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

Glimpses of Soliton TheoryThe Algebra and Geometry of Nonlinear PDEs

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

Glimpses of Soliton TheoryThe Algebra and Geometry of Nonlinear PDEs

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

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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

Problems 41Suggested Reading 43

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

Problems 63Suggested Reading 65

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

Problems 91Suggested Reading 93

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

Problems 109Suggested Reading 111

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

Problems 130Suggested Reading 132

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

Problems 145Suggested Reading 147

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

Problems 165Suggested Reading 171

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

Problems 193Suggested Reading 195

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

197

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

Problems 215Suggested Reading 217

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

Problems 232Suggested Reading 234

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

Problems 246Suggested Reading 248

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 butalso electromagnetic waves and sound waves. Because of tsunamis,microwave ovens, lasers, musical instruments, acoustic considerationsin auditoriums, ship design, the collapse of bridges due to vibration,solar energy, etc., this is clearly an important subject to study andunderstand. Generally, studying waves involves deriving and solv-ing some differential equations. Since these involve derivatives offunctions, they are a part of the branch of mathematics known toprofessors as analysis and to students as calculus. But, in general,the differential equations involved are so difficult to work with thatone needs advanced techniques to even get approximate informationabout their solutions.

It was therefore a big surprise in the late 20th century when itwas realized for the first time that some of these equations are mucheasier than they first appeared. These equations that are not asdifficult as people might have thought are called “soliton equations”

ix

x Preface

because among their solutions are some very interesting ones that wecall “solitons”. The original interest in solitons was just because theybehave a lot more like particles than we would have imagined. Butshortly after that, it became clear that there was something aboutthese soliton equations that made them not only interesting, but alsoridiculously easy as compared with most other wave equations.

As we will see, in some ways it is like a magic trick. Whenyou are impressed to see a magician pull a rabbit out of a hat orsaw an assistant in half it is because you imagine these things to beimpossible. You may later learn that these apparent miracles werereally 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. Justlike the magician’s secrets, these things are not obvious to a casualobserver, and so we can understand why it might have taken math-ematicians so long to realize that they were hiding behind some ofthese 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 easilythan we can for your average differential equation.

Just as solitons have revealed to us secrets about the nature ofwaves that we did not know before (and have therefore benefited sci-ence and engineering), the study of these “tricks” of soliton theoryhas revealed hidden connections between different branches of math-ematics that also were hidden before. All of these things fall underthe category of “soliton theory”, but it is the connections betweenanalysis, algebra and geometry (more than the physical significanceof solitons) that will be the primary focus of this book. Speakingpersonally, I find the interaction of these seemingly different mathe-matical disciplines as the underlying structure of soliton theory to beunbelievably beautiful. I know that some people prefer to work withthe more general – and more difficult – problems of analysis associ-ated with more general wave phenomena, but I hope that you will beable to appreciate the very specialized structure which is unique tothe 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 thebooks on the subject are written for researchers with doctorates inmath or physics (and experience with both). And even the handful ofbooks on soliton theory intended for an undergraduate audience tendto have expectations of prerequisites that will exclude many potentialreaders.

However, it is precisely this interdisciplinary nature of solitontheory – the way it brings together material that students wouldhave learned in different math courses and its connections to scienceand engineering – that make this subject an ideal topic for a singlesemester special topics class, “capstone” experience or reading course.

This textbook was written with that purpose in mind. It assumesa minimum of mathematical prerequisites (essentially only a calculussequence and a course in linear algebra) and aims to present thatmaterial at a level that would be accessible to any undergraduatemath major.

Correspondingly, it is not expected that this book alone will pre-pare the reader for actually working in this field of research as wouldmany of the more advanced textbooks on this subject. Rather, thegoal is only to provide a “glimpse” of some of the many facets ofthe mathematical gem that is soliton theory. Experts in the fieldare 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, highergenus algebro-geometric solutions, infinite dimensional Grassmannianmanifolds, and the method of inverse scattering are barely mentionedat all. Unfortunately, I could not see a way to include these topicswithout increasing the prerequisite assumptions and the length of thebook to the point that it could no longer serve its intended purpose.Suggestions of additional reading are included in footnotes and at theend of most chapters for those readers who wish to go beyond themere introduction to this subject that is provided here.

On the Use of Technology

This textbook assumes that the reader has access to the computerprogram Mathematica. For your convenience, an appendix to thebook is provided which explains the basic use of this software andoffers “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 tomake the subject of soliton theory accessible to undergraduates. Itserves three different roles:

The solutions we find to nonlinear PDEs are to be thought of asbeing waves which change in time. Although it is hoped that read-ers will develop the ability to understand some of the simplestexamples without computer assistance, Mathematica’s ability toproduce animations illustrating the dynamics of these waves al-lows us to visualize and “understand” solutions with complicatedformulae.

We rely on Mathematica to perform some messy (but otherwisestraightforward) computations. This simplifies exposition in thebook. (For example, in the proof of Theorem 10.6 it is much eas-ier to have Mathematica demonstrate without explanation that acertain combination of derivatives of four functions is equal to theWronskian of those four functions rather than to offer a more tra-ditional proof of this fact.) In addition, some homework problemswould be extremely tedious to answer correctly if the computationshad 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 merelynote that Mathematica knows the definition of this function, call-ing it WeierstrassP[], and can therefore graph or differentiateit for us. Although it would certainly be preferable to be ableto provide the rigorous mathematical definition of these functionsand to be able to prove that it has properties (such as being dou-bly periodic), doing so would involve too much advanced analysisand/or algebraic geometry to be compatible with the goals of thistextbook.Of course, there are other mathematical software packages avail-

able. If Mathematica is no longer available or if the reader wouldprefer to use a different program for any reason, it is likely that ev-erything could be equally achieved by the other program merely byappropriately “translating” the code. Moreover, by thinking of theMathematica code provided as merely being an unusual mathematicalnotation, patiently doing all computations by hand, and referring to

Preface xiii

the suggested supplemental readings on elliptic curves, it should bepossible to fully benefit from reading this book without any computerassistance at all.

Book Overview

Chapters 1 and 2 introduce the concepts of and summarize some ofthe key differences between linear and nonlinear differential equations.For those who have encountered differential equations before, someof this may appear extremely simple. However, it should be notedthat the approach is slightly different than what one would encounterin a typical differential equations class. The representation of lineardifferential equations in terms of differential operators is emphasized,as these will turn out to be important objects in understanding thespecial nonlinear equations that are the main object of study in laterchapters. The equivalence of differential equations under a certainsimple type of change of variables is also emphasized. The computerprogram Mathematica is used in these chapters to show animations ofexact solutions to differential equations as well as numerical approx-imations to those which cannot be solved exactly. Those requiringa more detailed introduction to the use of this software may wish toconsult Appendix A.

The story of solitons is then presented in Chapter 3, beginningwith the observation of a solitary wave on a canal in Scotland byJohn Scott Russell in 1834 and proceeding through to the modernuse of solitons in optical fibers for telecommunications. In addition,this chapter poses the questions which will motivate the rest of thebook: What makes the KdV Equation (which was derived to explainRussell’s observation) so different than most nonlinear PDEs, whatother equations have these properties, and what can we do with thatinformation?

The connection between solitary waves and algebraic geometryis introduced in Chapter 4, where the contribution of Korteweg andde Vries is reviewed. They showed that under a simple assumptionabout the behavior of its solutions, the wave equation bearing theirname transforms into a familiar form and hence can be solved usingknowledge of elliptic curves and functions. The computer programMathematica 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 maywish to consult Appendix B for an overview of the basic concepts.)

The n-soliton solutions of the KdV Equation are generalizationsof the solitary wave solutions discovered by Korteweg and de Vriesbased on Russell’s observations. At first glance, they appear to belinear combinations of those solitary wave solutions, although thenonlinearity of the equation and closer inspection reveal this not tobe the case. These solutions are introduced and studied in Chapter 5.

Although differential operators were introduced in Chapter 1 onlyin the context of linear differential equations, it turns out that theiralgebraic structure is useful in understanding the KdV equation andother nonlinear equations like it. Rules for multiplying and factoringdifferential 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 derivativewith respect to t is equal to AM −MA for a certain matrix A (so itsatisfies a differential equation). This digression into linear algebrais connected to the main subject of the book in Chapter 8. Therewe rediscover the important observation of Peter Lax that the KdVEquation can be produced by using the “trick” from Chapter 7 appliednot to matrices but to a differential operator (like those in Chapter 6)of order two. This observation is of fundamental importance not onlybecause 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 partialdifferential equations which, though different in other ways, share theKdV Equation’s remarkable properties of being exactly solvable andsupporting soliton solutions.

Chapter 9 introduces the KP Equation, which is a generalizationof the KdV Equation involving one additional spatial dimension (sothat it can model shallow water waves on the surface of the oceanrather than just waves in a canal). In addition, the Hirota Bilinearversion of the KP Equation and techniques for solving it are pre-sented. Like the discovery of the Lax form for the KdV Equation, theintroduction of the Bilinear KP Equation is more important than itmay at first appear. It is not simply a method for producing solu-tions to this one equation, but a key step towards understanding thegeometric structure of the solution space of soliton equations.

Preface xv

The wedge product of a pair of vectors in a 4-dimensional spaceis introduced in Chapter 10 and used to motivate the definition of theGrassmann Cone Γ2,4. Like elliptic curves, this is an object that wasstudied by algebraic geometers before the connection to soliton theorywas known. This chapter proves a finite dimensional version of thetheorem discovered by Mikio Sato who showed that the solution set tothe Bilinear KP Equation has the structure of an infinite dimensionalGrassmannian. This is used to argue that the KP Equation (andsoliton equations in general) can be understood as algebro-geometricequations which are merely disguised as differential equations.

Some readers may choose to stop at Chapter 10, as the connectionbetween the Bilinear KP Equation and the Plücker relation for Γ2,4makes a suitable “finale”, and because the material covered in thelast 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, asis done in Chapter 11, is only possible if the reader is comfortablewith the infinite. Pseudo-differential operators are infinite series andthe KP Hierarchy involves infinitely many variables. Yet, the readerwho persists is rewarded in Chapter 12 by the power and beautyof Sato’s theory which demonstrates a complete equivalence betweenthe soliton equations of the KP Hierarchy and the infinitely manyalgebraic equations characterizing all possible Grassmann Cones.

A concluding chapter reviews what we have covered, which is onlya small portion of what is known so far about soliton theory, andalso hints at what more there is to discover. The appendices whichfollow it are a Mathematica tutorial, supplementary information oncomplex numbers, a list of suggestions for independent projects whichcan be assigned after reading the book, the bibliography, a Glossaryof Symbols and an Index.

Acknowledgements

I am grateful to the students in my Math Capstone classes at theCollege of Charleston, who were the ‘guinea pigs’ for this experiment,and who provided me with the motivation and feedback needed toget 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 photoas Figure 9.1-4.

I am pleased to acknowledge the assistance and advice of my col-leagues Annalisa Calini, Benoit Charbonneau, Tom Ivey, StéphaneLafortune, 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 introducedme to this amazing subject and offered helpful advice regarding anearly draft of this book.

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Glossary of Symbols

Ṁ Placing a “dot” over a symbol indicates thederivative of that object with respect to thetime variable t. (See page 136.)

L ◦M Multiplication of differential operators andpseudo-differential operators is indicated by thesymbol “◦”. (See pages 115, 222.)

[·, ·] The commutator of two algebraic objects isachieved by computing their product in each ofthe two orders and subtracting one from theother. It is equal to zero if and only if theobjects 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). Whenn > k this agrees with the more commondefinition 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 bedecomposed into a wedge product of k elementsof V . (See pages 200, 238.)

297

298 Glossary of Symbols

℘(z; k1, k2) The Weierstrass ℘-function is adoubly-periodic, complex analytic functionassociated to the elliptic curvey2 = 4x3 − k1x− k2. (See page 71.)

ΨDO This is the abbreviation for “pseudo-differentialoperator”, which is a generalization of thenotion 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.)

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

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

t The collection of infinitely many “timevariables” (t1, t2, t3, t4, . . .) on which solutionsof the KP and Bilinear KP Hierarchies depend.The first three are identified with the variablesx, y and t, respectively. (See page 227.)

usol(k)(x, t) The pure 1-soliton solution to the KdVEquation (3.1) which translates with speed k2

and such that the local maximum occurs atx = 0 and time t = 0. (See page 50.)

uell(c,ω,k1,k2)(x, t) A solution to the KdV Equation (3.1) written interms of the Weierstrass ℘-function ℘(z; k1, k2)which translates with speed c. (See page 85.)

Wr(f1, . . . , fn) The Wronskian determinant of the functionsf1, . . . , fn with respect to the variable x = t1.(See page 267.)

V An n-dimensional vector space with basiselements φi (1 ≤ i ≤ n). (In Chapter 10,n = 4.) (See pages 197, 235.)

φi One of the basis elements for the n-dimensionalvector 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 basisvector) 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 basisvector) of the

(nk

)-dimensional vector space W .

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

Index

Airy, George Biddell, 47, 48algebraic geometry, 53, 67, 164,

203, 248, 255AnimBurgers[], 37arXiv.org, 64autonomous 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, 222Boussinesq Equation, 159, 167,

168, 174, 193, 284Boussinesq, Joseph Valentin, 50,

159, 276Burchnall 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, 32DAlembert[], 64de Vries, Gustav, 50, 62decomposability, 200, 202, 215,

216, 238, 247differential algebra, 113differential equations, 1

animating solutions of, 13autonomous, 4, 5dispersive, 35equivalence of, 9–11linear, 4, 23, 25, 26, 29, 30,

40, 48nonlinear, 4, 35, 38, 40numerical solution, 15, 280,

282ordinary, 4partial, 4solution, 2symmetries, 66, 275

differential operators, 23–25, 52,113, 138, 140, 154, 164

addition, 115algebra of, 113

301

302 Index

factoring, 121, 132kernel, 24, 27, 28multiplication, 115, 118

dispersion, 35, 40, 48, 52dressing, 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, 82singular, 69, 71

evolution equation, 15, 226, 228Exp[], 260

Fermi, Enrico, 53Fermi-Pasta-Ulam Experiment, 53,

54, 280findK[], 128, 141, 143Fourier 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, 286Great Eastern (The), 49

Hirota derivatives, 187, 283Hirota, 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, 150inviscid, 40Inviscid Burgers’ Equation, 51Inviscid Burgers’ Equation, 36,

38isospectrality, 134, 137, 144, 145,

149, 254

Jacobian, 90, 165, 254

Kadomtsev, B.B., 178KdV[], 64, 99KdV Equation, 51KdV Equation, 50, 51, 54, 59,

62–64, 84, 85, 89, 95, 96,106, 150, 154, 165, 173

rational solutions, 63stationary solutions, 63

kernel, 24, 27, 28, 42, 122, 131,132, 138, 140, 141, 143–145, 166, 170, 225, 228–230

Korteweg, Diederik, 50, 62KP[], 175KP Equation, 90, 173, 178, 181,

183, 191, 192, 228, 233,285

rational solutions, 193, 195KP Hierarchy, 227–231, 233, 244,

245Kruskal, 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 operatorLax, Peter, 150linear differential equation, 4, 23,

25, 26, 29, 30, 40linear independence, 27, 109, 127,

129, 130, 207, 236, 237,267, 286

maketau[], 98, 99makeu[], 98, 99, 183Mathematica, 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, 259capitalization, 267complex numbers, 270defining functions, 261graphics, 263making animations, 13matrices and vectors, 265numerical approximation, 263simplifying expressions, 262

matrix exponentiation, 279MatrixExp[], 279method of characteristics, 36Module[], 262MyAnimate[], 13, 85, 174

N[], 263n-KdV Hierarchy, 156, 168, 220,

226, 227, 231, 232n-soliton, see solitonNavier-Stokes Equations, 38nicely weighted functions, 170,

187–189, 191, 192, 194,195, 204, 208, 212, 216,217, 241–243, 245, 298

nonlinear differential equation, 4Nonlinear Schrödinger Equation,

277Novikov, Sergei, 65numerical approximation, 54, 280,

282

ocean waves, 178, 179odoapply[], 119odomult[], 119odosimp[], 119–121optical solitons, 63, 278ordinary differential equation, 4

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

ParametricPlot[], 77partial differential equation, 4,

18, 51, 59, 61, 62, 89Pasta, John, 53Perring, J.K., 277Petviashvili, V.I., 178phase shift, 106, 107, 109, 111,

176phi[], 188Plot[], 263Plot3D[], 264Plücker relations, 200, 202, 204,

206, 238, 239potential function, 139, 150projective space, 71

304 Index

projective space, 286pseudo-differential operators, viii,

219–221, 224, 225, 232,298

quantum physics, 52, 53, 59, 60,139

rogue waves, 281Russell, John Scott, 45–50, 54,

55, 59, 62, 63

Sato, Mikio, 212, 248Schrödinger Operator, 53, 139,

145, 149, 150shock wave, 38SimpleEvolver[], 16Simplify[], 262Sine-Gordon Equation, 160, 169,

171singular soliton, 99, 100, 284singularity, 99, 100Skyrme, T.H.R., 277solitary wave, 46, 48, 50, 53–55,

58soliton, 55, 56, 58, 89, 177

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

interaction, 58, 103, 106singular, 99, 100, 284theory, ix, 60, 251, 253, 255

solution, differential equation, 2Spectral Curve, 165Sqrt[], 260Stokes, George Gabriel, 47, 48, 51superposition, 26, 31, 33, 40

symmetries, 66, 275

τ -function, 96, 99, 178, 181, 192,194, 205, 206, 208, 209,244

Table[], 98, 265tau-function, see τ -functionToda Lattice, 161translation, 19, 33, 46, 50traveling wave, 32, 84Tsingou, Mary, 53

Ulam, Stanislaw, 53, 54

viscosity, 39, 40

Wave Equation, 30, 32, 55, 64wedge product, 197, 198, 235Weierstrass p-function, see ℘-

functionWeierstrassHalfPeriods[], 73,

75WeierstrassInvariants[], 76,

86, 179WeierstrassP[], 72, 74, 75, 77,

86, 179WeierstrassPPrime[], 72Wronskian, 98, 127, 128, 132,

189, 192, 195, 204–208,212, 216, 217, 230, 241,245, 266, 267, 298

Wronskian[], 98Wronskian Matrix, 128, 266WronskianMatrix[], 266

Zabusky, Norman, 54, 277

For additional informationand updates on this book, visit

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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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