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Glimpses of Soliton TheoryThe Algebra and Geometry of Nonlinear PDEs

Alex Kasman

STUDENT MATHEMAT ICAL L IBRARYVolume 54

stml-54-kasman-cov.indd 1 9/2/10 11:19 AM


http://dx.doi.org/10.1090/stml/054


Alex Kasman

STUDENT MATHEMAT ICAL L IBRARYVolume 54

American Mathematical SocietyProvidence, Rhode Island

Editorial Board

Gerald B. FollandRobin Forman

Brad G. Osgood (Chair)John Stillwell

2010 Mathematics Subject Classification. Primary 35Q53, 37K10, 14H70,14M15, 15A75.

Figure 9.1-6 on page 180 by Terry Toedtemeier, “Soliton in Shallow Wa-ter Waves, Manzanita-Neahkahnie, Oregon”, c©1978, used with permissionof the photographer’s estate.

For additional information and updates on this book, visitwww.ams.org/bookpages/stml-54

Library of Congress Cataloging-in-Publication Data

Kasman, Alex, 1967–Glimpses of soliton theory : the algebra and geometry of nonlinear PDEs /

Alex Kasman.p. cm. – (Student mathematical library ; v. 54)

Includes bibliographical references and index.ISBN 978-0-8218-5245-3 (alk. paper)1. Korteweg-de Vries equation. 2. Geometry, Algebraic. 3. Differential equa-

tions, Partial. I. Title.

QA377.K367 2010515′.353–dc22 2010024820

Copying and reprinting. Individual readers of this publication, and nonprofitlibraries acting for them, are permitted to make fair use of the material, such as tocopy a chapter for use in teaching or research. Permission is granted to quote briefpassages from this publication in reviews, provided the customary acknowledgment ofthe source is given.

Republication, systematic copying, or multiple reproduction of any material in thispublication is permitted only under license from the American Mathematical Society.Requests for such permission should be addressed to the Acquisitions Department,American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected].

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Printed in the United States of America.

©∞ The paper used in this book is acid-free and falls within the guidelinesestablished to ensure permanence and durability.

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10 9 8 7 6 5 4 3 2 1 15 14 13 12 11 10

ContentsPreface ix

Chapter 1. Differential Equations 1§1.1. Classification of Differential Equations 4§1.2. Can we write solutions explicitly? 5§1.3. Differential Equations as Models of Reality

§1.4. Named Equations 8§1.5. When are two equations equivalent? 9§1.6. Evolution in Time 12

Problems 18Suggested Reading 22

Chapter 2. Developing PDE Intuition 23§2.1. The Structure of Linear Equations 23§2.2. Examples of Linear Equations 30§2.3. Examples of Nonlinear Equations 35


Chapter 3. The Story of Solitons 45§3.1. The Observation 45§3.2. Terminology and Backyard Study 46§3.3. A Less-than-enthusiastic Response 47§3.4. The Great Eastern 49§3.5. The KdV Equation 49§3.6. Early 20th Century 52

v

and ealityUnr 7

vi

§3.7. Numerical Discovery of Solitons 53§3.8. Hints of Nonlinearity 57§3.9. Explicit Formulas for n-soliton Solutions 59§3.10. Soliton Theory and Applications 60§3.11. Epilogue 62


Chapter 4. Elliptic Curves and KdV Traveling Waves 67§4.1. Algebraic Geometry 67§4.2. Elliptic Curves and Weierstrass ℘-functions 68§4.3. Traveling Wave Solutions to the KdV Equation 84


Chapter 5. KdV n-Solitons 95§5.1. Pure n-soliton Solutions 95§5.2. A Useful Trick: The τ -function 96§5.3. Some Experiments 99§5.4. Understanding the 2-soliton Solution 103§5.5. General Remarks and Conclusions 109


Chapter 6. Multiplying and Factoring Differential Operators 113§6.1. Differential Algebra 113§6.2. Factoring Differential Operators 121§6.3. Almost Division 124§6.4. Application to Solving Differential Equations 125§6.5. Producing an ODO with a Specified Kernel 127


Chapter 7. Eigenfunctions and Isospectrality 133

vii

§7.1. Isospectral Matrices 133§7.2. Eigenfunctions and Differential Operators 138§7.3. Dressing for Differential Operators 140


Chapter 8. Lax Form for KdV and Other Soliton Equations 149§8.1. KdV in Lax Form 150§8.2. Finding Other Soliton Equations 154§8.3. Lax Equations Involving Matrices 159§8.4. Connection to Algebraic Geometry 164


Chapter 9. The KP Equation and Bilinear KP Equation 173§9.1. The KP Equation 173§9.2. The Bilinear KP Equation 181


Chapter 10. The Grassmann Cone Γ2,4 and the Bilinear KPEquation

197

§10.1. Wedge Products 197§10.2. Decomposability and the Plucker Relation 200§10.3. The Grassmann Cone Γ2,4 as a Geometric Object 203§10.4. Bilinear KP as a Plucker Relation 204§10.5. Geometric Objects


Chapter 11. Pseudo-Differential Operators and the KPHierarchy

219

§11.1. Motivation 219§11.2. The Algebra of Pseudo-Differential Operators 220

Nonlinear PDEs

Solutionthe ofin Sp209

aces

viii

§11.3. ΨDOs are Not Really Operators 224§11.4. Application to Soliton Theory 225


Chapter 12. The Grassmann Cone Γk,n and the Bilinear KPHierarchy

235

§12.1. Higher Order Wedge Products 235§12.2. The Bilinear KP Hierarchy 240


Chapter 13. Concluding Remarks 251§13.1. Soliton Solutions and their Applications 251§13.2. Algebro-Geometric Structure of Soliton Equations 252

Appendix A. Mathematica Guide 257§A.1. Basic Input 257§A.2. Some Notation 259§A.3. Graphics 263§A.4. Matrices and Vectors 265§A.5. Trouble Shooting: Common Problems and Errors 267

Appendix B. Complex Numbers 269§B.1. Algebra with Complex Numbers 269§B.2. Geometry with Complex Numbers 270§B.3. Functions and Complex Numbers 272

Problems 274

Appendix C. Ideas for Independent Projects 275References 289Glossary of Symbols 297Index 301

PrefaceBy covering a carefully selected subset of topics, offering detailed

explanations and examples, and with the occasional assistance of

technology, this book aims to introduce undergraduate students to a

subject normally only encountered by graduate students and

researchers. Because of its interdisciplinary nature (bringing

together different branches of mathematics as well as having

connections to science and engineering), it is hoped that this book

would be ideal for a one semester special topics class, “capstone” or

reading course.

About Soliton Theory

There are many different phenomena in the real world which we de-

scribe as “waves”. For example, consider not only water waves but

also electromagnetic waves and sound waves. Because of tsunamis,

microwave ovens, lasers, musical instruments, acoustic considerations

in auditoriums, ship design, the collapse of bridges due to vibration,

solar energy, etc., this is clearly an important subject to study and

understand. Generally, studying waves involves deriving and solv-

ing some differential equations. Since these involve derivatives of

functions, they are a part of the branch of mathematics known to

professors as analysis and to students as calculus. But, in general,

the differential equations involved are so difficult to work with that

one needs advanced techniques to even get approximate information

about their solutions.

It was therefore a big surprise in the late 20th century when it

was realized for the first time that some of these equations are much

easier than they first appeared. These equations that are not as

difficult as people might have thought are called “soliton equations”

ix

x Preface

because among their solutions are some very interesting ones that we

call “solitons”. The original interest in solitons was just because they

behave a lot more like particles than we would have imagined. But

shortly after that, it became clear that there was something about

these soliton equations that made them not only interesting, but also

ridiculously easy as compared with most other wave equations.

As we will see, in some ways it is like a magic trick. When

you are impressed to see a magician pull a rabbit out of a hat or

saw an assistant in half it is because you imagine these things to be

impossible. You may later learn that these apparent miracles were

really the result of the use of mirrors or a jacket with hidden pockets.

In soliton theory, the role of the “mirrors” and “hidden pockets”

is played by a surprising combination of algebra and geometry. Just

like the magician’s secrets, these things are not obvious to a casual

observer, and so we can understand why it might have taken math-

ematicians so long to realize that they were hiding behind some of

these wave equations. Now that the tricks have been revealed to us,

however, we can do amazing things with soliton equations. In par-

ticular, we can find and work with their solutions much more easily

than we can for your average differential equation.

Just as solitons have revealed to us secrets about the nature of

waves that we did not know before (and have therefore benefited sci-

ence and engineering), the study of these “tricks” of soliton theory

has revealed hidden connections between different branches of math-

ematics that also were hidden before. All of these things fall under

the category of “soliton theory”, but it is the connections between

analysis, algebra and geometry (more than the physical significance

of solitons) that will be the primary focus of this book. Speaking

personally, I find the interaction of these seemingly different mathe-

matical disciplines as the underlying structure of soliton theory to be

unbelievably beautiful. I know that some people prefer to work with

the more general – and more difficult – problems of analysis associ-

ated with more general wave phenomena, but I hope that you will be

able to appreciate the very specialized structure which is unique to

the mathematics of solitons.

About This Book

Because it is such an active area of research, because it has deep con-

nections to science and engineering, and because it combines many

Preface xi

different areas of mathematics, soliton theory is generally only en-

countered by specialists with advanced training. So, most of the

books on the subject are written for researchers with doctorates in

math or physics (and experience with both). And even the handful of

books on soliton theory intended for an undergraduate audience tend

to have expectations of prerequisites that will exclude many potential

readers.

However, it is precisely this interdisciplinary nature of soliton

theory – the way it brings together material that students would

have learned in different math courses and its connections to science

and engineering – that make this subject an ideal topic for a single

semester special topics class, “capstone” experience or reading course.

This textbook was written with that purpose in mind. It assumes

a minimum of mathematical prerequisites (essentially only a calculus

sequence and a course in linear algebra) and aims to present that

material at a level that would be accessible to any undergraduate

math major.

Correspondingly, it is not expected that this book alone will pre-

pare the reader for actually working in this field of research as would

many of the more advanced textbooks on this subject. Rather, the

goal is only to provide a “glimpse” of some of the many facets of

the mathematical gem that is soliton theory. Experts in the field

are likely to note that many truly important topics have been ex-

cluded. For example, symmetries of soliton equations, the Hamil-

tonian formulation, applications to science and engineering, higher

genus algebro-geometric solutions, infinite dimensional Grassmannian

manifolds, and the method of inverse scattering are barely mentioned

at all. Unfortunately, I could not see a way to include these topics

without increasing the prerequisite assumptions and the length of the

book to the point that it could no longer serve its intended purpose.

Suggestions of additional reading are included in footnotes and at the

end of most chapters for those readers who wish to go beyond the

mere introduction to this subject that is provided here.

On the Use of Technology

This textbook assumes that the reader has access to the computer

program Mathematica. For your convenience, an appendix to the

book is provided which explains the basic use of this software and

offers “troubleshooting” advice. In addition, at the time of this writ-

xii Preface

ing, a file containing the code for many of the commands and exam-

ples in the textbook can be downloaded from the publisher’s website:

www.ams.org/bookpages/stml-54.

It is partly through this computer assistance that we are able to

make the subject of soliton theory accessible to undergraduates. It

serves three different roles:

The solutions we find to nonlinear PDEs are to be thought of as

being waves which change in time. Although it is hoped that read-

ers will develop the ability to understand some of the simplest

examples without computer assistance, Mathematica’s ability to

produce animations illustrating the dynamics of these waves al-

lows us to visualize and “understand” solutions with complicated

formulae.

We rely on Mathematica to perform some messy (but otherwise

straightforward) computations. This simplifies exposition in the

book. (For example, in the proof of Theorem 10.6 it is much eas-

ier to have Mathematica demonstrate without explanation that a

certain combination of derivatives of four functions is equal to the

Wronskian of those four functions rather than to offer a more tra-

ditional proof of this fact.) In addition, some homework problems

would be extremely tedious to answer correctly if the computations

had to be computed by hand.

Instead of providing a definition of the elliptic function ℘(z; k1, k2)that is used in Chapter 4 and deriving its properties, we merely

note that Mathematica knows the definition of this function, call-

ing it WeierstrassP[], and can therefore graph or differentiate

it for us. Although it would certainly be preferable to be able

to provide the rigorous mathematical definition of these functions

and to be able to prove that it has properties (such as being dou-

bly periodic), doing so would involve too much advanced analysis

and/or algebraic geometry to be compatible with the goals of this

textbook.

Of course, there are other mathematical software packages avail-

able. If Mathematica is no longer available or if the reader would

prefer to use a different program for any reason, it is likely that ev-

erything could be equally achieved by the other program merely by

appropriately “translating” the code. Moreover, by thinking of the

Mathematica code provided as merely being an unusual mathematical

notation, patiently doing all computations by hand, and referring to

Preface xiii

the suggested supplemental readings on elliptic curves, it should be

possible to fully benefit from reading this book without any computer

assistance at all.

Book Overview

Chapters 1 and 2 introduce the concepts of and summarize some of

the key differences between linear and nonlinear differential equations.

For those who have encountered differential equations before, some

of this may appear extremely simple. However, it should be noted

that the approach is slightly different than what one would encounter

in a typical differential equations class. The representation of linear

differential equations in terms of differential operators is emphasized,

as these will turn out to be important objects in understanding the

special nonlinear equations that are the main object of study in later

chapters. The equivalence of differential equations under a certain

simple type of change of variables is also emphasized. The computer

program Mathematica is used in these chapters to show animations of

exact solutions to differential equations as well as numerical approx-

imations to those which cannot be solved exactly. Those requiring

a more detailed introduction to the use of this software may wish to

consult Appendix A.

The story of solitons is then presented in Chapter 3, beginning

with the observation of a solitary wave on a canal in Scotland by

John Scott Russell in 1834 and proceeding through to the modern

use of solitons in optical fibers for telecommunications. In addition,

this chapter poses the questions which will motivate the rest of the

book: What makes the KdV Equation (which was derived to explain

Russell’s observation) so different than most nonlinear PDEs, what

other equations have these properties, and what can we do with that

information?

The connection between solitary waves and algebraic geometry

is introduced in Chapter 4, where the contribution of Korteweg and

de Vries is reviewed. They showed that under a simple assumption

about the behavior of its solutions, the wave equation bearing their

name transforms into a familiar form and hence can be solved using

knowledge of elliptic curves and functions. The computer program

Mathematica here is used to introduce the Weierstrass ℘-functionand its properties without requiring the background in complex anal-

ysis which would be necessary to work with this object unassisted.

xiv Preface

(Readers who have never worked with complex numbers before may

wish to consult Appendix B for an overview of the basic concepts.)

The n-soliton solutions of the KdV Equation are generalizations

of the solitary wave solutions discovered by Korteweg and de Vries

based on Russell’s observations. At first glance, they appear to be

linear combinations of those solitary wave solutions, although the

nonlinearity of the equation and closer inspection reveal this not to

be the case. These solutions are introduced and studied in Chapter 5.

Although differential operators were introduced in Chapter 1 only

in the context of linear differential equations, it turns out that their

algebraic structure is useful in understanding the KdV equation and

other nonlinear equations like it. Rules for multiplying and factoring

differential operators are provided in Chapter 6.

Chapter 7 presents a method for making an n × n matrix Mdepending on a variable t with two interesting properties: its eigen-

values do not depend on t (the matrix is isospectral) and its derivative

with respect to t is equal to AM −MA for a certain matrix A (so it

satisfies a differential equation). This digression into linear algebra

is connected to the main subject of the book in Chapter 8. There

we rediscover the important observation of Peter Lax that the KdV

Equation can be produced by using the “trick” from Chapter 7 applied

not to matrices but to a differential operator (like those in Chapter 6)

of order two. This observation is of fundamental importance not only

because it provides an algebraic method for solving the KdV Equa-

tion, but also because it can be used to produce and recognize othersoliton equations. By applying the same idea to other types of oper-

ators, we briefly encounter a few other examples of nonlinear partial

differential equations which, though different in other ways, share the

KdV Equation’s remarkable properties of being exactly solvable and

supporting soliton solutions.

Chapter 9 introduces the KP Equation, which is a generalization

of the KdV Equation involving one additional spatial dimension (so

that it can model shallow water waves on the surface of the ocean

rather than just waves in a canal). In addition, the Hirota Bilinear

version of the KP Equation and techniques for solving it are pre-

sented. Like the discovery of the Lax form for the KdV Equation, the

introduction of the Bilinear KP Equation is more important than it

may at first appear. It is not simply a method for producing solu-

tions to this one equation, but a key step towards understanding the

geometric structure of the solution space of soliton equations.

Preface xv

The wedge product of a pair of vectors in a 4-dimensional space

is introduced in Chapter 10 and used to motivate the definition of the

Grassmann Cone Γ2,4. Like elliptic curves, this is an object that was

studied by algebraic geometers before the connection to soliton theory

was known. This chapter proves a finite dimensional version of the

theorem discovered by Mikio Sato who showed that the solution set to

the Bilinear KP Equation has the structure of an infinite dimensional

Grassmannian. This is used to argue that the KP Equation (and

soliton equations in general) can be understood as algebro-geometric

equations which are merely disguised as differential equations.

Some readers may choose to stop at Chapter 10, as the connection

between the Bilinear KP Equation and the Plucker relation for Γ2,4

makes a suitable “finale”, and because the material covered in the

last two chapters necessarily involves a higher level of abstraction.

Extending the algebra of differential operators to pseudo-differen-

tial operators and the KP Equation to the entire KP Hierarchy, as

is done in Chapter 11, is only possible if the reader is comfortable

with the infinite. Pseudo-differential operators are infinite series and

the KP Hierarchy involves infinitely many variables. Yet, the reader

who persists is rewarded in Chapter 12 by the power and beauty

of Sato’s theory which demonstrates a complete equivalence between

the soliton equations of the KP Hierarchy and the infinitely many

algebraic equations characterizing all possible Grassmann Cones.

A concluding chapter reviews what we have covered, which is only

a small portion of what is known so far about soliton theory, and

also hints at what more there is to discover. The appendices which

follow it are a Mathematica tutorial, supplementary information on

complex numbers, a list of suggestions for independent projects which

can be assigned after reading the book, the bibliography, a Glossary

of Symbols and an Index.

Acknowledgements

I am grateful to the students in my Math Capstone classes at the

College of Charleston, who were the ‘guinea pigs’ for this experiment,

and who provided me with the motivation and feedback needed to

get it in publishable form.

Thanks to Prudence Roberts for permission to use Terry Toedte-

meier’s 1978 photo “Soliton in Shallow Water Waves, Manzanita-

Neahkahnie, Oregon” as Figure 9.1-6 and to the United States Army

xvi Preface

Air Corps whose public use policy allowed me to reproduce their photo

as Figure 9.1-4.

I am pleased to acknowledge the assistance and advice of my col-

leagues Annalisa Calini, Benoit Charbonneau, Tom Ivey, Stephane

Lafortune, Brenton Lemesurier, Hans Lundmark, and Oleg Smirnov.

This book would not have been possible without the advice and sup-

port of Ed Dunne, Cristin Zannella, Luann Cole, the anonymous ref-

erees and the rest of the editorial staff at the AMS. And thanks espe-

cially to Emma Previato, my thesis adviser, who originally introduced

me to this amazing subject and offered helpful advice regarding an

early draft of this book.

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Glossary of Symbols

M Placing a “dot” over a symbol indicates the

derivative of that object with respect to the

time variable t. (See page 136.)

L ◦M Multiplication of differential operators and

pseudo-differential operators is indicated by the

symbol “◦”. (See pages 115, 222.)

[·, ·] The commutator of two algebraic objects is

achieved by computing their product in each of

the two orders and subtracting one from the

other. It is equal to zero if and only if the

objects commute. (See page 118.)

(n

k

)The binomial coefficient is defined asn(n−1)(n−2)···(n−k+1)

k! (or 1 if k = 0). When

n > k this agrees with the more common

definition n!k!(n−k)! but extends it to the case

n < k. (See pages 115, 222.)

v ∧ w The “wedge product” of vectors takes kelements of V to an element of W . (See pages198, 235.)

Γk,n The set of vectors in W which can be

decomposed into a wedge product of k elements

of V . (See pages 200, 238.)

297

298 Glossary of Symbols

℘(z; k1, k2) The Weierstrass ℘-function is a

doubly-periodic, complex analytic function

associated to the elliptic curve

y2 = 4x3 − k1x− k2. (See page 71.)

ΨDO This is the abbreviation for “pseudo-differential

operator”, which is a generalization of the

notion of a differential operator. (See page 220.)

ϕ(n)λ A “nicely weighted function” of the variables x,

y and t satisfying (9.6). (See page 188.)

ϕ(n)λ A “nicely weighted function” of the variables

t = (t1, t2, . . .) satisfying (12.2). (See page 241.)

t The collection of infinitely many “time

variables” (t1, t2, t3, t4, . . .) on which solutions

of the KP and Bilinear KP Hierarchies depend.

The first three are identified with the variables

x, y and t, respectively. (See page 227.)

usol(k)(x, t) The pure 1-soliton solution to the KdV

Equation (3.1) which translates with speed k2

and such that the local maximum occurs at

x = 0 and time t = 0. (See page 50.)

uell(c,ω,k1,k2)(x, t) A solution to the KdV Equation (3.1) written in

terms of the Weierstrass ℘-function ℘(z; k1, k2)which translates with speed c. (See page 85.)

Wr(f1, . . . , fn) The Wronskian determinant of the functions

f1, . . . , fn with respect to the variable x = t1.(See page 267.)

V An n-dimensional vector space with basis

elements φi (1 ≤ i ≤ n). (In Chapter 10,

n = 4.) (See pages 197, 235.)

φi One of the basis elements for the n-dimensional

vector space V (1 ≤ i ≤ n). (In Chapter 10,

n = 4.) (See pages 197, 235.)

Glossary of Symbols 299

Φ An arbitrary element (not necessarily a basis

vector) of the n-dimensional vector space V .

(In Chapter 10, n = 4.) (See pages 197, 235.)

W An(nk

)-dimensional vector space with basis

elements ωi1···ik (1 ≤ i1 < i1 < · · · < ik ≤ n).(In Chapter 10, k = 2 and n = 4.) (See pages197, 235.)

ωi1···ik One of the basis elements for the(nk

)-dimensional vector space W

(1 ≤ i1 < i1 < · · · < ik ≤ n). (In Chapter 10,

k = 2 and n = 4.) (See pages 197, 235.)

Ω An arbitrary element (not necessarily a basis

vector) of the(nk

)-dimensional vector space W .

(In Chapter 10, k = 2 and n = 4.) (See pages197, 235.)

Index

Airy, George Biddell, 47, 48

algebraic geometry, 53, 67, 164,

203, 248, 255

AnimBurgers[], 37

arXiv.org, 64

autonomous differential equation,

4, 5, 21, 81

Bilinear KP Equation, 181–183,

185, 187, 188, 204, 206–

210, 212, 214, 215, 240,

242

Bilinear KP Hierarchy, 240–244,

283

bilinearKP[], 182, 187, 196,

206, 207, 214

binomial coefficient, 115, 222

Boussinesq Equation, 159, 167,

168, 174, 193, 284

Boussinesq, Joseph Valentin, 50,

159, 276

Burchnall and Chaundy, 53, 164,

165

commutator, 118, 121, 130, 137,

139, 146, 152, 155, 160,

162

complex conjugate, 277

complex numbers, 78, 86, 269,

270, 272, 277, 284

cross product, 237, 246

D’Alembert, 30, 32

DAlembert[], 64

de Vries, Gustav, 50, 62

decomposability, 200, 202, 215,

216, 238, 247

differential algebra, 113

differential equations, 1

animating solutions of, 13

autonomous, 4, 5

dispersive, 35

equivalence of, 9–11

linear, 4, 23, 25, 26, 29, 30,

40, 48

nonlinear, 4, 35, 38, 40

numerical solution, 15, 280,

282

ordinary, 4

partial, 4

solution, 2

symmetries, 66, 275

differential operators, 23–25, 52,

113, 138, 140, 154, 164

addition, 115

algebra of, 113

301

302 Index

factoring, 121, 132

kernel, 24, 27, 28

multiplication, 115, 118

dispersion, 35, 40, 48, 52

dressing, 131, 136, 140, 146, 152,

154, 174, 220, 228, 245,

252

eigenfunction, 138–140, 143, 144,

146, 147

elliptic curves, 50, 53, 68, 70, 77,

80, 89

group law, 82

singular, 69, 71

evolution equation, 15, 226, 228

Exp[], 260

Fermi, Enrico, 53

Fermi-Pasta-Ulam Experiment, 53,

54, 280

findK[], 128, 141, 143

Fourier Analysis, 32

gauge transformation, 185, 186,

194, 248

Gelfand-Levitan-Marchenko Inver-

sion Formula, 150

Grassmann Cone, 200, 203, 204,

206, 209, 238, 253, 254

Grassmannian, 204, 205, 286

Great Eastern (The), 49

Hirota derivatives, 187, 283

Hirota, Ryogo, 187

initial profile, 13, 15–20, 23, 34,

36–38, 41, 51, 54, 86,

103, 157, 168

internal waves, 281

intertwining, 131, 134, 135, 140,

141, 143, 146, 166, 170,

278

invariant subspace, 140, 152, 166,

191

inverse scattering, 59, 150

inviscid, 40

Inviscid Burgers’ Equation, 51

Inviscid Burgers’ Equation, 36,

38

isospectrality, 134, 137, 144, 145,

149, 254

Jacobian, 90, 165, 254

Kadomtsev, B.B., 178

KdV[], 64, 99

KdV Equation, 51

KdV Equation, 50, 51, 54, 59,

62–64, 84, 85, 89, 95, 96,

106, 150, 154, 165, 173

rational solutions, 63

stationary solutions, 63

kernel, 24, 27, 28, 42, 122, 131,

132, 138, 140, 141, 143–

145, 166, 170, 225, 228–

230

Korteweg, Diederik, 50, 62

KP[], 175

KP Equation, 90, 173, 178, 181,

183, 191, 192, 228, 233,

285

rational solutions, 193, 195

KP Hierarchy, 227–231, 233, 244,

245

Kruskal, Martin, 54, 277

Lax Equation, 150, 152, 155, 158,

165, 219, 225, 226, 229,

252, 277

Index 303

Lax operator, 151, 160, 162, 220,

277

Lax Pair, see Lax operator

Lax, Peter, 150

linear differential equation, 4, 23,

25, 26, 29, 30, 40

linear independence, 27, 109, 127,

129, 130, 207, 236, 237,

267, 286

maketau[], 98, 99

makeu[], 98, 99, 183

Mathematica, xi–xiii, xv, 13, 16,

17, 19, 20, 30, 37, 38,

64, 72–74, 76–78, 80,

85–87, 89, 92, 98, 99,

101, 103, 104, 110, 119–

121, 127–129, 143, 145–

147, 167, 168, 175, 176,

179, 182–184, 186–188,

191–195, 206, 207, 209,

210, 213, 214, 217, 233,

242, 244, 247, 257–268,

270, 272, 274, 278, 279,

281, 282, 284, 285

arithmetic, 259

capitalization, 267

complex numbers, 270

defining functions, 261

graphics, 263

making animations, 13

matrices and vectors, 265

numerical approximation, 263

simplifying expressions, 262

matrix exponentiation, 279

MatrixExp[], 279

method of characteristics, 36

Module[], 262

MyAnimate[], 13, 85, 174

N[], 263

n-KdV Hierarchy, 156, 168, 220,

226, 227, 231, 232

n-soliton, see soliton

Navier-Stokes Equations, 38

nicely weighted functions, 170,

187–189, 191, 192, 194,

195, 204, 208, 212, 216,

217, 241–243, 245, 298

nonlinear differential equation, 4

Nonlinear Schrodinger Equation,

277

Novikov, Sergei, 65

numerical approximation, 54, 280,

282

ocean waves, 178, 179

odoapply[], 119

odomult[], 119

odosimp[], 119–121

optical solitons, 63, 278

ordinary differential equation, 4

℘-function, 71, 72, 74–77, 80, 84,95, 179

ParametricPlot[], 77

partial differential equation, 4,

18, 51, 59, 61, 62, 89

Pasta, John, 53

Perring, J.K., 277

Petviashvili, V.I., 178

phase shift, 106, 107, 109, 111,

176

phi[], 188

Plot[], 263

Plot3D[], 264

Plucker relations, 200, 202, 204,

206, 238, 239

potential function, 139, 150

projective space, 71

304 Index

projective space, 286

pseudo-differential operators, viii,

219–221, 224, 225, 232,

298

quantum physics, 52, 53, 59, 60,

139

rogue waves, 281

Russell, John Scott, 45–50, 54,

55, 59, 62, 63

Sato, Mikio, 212, 248

Schrodinger Operator, 53, 139,

145, 149, 150

shock wave, 38

SimpleEvolver[], 16

Simplify[], 262

Sine-Gordon Equation, 160, 169,

171

singular soliton, 99, 100, 284

singularity, 99, 100

Skyrme, T.H.R., 277

solitary wave, 46, 48, 50, 53–55,

58

soliton, 55, 56, 58, 89, 177

n-soliton, 56, 59, 60, 95, 96,178, 284

interaction, 58, 103, 106

singular, 99, 100, 284

theory, ix, 60, 251, 253, 255

solution, differential equation, 2

Spectral Curve, 165

Sqrt[], 260

Stokes, George Gabriel, 47, 48, 51

superposition, 26, 31, 33, 40

symmetries, 66, 275

τ -function, 96, 99, 178, 181, 192,194, 205, 206, 208, 209,

244

Table[], 98, 265

tau-function, see τ -functionToda Lattice, 161

translation, 19, 33, 46, 50

traveling wave, 32, 84

Tsingou, Mary, 53

Ulam, Stanislaw, 53, 54

viscosity, 39, 40

Wave Equation, 30, 32, 55, 64

wedge product, 197, 198, 235

Weierstrass p-function, see ℘-function

WeierstrassHalfPeriods[], 73,

75

WeierstrassInvariants[], 76,

86, 179

WeierstrassP[], 72, 74, 75, 77,

86, 179

WeierstrassPPrime[], 72

Wronskian, 98, 127, 128, 132,

189, 192, 195, 204–208,

212, 216, 217, 230, 241,

245, 266, 267, 298

Wronskian[], 98

Wronskian Matrix, 128, 266

WronskianMatrix[], 266

Zabusky, Norman, 54, 277

For additional informationand updates on this book, visit

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AMS on the Webwww.ams.orgSTML/54

Solitons are explicit solutions to nonlinear partial differential equa-tions exhibiting particle-like behavior. This is quite surprising, both mathematically and physically. Waves with these properties were once believed to be impossible by leading mathematical physicists, yet they are now not only accepted as a theoretical possibility but are regularly observed in nature and form the basis of modern fi ber-optic commu-nication networks.

Glimpses of Soliton Theory addresses some of the hidden mathematical connections in soliton theory which have been revealed over the last half-century. It aims to convince the reader that, like the mirrors and hidden pockets used by magicians, the underlying algebro-geometric structure of soliton equations provides an elegant and surprisingly simple explanation of something seemingly miraculous.

Assuming only multivariable calculus and linear algebra as prereq-uisites, this book introduces the reader to the KdV Equation and its multisoliton solutions, elliptic curves and Weierstrass -functions, the algebra of differential operators, Lax Pairs and their use in discov-ering other soliton equations, wedge products and decomposability, the KP Equation and Sato’s theory relating the Bilinear KP Equation to the geometry of Grassmannians.

Notable features of the book include: careful selection of topics and detailed explanations to make this advanced subject accessible to any undergraduate math major, numerous worked examples and thought-provoking but not overly-diffi cult exercises, footnotes and lists of suggested readings to guide the interested reader to more information, and use of the software package Mathematica® to facili-tate computation and to animate the solutions under study. This book provides the reader with a unique glimpse of the unity of math-ematics and could form the basis for a self-study, one-semester special topics, or “capstone” course.

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