Important Developments in Soliton Theory

562 Pages·1993·16.82 MB·English

by  A. S. Fokas, V. E. Zakharov (auth.), Professor A. S. Fokas, Professor V. E. Zakharov (eds.)

Spiinger Series in Nonlinear Dynamics
Springer Series In Nonlinear Dynamics
Solitons - Introduction and Applications
Editor: M. Lakshmanan
What Is Integrability ?
Editor: V. E. Zakharov
Rossby Vortices and Spiral Structures
By M. V. Nezlin and E. N. Snezhkin
Algebro-Geometrical Approach to Nonlinear Evolution Equations
By E. D. Belokolos, A. I. Bobenko, V. Z. Enolsky, A. R. Its and V. B.
Matveev
Darboux Transformations and Solitons
By V. B. Matveev and M. A. Salle
Optical Solitons
By F. Abdullaev, S. Darmanyan and P. Khabibullaev
Wave Turbulence Under Parametric Excitation
Applications to Magnetics
By V. S. LVov
Kolmogorov Spectra of Turbulence I Wave Turbulence
By V. E. Zakharov, V. S. LVov and G. Falkovich
Nonlinear Processes in Physics
Editors: A. S. Fokas, D. J. Kaup, A. C. Newell and V. E. Zakharov
Important Developments in Soliton Theory
Editors: A. S. Fokas and V. E. Zakharov
A. S. Fokas V. E. Zakharov (Eds.)
Important Developments
in Soliton Theory
With 59 Figures
Springer-Verlag Berlin Heidelberg GmbH
Professor A. S. Fokas
Clarkson University, Potsdam, NY 13699-5815, USA
Professor V. E. Zakharov
Landau Institute for Theoretical Physics, ul. Kosygina 2
117334 Moscow, Russia and
University of Arizona, Tucson, AZ 85721, USA
ISBN 978-3-642-63450-5 ISBN 978-3-642-58045-1 (eBook)
DOI 10.1007/978-3-642-58045-1
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© Springer-Verlag Berlin Heidelberg 1993
Originally published by Springer-Verlag Berlin Heidelberg New York in 1993
Softcover reprint of the hardcover 1st edition 1993
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Preface
The mathematical modeling of a great variety of nonlinear phenomena
arising in physics and biology leads to certain nonlinear equations. It is
quite remarkable that many of these universal equations are "integrable".
Roughly speaking, this means that one can extract from these equations
almost as much information as one can extract from the correspond
ing linear equations. Furthermore, precisely because these equations are
nonlinear, they exhibit richer phenomenology than the linear equations.
In particular many of them support solitons, i.e. localized solutions with
particle-like properties.
Following the discovery of the concepts of solitons and of integrable
behavior by Kruskal and Zabusky in the late 1960s, there was an explo
sion of results associated with integrable equations. Most of the results
obtained in the decade 1970--1980 can be found in several books.
Important developments in the theory of integrable equations con
tinued in the decade of 1980--1990. In particular this decade has been
marked by the impact of soliton theory in many diverse areas of math
ematics and physics such as algebraic geometry (the solution of the
Schottky problem), group theory (the discovery of quantum groups),
topology (the connection of Jones polynomials with integrable mod
els), quantum gravity (the connection of the KdV with matrix models)
etc. There exists no book covering the important developments of the
last decade. We have collected in this volume most of these important
developments.
This book should be of interest to a wide audience. This includes:
Experts in the field of integrable equations, since it provides a com
pendium of the up to date developments; experts in the field of nonlinear
phenomena in general, since integrable equations can be used as a first
approximation to more complex equations including those exhibiting
chaotic behavior; applied mathematicians, mathematical physists, engi
neers, and biologists, since it gives a comprehensive picture of a new
area in mathematical physics with a tremendous range of applications.
Part of this project has been supported by the Air Force Office of
Scientific Research - Arje Nachman (Program Manager), whose support
v
is gratefully acknowledged. The authors are thankful to Cindy Smith for
her help in completing this volume.
Spring 1993 .4.8. Fokas
V.E. Zakharov
VI
Contents
Introduction
By A.S. Fokas and V.E. Zakharov 1
Part I Methods of Solution
The Inverse Scattering Transform on the Line
By R. Beals, P. Deift, and X. Zhou . . . . . . . . . . . . . . . . . . . 7
C-Integrable Nonlinear Partial Differential Equations
By F. Calogero ................................ 33
Integrable Lattice Equations
By H.W. Capel and F. Nijhoff (With 1 Figure) ........... 38
The Inverse Spectral Method on the Plane
By R. Coifman and A.S. Fokas . . . . . . . . . . . . . . . . . . . . . . 58
Dispersion Relations for Nonlinear Waves
and the Schottky Problem
By B. Dubrovin ................................ 86
The Isomonodromy Method and the Painleve Equations
By A.S. Fokas and A.R. Its (With 4 Figures) ............ 99
The Cauchy Problem for Doubly Periodic Solutions
of KP-II Equation
By I. Krichever ................................ 123
Integrable Singular Integral Evolution Equations
By P.M. Santini ................................ 147
Part II Asymptotic Results
Long-Time Asymptotics
for Integrable Nonlinear Wave Equations
By P.A. Deift, A.R. Its, and X. Zhou (With 11 Figures) 181
VII
The Generation and Propagation of Oscillations
in Dispersive Initial Value Problems
and Their Limiting Behavior
By P.O. Lax, C.D. Levermore, and S. Venakides
(With 14 Figures) ............................... 205
Differential Geometry and Hydrodynamics of Soliton Lattices
By S.P. Novikov ............................... 242
Part ITI Algebraic Aspects
Bi-Hamiltonian Structures and Integrability
By A.S. Fokas and I.M. Gel'fand .................... 259
On the Symmetries of Integrable Systems
By P.G. Grinevich, A.Yu. Orlov, and E.I. Schulman ....... 283
The n-Component KP Hierarchy and Representation Theory
By V.G. Kac and J.W. van de Leur . . . . . . . . . . . . . . . . . .. 302
Compatible Brackets in Hamiltonian Mechanics
By H.P. McKean (With 1 Figure) .................... 344
Symmetries - Test of Integrability
By A.B. Shabat and A.V. Mikhailov .................. 355
Conservation and Scattering in Nonlinear Wave Systems
By V. Zakharov, A. Balk, and E. Schulman ............. 375
Part IV Quantum and Statistical Mechanical Models
The Quantum Correlation Function as the T Function
of Classical Differential Equations
By A.R. Its, A.G. Izergin, V.E. Korepin, and N.A. Slavnov 407
Lattice Models in Statistical Mechanics and Soliton Equations
By B.M. McCoy ............................... 418
Elementary Introduction to Quantum Groups
By L.A. Takhtajan .............................. 441
Knot Theory and Integrable Systems
By M. Wadati (With 11 Figures) .. . . . . . . . . . . . . . . . . . .. 468
VIII
Part V Near-Integrable Models and Computational
Aspects
Solitons and Computation
By M.J. Ablowitz and B.M. Herbst (With 10 Figures) 489
Symplectic Aspects of Some Eigenvalue Algorithms
By P. Deift, L.-c. Li, and C. Tomei (With 2 Figures) 511
Whiskered Tori for NLS Equations
By D.W. McLaughlin (With 4 Figures) ................ 537
Index of Contributors. . . . . . . . . . . . . . . . . . . . . . . . . . .. 559
IX
Introduction
A.S. Fokas and V.E. Zakharov
Many apparently disparate nonlinear systems, have the commonality that
they exhibit a similar behavior. "Chaotic" and "integrable" behaviors are
among the best studied such behaviors arising in a variety of physically impor
tant equations. This book contains some of the most important developments
in the theory of integrable equations that occurred in the last decade.
The fascinating new world of solitons and of integrable behavior was dis
covered by Kruskal and Zabusky. These investigators, in their effort to explain
some curious numerical results of Fermi-Pasta-Ulam, were lead to study the
Korteweg-deVries (KdV) equation. This equation is a nonlinear evolution
equation in one spatial and one temporal dimension. It was first derived in
1885 by Korteweg and deVries (who were trying to explain the famous exper
iments of J. Scott Russell of 1834) and had already reappeared in the 1960s
in a number of other physical contexts (plasma physics, stratified internal
waves, etc.). The KdV possesses an exponentially localized traveling-wave so
lution. Kruskal and Zabusky studied numerically the interaction properties
of these localized waves and found a unexpected behavior: After interaction,
these waves regained their initial amplitude and velocity, and the only residual
effect of interaction was a "phase shift" (i.e. a change in the position they
would have reached without interaction). This particle-like property led them
to call these waves Solitons. The next challenge was the analytical under
standing of these numerical results. The search for additional conservation
laws (which perhaps were responsible for the stability properties of solitons)
led Miura to the discovery of the modified KdV as well as to a Riccati-type
relationship between KdV and modified KdV. The linearization of this Riccati
equation, led Gardner, Green, Kruskal, and Miura to discover the connection
between the KdV and the time-independent Schrodinger scattering problem.
In this way, the solution of the Cauchy problem for the KdV had been reduced
to the reconstruction of a potential from knowledge of appropriate scattering
data: Let KdV describe the propagation of a water wave and suppose that
this wave is frozen at a given instant of time. By bombarding this water wave
with quantum particles, one can reconstruct its shape from knowledge of how
these particles scatter. In other words, the scattering data provide an alter
native description of the wave at a fixed time. The mathematical expression
of this description takes the form of a linear integral equation and was found
by Faddeev, who generalized the earlier results of Gel'fand and Levitan and
of Marchenko on the radial Schrodinger equation. The time evolution of the