Solitons, nonlinear evolution equations and inverse scattering / M.J. Ablowitz and P.A. Clark.

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Includes bibliographical references.

Solitons have been of considerable interest to mathematicians since their discovery by Kruskal and Zabusky. This book brings together several aspects of soliton theory currently only available in research papers. Emphasis is given to the multi-dimensional problems arising and includes inverse scattering in multi-dimensions, integrable nonlinear evolution equations in multi-dimensions and the ∂ method. Thus, this book will be a valuable addition to the growing literature in the area and essential reading for all researchers in the field of soliton theory.

Cover; Title; Copyright; Contents; 1 Introduction; 1.1 Historical remarks and applications; 1.2 Physical Derivation of the Kadomtsev-Petviashvili equation; 1.3 Travelling wave solutions of the Korteweg-de Vries equation; 1.4 The discovery of the soliton; 1.5 An infinite number of conserved quantities; 1.6 Fourier transforms; 1.7 The associated linear scattering problem and inverse scattering; 1.7.1 The inverse scattering method; 1.7.2 Reflectionless potentials; 1.8 Lax's generalization; 1.9 Linear scattering problems and associated nonlinear evolution equations

1.10 Generalizations of the I.S.T. in one spatial dimension1.11 Classes of integrable equations; 1.11.1 Ordinary differential equations; 1.11.2 Partial differential equations in one spatial dimension; 1.11.3 Differential-difference equations; 1.11.4 Singular integro-differential equations; 1.11.5 Partial differential equations in two spatial dimensions; 1.11.6 Multidimensional scattering equations; 1.11.7 Multidimensional differential geometric equations; 1.11.8 The Self-dual Yang-Mills equations; 2 Inverse Scattering for the Korteweg-de Vries Equation; 2.1 Introduction

2.2 The direct scattering problem2.3 The inverse scattering problem; 2.4 The time dependence; 2.5 Further remarks; 2.5.1 Soliton solutions; 2.5.2 Delta-function initial profile; 2.5.3 A general class of solutions of the Korteweg-de Vries equation; 2.5.4 The Gel'fand-Levitan-Marchenko integral equation; 2.6 Properties of completely integrable equations; 2.6.1 Solitons; 2.6.2 Infinite number of conservation laws; 2.6.3 Compatibility of linear operators; 2.6.4 Completely integrable Hamiltonian system and action-angle variables; 2.6.5 Bilinear representation; 2.6.6 Backland transformations

2.6.7 Painleve property2.6.8 Prolongation structure; 3 General Inverse Scattering in One Dimension; 3.1 Inverse scattering and Riemann-Hilbert problems for N x N matrix systems; 3.1.1 The direct and inverse scattering problems: 2nd order case; 3.1.2 The direct and inverse scattering problems: iVth order case; 3.1.3 The time dependence; 3.1.4 Hamiltonian system and action-angle variables for the nonlinear Schrodinger equation; 3.1.5 Riemann-Hilbert problems for iVth order Sturm-Liouville scattering problems; 3.2 Riemann-Hilbert problems for discrete scattering problems

3.2.1 Differential-difference equations: discrete Schrodinger scattering problem3.2.2 Differential-difference equations: discrete 2 x 2 scattering problem; 3.2.3 Partial-difference equations; 3.3 Homoclinic structure and numerically induced chaos for the nonlinear Schrodinger equation; 3.3.1 Introduction; 3.3.2 A linearized stability analysis; 3.3.3 Hirota's method for the single homoclinic orbit; 3.3.4 Combination homoclinic orbits; 3.3.5 Numerical homoclinic instability; 3.3.6 Duffmg's equations and Mel'nikov analysis; 3.4 Cellular Automata

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