孤立子、非線性發(fā)展方程和逆散射

內(nèi)容概要

An exciting and extremely active area of research investigation during the past twenty years has been the study of Solitons and the related issue of the construction of solutions to a wide class of nonlinear equations. Indeed there have been a few books written which serve to review aspect of this burgeoning field. A book coauthored by one of us (MJA) exactly ten years ago, discussed many of the relevant viewpoints as well as a variety of applications. Certain important and novel subareas of research such as the the application of the Inverse Scattering Transform (I.S.T.) to solve nonlinear wave equations on the infinite interval, in one spatial and one temporal dimension (1 + 1), were described in detail. At that time the complete inverse scattering methodology had been carried out primarily for those nonlinear equations related to second order scattering problems. Although it was known that certain nonlinear evolution equations in one and two spatial dimensions were related to suitable (higher order and two dimensional) linear scattering problems, and special techniques were available, nevertheless it was not yet clear that a unified and effective procedure could be applied to all of these nonlinear equations. The main purpose of this book is the description of how the I.S.T. technique can be applied to these situations.

書籍目錄

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 dimension 　1.11 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 equations2  Inverse Scattering for the Korteweg-de Vries Equation　2.1  Introduction　2.2  The direct scattering problem　2.3  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　　2.5.4  The Gel'fand-Levitan-Mar henkcintegral equation　　2.5.3  A general class of solutions of the Korteweg-de Vries equation　2.6  Properties of completely integrable equations　　2.6.1  Solitons　　……3 General Inverse Scattering in One Dimension4 Inverse Scattering for Integro-Differential Equation5 Inverse Scattering in Two Dimensions6 Inverse Scattering in Multidimensions7 The Painleve Equations8 Further Remarks and Open ProblemsAppendixReferencesSubject Index

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