–87

Q2 = −56(90α2

1 − 127α1α2 − 21α22 − 48α1α3

− 40α2α3 + 16α23

+ 13α1α4 + 168α2α4 + 30α3α4 − 81α24)

+72(12β1 − 21β2 − 12β3 + 23β4 + 3β5

− 12β6 + 8β7 − β8)

−74

C +32

G − 3I, (A.21)

Q3 = −256

(54α21 − 77α1α2 − 15α2

2 − 24α1α3

− 40α2α3 + 16α23

− α1α4 + 128α2α4 + 18α3α4 − 59α24)

+352

(8β1 − 15β2 − 12β3 + 21β4 + 5β5 − 12β6

+ 8β7 − 3β8)

−194

C +92

G − 10I, (A.22)

Q4 =56(90α2

1 − 107α1α2 − 21α22 − 48α1α3

− 40α2α3 + 16α23 − 7α1α4

+ 168α2α4 + 30α3α4 − 81α24)

−72(12β1 − 21β2 − 12β3 + 27β4 + 3β5 − 12β6

+ 8β7 − 5β8)

+54

C −32

G + 3I. (A.23)

A.8. Choice of the coefficients in Eq. (3.3) that eliminates Q2,Q3, Q4 of Eq. (8.2)

C =100

3α1(α2 − α4) −

843

(β4 − β8), (A.24)

G =259

(18α21 + 3α1α2 + 3α2

2 − 24α1α3 + 40α2α3 − 16α23

+ 3α1α4 − 48α2α4 + 6α3α4 + 15α24)

−73(15β2 + 60β3 − 59β4 − 45β5 + 60β6 − 40β7

+ 9β8), (A.25)

I =103

(6α1α2 + 3α22 − 6α1α3 + 20α2α3 − 8α2

3

+ 6α1α4 − 34α2α4 + 13α24)

+ 14(β1 − 3β2 − 6β3 + 8β4 + 4β5 − 6β6+ 4β7 − 2β8). (A.26)

A.9. Third-order perturbation added to Eq. (2.1)

630γ1u4ux + 1260γ2u(ux )3+ 2520γ3u2ux uxx

+ 1302γ4ux (uxx )2+ 420γ5u3uxxx

+ 966γ6(ux )2uxxx + 1260γ7uuxx uxxx + 756γ8uux uxxxx

+ 252γ9uxxx uxxxx + 126γ10u2uxxxxx

+ 168γ11uxx uxxxxx + 72γ12ux uxxxxxx + 18γ13uuxxxxxxx

+ γ14uxxxxxxxxx . (A.27)

The term γ14S5 is to be added to the NF (Eq. (2.8)). Thesymmetry S5 is obtained from Eq. (A.27) by setting all γk = 1.

A.10. Differential polynomial part of the third-order correc-tion, u(3), to the NIT (Eq. (2.7))

u(3)d = A01u4

+ A02u3(q(1))2+ A03u2(q(1))4

+ A04u2q(1)q(2)+ A05u2ux q(1)

+ A06u2uxx + A07u(q(1))6+ A08u(q(1))3q(2)

+ A09uux (q(1))3

+ A10uuxx (q(1))2

+ A11uq(1)q(3)+ A12uuxxx q(1)

+ A13uux q(2)

+ A14u(ux )2+ A15uuxxxx + A16(ux )

2(q(1))2

+ A17ux q(3)+ A18ux (q

(1))5+ A19ux (q

(1))2q(2)

+ A20ux uxx q(1)+ A21ux uxxx + A22(uxx )

2

+ A23uxx (q(1))4

+ A24uxx q(1)q(2)

+ A25uxxx (q(1))3

+ A26uxxx q(2)

+ A27uxxxx (q(1))2

+ A28uxxxxx q(1)+ A29uxxxxxx . (A.28)

A.11. Terms that can contribute to the differential polynomialpart of the third-order obstacle, R(3), in Eqs. (10.3) and (10.4)

q(1)q(2) R21

q(2)∂−1x R31, q(2)∂x R21, q(2)u∂−1

x R21

(q(1))4 R21

(q(1))3∂x R21, u(q(1))3∂−1x R21, q(1)∂3

x R21

(q(1))2 R31, (q(1))2u R21, (q(1))2ux∂−1x R21,

(q(1))2∂2x R21

q(1)∂x R31, q(1)∂−1x R41, q(1)∂−1

x R32, q(1)u∂x R21,

q(1)u∂−1x R31

q(1)ux R21, q(1)u2∂−1x R21

R41, R32, u R31, u∂2x R21, ux∂x R21, ux∂

−1x R31,

u2 R21

uxx R21, uux∂−1x R21, uxxx∂

−1x R21.

(A.29)

In Eq. (A.29), q(l), l = 1, 2, are defined in Eq. (3.3).

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