glimpses of soliton theory - american mathematical society · glimpses of soliton theory the...
TRANSCRIPT
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
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].
c© 2010 by the American Mathematical Society. All rights reserved.The American Mathematical Society retains all rightsexcept those granted to the United States Government.
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.
Visit the AMS home page at http://www.ams.org/
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
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 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
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 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.
References
References[1] Adler, M.; van Moerbeke, P. “Hermitian, symmetric and symplectic
random ensembles: PDEs for the distribution of the spectrum”, Ann.of Math., (2) 153 (2001), no. 1, 149–189.
[2] Airy, G.B. Tides and waves, Encyc. Metrop., 192 (1845), pp. 241–
396.
[3] Andrianov, A.A.; Borisov, N.V.; Eides, M.I., Ioffe, M.V. “The Factoriza-
tion Method and Quantum Systems with Equivalent Energy Spectra”,
Physics Letters, 105A (1984), pp. 19–22.
[4] Ascher, U. M.; McLachlan, R. I. “On symplectic and multisymplectic
schemes for the KdV equation”, J. Sci. Comput., 25 (2005), no. 1–2,
83–104.
[5] Belokolos, E.D; Bobenko, A.I.; Enol’skii, V.Z; Its, A.R; Matveev, V. B.
Algebro-Geometric Approach to Nonlinear Integrable Equa-tions, Springer Series in Nonlinear Dynamics, Springer-Verlag. 1994.
[6] N. Benes, A. Kasman, K. Young “On Decompositions of the KdV 2-
Soliton”, The Journal of Nonlinear Science, 16 (2006), 2, 179–200.
[7] Biondini, G.; Kodama, Y. “On a family of solutions of the Kadomtsev-
Petviashvili equation which also satisfy the Toda lattice hierarchy”, J.Phys. A, 36 (2003), no. 42, 10519—10536.
[8] Biondini, G.; Chakravarty, S. “Elastic and inelastic line-soliton solutions
of the Kadomtsev-Petviashvili II equation”, Math. Comput. Simulation,74 (2007), no. 2–3, 237–250.
[9] Biondini, G.; Maruno, K.-I.; Oikawa, Mo.; Tsuji, H. “Soliton interac-
tions of the Kadomtsev-Petviashvili equation and generation of large-
amplitude water waves”, Stud. Appl. Math., 122 (2009), no. 4, 377–394.
289
290 References
[10] Bocher, M. “The theory of linear dependence”, Ann. of Math., (2) 2
(1900/01), 81–96.
[11] Bostan, A.; Dumas, Ph. “Wronskians and Linear Independence”, to
appear in American Mathematical Monthly,.
[12] Bogdanov, L. V.; Zakharov, V. E. “The Boussinesq equation revisited”,
Phys. D, 165 (2002), no. 3–4, 137–162.
[13] Boussinesq, J.V. Theorie de l’ecoulement tourbillonnant et tu-multueux des liquides dans les lits rectilignes a grande section,Gauthier-Villars et fils (1897).
[14] Boyce, William E.; DiPrima, Richard C. Elementary differentialequations and boundary value problems, John Wiley & Sons, Inc.,
New York-London-Sydney 1965.
[15] Bullough, R. K.; Caudrey, P. J. “Solitons and the Korteweg-de Vries
equation”, Acta Appl. Math., 39 (1995), no. 1–3, 193–228.
[16] Burchnall, J.L.; Chaundy; T.W. “Commutative Ordinary Differential
Operators”, Proc. R. Soc. Lond. A, April 2, 1928, 118, 557–583.
[17] Crighton, D.G.. “Applications of KdV”, Acta Appl. Math., 39 (1995),
no. 1-3, 39–67.
[18] Devaney, R.L. An introduction to chaotic dynamical systems,Second edition. Addison-Wesley Studies in Nonlinearity. Addison-
Wesley Publishing Company, Advanced Book Program, Redwood City,
CA, 1989.
[19] Devaney, R.; Blanchard, P.; Hall, G. Differential Equations, Brooks/ Cole Publishing, 2004.
[20] Dickey, L.A. Soliton Equations and Hamiltonian Systems, World
Scientific Press, 1991.
[21] Doktorov, E.V.; Leble, S.B. A dressing method in mathematicalphysics, Mathematical Physics Studies, 28. Springer, Dordrecht, 2007.
[22] Drazin, P.G., Johnson, R.S., Solitons: an introduction, Cambridge
Univ. Press, 1989.
[23] Fefferman, C.L. “Existence and Smoothness of the Navier-Stokes Equa-
tion”, http://claymath.org/millennium/Navier-Stokes Equations/.
[24] Fermi, E., Pasta, J., Ulam, S. “Studies of Nonlinear Problems. I”, in
Nonlinear Wave Motion, Lectures in Applied Math., vol. 15, Amer.
Math. Soc., 1974, pp. 143-145.
[25] Filippov, A.V., The Versatile Soliton, Birkhauser, (2000).
[26] Foda, O.; Wheeler, M.; Zuparic, M. “XXZ scalar products and KP”,
Nucl. Phys., B820, (2009), 649–663.
References 291
[27] Francoise, J.-P. “Symplectic geometry and soliton theory”, Topics insoliton theory and exactly solvable nonlinear equations (Ober-wolfach, 1986), 300–306, World Sci. Publishing, Singapore, 1987.
[28] Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M., “Method for
solving the Korteweg-de Vries equation”, Physics Rev. Lett., 19, (1967),1095–1097.
[29] Gekhtman, M.; Kasman, A. “On KP generators and the geometry of
the HBDE”, J. Geom. Phys., 56, (2006), no. 2, 282–309.
[30] Gesztesy, F.; Holden, H. Soliton equations and their algebro-geometric solutions. Vol. I. (1 + 1)-dimensional continuousmodels, Cambridge Studies in Advanced Mathematics, 79, Cambridge
University Press, Cambridge, 2003.
[31] Goff, D.R.; Hansen, K.S Fiber Optic Reference Guide: a practicalguide to communications technology, Focal Press (2002).
[32] Griffiths, P.; Harris, J. Principles of algebraic geometry, Reprint
of the 1978 original, Wiley Classics Library, John Wiley & Sons, Inc.,
New York, 1994. .
[33] Hasegawa, A. “Optical solitons in communications: From integrability
to controllability”, Acta Appl. Math., 39, (1995), no. 1-3, 85–90.
[34] Hodge, W. V. D.; Pedoe, D. Methods of algebraic geometry. Vol.II. Book III: General theory of algebraic varieties in projectivespace. Book IV: Quadrics and Grassmann varieties, Reprint of
the 1952 original, Cambridge Mathematical Library, Cambridge Univer-
sity Press, Cambridge, 1994.
[35] Holmes, M.H. Introduction to numerical methods in differen-tial equations, Texts in Applied Mathematics, 52, Springer, New York,
2007.
[36] Ince, E. L. Ordinary Differential Equations, Dover Publications,
New York, 1944.
[37] Hammack J., Scheffner N. and Segur H., “Two-dimensional periodic
waves in shallow water”, J. Fluid Mech., 209, 567–589, (1989).
[38] Hammack J., McCallister D., Scheffner N. and Segur H., “Two-dimensional
periodic waves in shallow water. II. Asymmetric waves”, J. Fluid Mech.,285, 95-122 (1995) .
[39] Hartshorne, R. Algebraic geometry, Graduate Texts in Mathematics,
No. 52, Springer-Verlag, New York-Heidelberg, 1977.
[40] Hirota, R. The direct method in soliton theory, Cambridge Tracts
in Mathematics, 155, Cambridge University Press, Cambridge, 2004.
292 References
[41] Hulek, K. Elementary Algebraic Geometry, Amer. Math. Soc.,
Providence, 2003.
[42] Kac, V. G.; Raina, A. K. Bombay lectures on highest weight rep-resentations of infinite-dimensional Lie algebras, Advanced Seriesin Mathematical Physics, 2, World Scientific Publishing Co., Inc., Tea-
neck, NJ, 1987.
[43] Kadomtsev B.B. and Petviashvili V.I., “On the stability of solitary
waves in weakly dispersive media”, Sov. Phys. Dokl., 15, 539-541
(1970).
[44] Kaplansky, I. An Introduction to Differential Algebra, ActualitesSci. Ind., No. 1251 Publ. Inst. Math. Univ. Nancago, No. 5 Hermann,
Paris 1957.
[45] Kasman, A. “Bispectral KP solutions and linearization of Calogero-
Moser particle systems”, Comm. Math. Phys., 172, (1995), no. 2,
427–448.
[46] Kasman, A. “Orthogonal polynomials and the finite Toda lattice”, J.Math. Phys., 38, (1997), no. 1, 247–254.
[47] Kasman, A. “Kernel inspired factorizations of partial differential oper-
ators”, J. Math. Anal. Appl., 234, (1999), no. 2, 580–591.
[48] Kasman, A.; Pedings, K.; Reiszl, A.; Shiota, T. “Universality of rank 6
Plucker relations and Grassmann cone preserving maps”, Proc. Amer.Math. Soc., 136 (2008), no. 1, 77–87.
[49] Knobel, R.An introduction to the mathematical theory of waves,Student Mathematical Library, 3, IAS/Park City Mathematical Sub-
series, American Mathematical Society, Providence, RI; Institute for
Advanced Study (IAS), Princeton, NJ, 2000.
[50] Korteweg, D.J., de Vries, G., “On the change of form of long waves
advancing in a rectangular canal, and on a new type of long stationary
waves”, Philos. Mag. Ser. 5, 39 (1895), 422-443.
[51] Lakshmanan, M. “Integrable nonlinear wave equations and possible con-
nections to tsunami dynamics”, Tsunami and nonlinear waves, 31–49, Springer, Berlin, 2007.
[52] Lax, P.D., “Integrals of nonlinear equations of evolution and solitary
waves”, Comm. Pure. Appl. Math., 21 (1968), 467-490.
[53] Lax, P.D.; Phillips R.S., Scattering Theory for Automorphic Func-tions, Princeton University Press, 1976.
[54] Levi, D.; Ragnisco, O. “Dressing method and Bcklund and Darboux
transformations”, Backlund and Darboux transformations. Thegeometry of solitons (Halifax, NS, 1999), 29–51, CRM Proc. Lec-ture Notes, 29, Amer. Math. Soc., Providence, RI, 2001.
References 293
[55] Macdonald, I. G. Symmetric functions and Hall polynomials, Ox-
ford Mathematical Monographs. The Clarendon Press, Oxford Univer-
sity Press, New York, 1979.
[56] Matveev, V. B.; Salle, M. A. Darboux transformations and soli-tons, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin,1991.
[57] May, R. The best possible time to be alive. The logistic map, It mustbe beautiful, 28–45, Granta Pub., London, 2003.
[58] McKean, H.; Moll, V. Elliptic curves. Function theory, geometry,arithmetic, Cambridge University Press, Cambridge, 1997.
[59] Mason L.J.; Singer M.A.; Woodhouse, N.M.J. “Tau functions and the
twistor theory of integrable systems”, J. Geom. and Phys., 32 (2000),
397–430.
[60] Miwa, T.; Jimbo, M.; Date, E. Solitons. Differential equations, sym-metries and infinite-dimensional algebras. Translated from the1993 Japanese original by Miles Reid, Cambridge Tracts in Math-ematics, 135. Cambridge University Press, Cambridge, 2000.
[61] McAdams, A.; Osher, S.; Teran, J. “Awesome Explosions, Turbulent
Smoke, and Beyond: Applied Mathematics and Scientific Computing in
the Visual Effects Industry”, Notices of the AMS, 57 (2010) 614–623.
[62] Munoz Porras, J. M.; Plaza Martın, F. J. “Equations of the moduli of
pointed curves in the infinite Grassmannian”, J. Differential Geom., 51,no. 3, 431–469 (1999).
[63] Novikov, S.P., Integrability in Mathematics and Theoretical Physics:Solitons, Mathematical Intelligencer 4 (1992) 13–21.
[64] Olver, P.J. Applications of Lie groups to differential equations,Graduate Texts in Mathematics, 107. Springer-Verlag, New York, 1986.
[65] Okounkov, A. “Infinite wedge and random partitions”, Selecta Math.(N.S.), 7 (2001), no. 1, 57–81.
[66] Osborne, A. R. “The generation and propagation of internal solitons in
the Andaman Sea”, Soliton theory: a survey of results, 152–173,Nonlinear Sci. Theory Appl., Manchester Univ. Press, Manchester,
1990.
[67] Palais, R.S. “The symmetries of solitons”, Bull. Amer. Math. Soc.(N.S.), 34 (1997), no. 4, 339–403. .
[68] Pego, R. Letter to the Editor, Notices of AMS, vol. 45, March 1998,
358.
[69] Previato, E. “Seventy years of spectral curves: 1923–1993”, Integrablesystems and quantum groups (Montecatini Terme, 1993), 419–481, Lecture Notes in Math., 1620, Springer, Berlin, 1996.
294 References
[70] Rebbi, C.; Soliani, G.; Solitons and Particles, World Scientific Pub.,
1985.
[71] Reid, M. Undergraduate algebraic geometry, London Mathemati-
cal Society Student Texts, 12, Cambridge University Press, Cambridge,
1988.
[72] Remoissenet, M. Waves called solitons. Concepts and experi-ments. Third edition Springer-Verlag, Berlin, 1999.
[73] Ritt, J.F. Differential Algebra, American Mathematical Society Col-
loquium Publications, Vol. XXXIII, AMS, 1950.
[74] Russell, J.S. “Report on Waves”, Report of the fourteenth meeting of theBritish Association for the Advancement of Science, York, September
1844 (London 1845), pp. 311-390.
[75] Sagan, B.E. The Symmetric Group: Representations, Combi-natorial Algorithms, and Symmetric Functions, (2nd Edition),
Graduate Texts in Mathematics, Springer, 2001.
[76] Sato, M.; Sato, Y. “Soliton equations as dynamical systems on infinite-
dimensional Grassmann manifold”, Nonlinear partial differentialequations in applied science (Tokyo, 1982), 259–271, North-Holland
Math. Stud., 81, North-Holland, Amsterdam, 1983.
[77] Shiota, T. “Characterization of Jacobian varieties in terms of soliton
equations”, Invent. Math., 83 (1986), 333–382.
[78] Scott, A. C. Davydov’s soliton Solitons (Tiruchirapalli, 1987), 343–358, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1988.
[79] Segal, G. “Integrable systems and inverse scattering”, Integrable sys-tems, (Oxford, 1997), 53–119, Oxf. Grad. Texts Math., 4, Oxford
Univ. Press, New York, 1999.
[80] Segal, G.; Wilson, G. “Loop groups and equations of KdV type”, Inst.Hautes Etudes Sci. Publ. Math., No. 61, (1985), 5–65.
[81] Silverman, J.H. The arithmetic of elliptic curves, Graduate Texts
in Mathematics, 106, Springer-Verlag, New York, 1986.
[82] Sogo, K “A Way from String to Soliton: Introduction of KP Coordinate
to String Amplitudes”, J. Phys. Soc. Japan, 56 (1987), 2291–2297.
[83] Sreelatha, K.S.; Parameswar, L.; Joseph, K. Babu; “Optical Computing
and Solitons”, AIP Conf. Proc, 1004, (2008), 294–298.
[84] Stokes G.G. “On the theory of oscillatory waves”, Transactions of theCambridge Philosophical Society, 8, (1847), 441–455.
[85] Stuhlmeier, R. “KdV theory and the Chilean tsunami of 1960”, DiscreteContin. Dyn. Syst. Ser. B, 12, (2009), no. 3, 623–632.
References 295
[86] Takhtajan, L.A. Quantum mechanics for mathematicians, Gradu-ate Studies in Mathematics, 95, American Mathematical Society, Prov-
idence, RI, 2008.
[87] Toda, M. “Solitons in Discrete Systems”, Future Directions of Non-linear Dynamics in Physical and biological Systems, (NATO ASI
Series B Physics Volume 312), Plenum Press, 1993, 37–43.
[88] Tracy, C.A.; Widom, H. “Introduction to random matrices”, LectureNotes in Physics,, 424, (1993), 103–130.
[89] Treves, F. Introduction to Pseudo Differential and Fourier Inte-gral Operators, University Series in Mathematics, Plenum Publ. Co.,
1981.
[90] Tsuchiya, S; Dalfovo, F.; Pitaevski, L. “Solitons in two-dimensional
Bose-Einstein condensates”, Phys. Rev. A, 77, 045601, (2008).
[91] Ulam, S. Collected Papers of Enrico Fermi, University of Chicago
Press, (1965).
[92] Whitham, G.B. Linear and Nonlinear Waves, Pure & Applied Math-
ematics, A Wiley Interscience Series of Texts, Monographs & Tracts,
John Wiley & Sons, 1973.
[93] Vanhaecke, P. “Stratifications of hyperelliptic Jacobians and the Sato
Grassmannian”, Acta Appl. Math., 40, (1995), 143–172.
[94] Wallace, P. Paradox Lost: Images of the Quantum, Springer-
Verlag, 1996.
[95] Whittaker, E. T.; Watson, G. N. A course of modern analysis. Anintroduction to the general theory of infinite processes and ofanalytic functions; with an account of the principal transcen-dental functions., Reprint of the fourth (1927) edition, Cambridge
Mathematical Library, Cambridge University Press, Cambridge, 1996.
[96] Yakushevich, L. V.; Savin, A. V.; Manevitch, L. I. “Nonlinear dynamics
of topological solitons in DNA”, Phys. Rev. E, (3), 66, (2002), no. 1,
016614, 14 pp..
[97] Zabrodin, A. “Discrete Hirota’s equation in integrable models”, Int. J.Mod. Phys., B11, (1997), 3125–3158.
[98] Zabrodin, A. “A Survey of Hirota’s Bilinear Difference Equation”,
www.arxiv.org/solv-int/9704001.
[99] Zabusky, N.J., Kruskal, M.D., “Interaction of solitons in a collisionless
plasma and the recurrence of initial states”, Physics Rev. Lett., 15,
(1965), 240–243.
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
www.ams.org/bookpages/stml-54
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.
stml-54-kasman-cov.indd 1 9/2/10 11:19 AM