Asymptotic analysis of soliton problems. An inverse scattering approach
type: Item
instance of: scholarly article
title: Asymptotic analysis of soliton problems. An inverse scattering approach
zbMATH Open document ID: 0643.35003
author: Peter Cornelis Schuur
published in: Lecture Notes in Mathematics
publication date: 1986-01-01T00:00:00Z
review text: This book is devoted to the inverse scattering technique applied to the analysis of the asymptotic behaviour for large time of solutions of soliton problems for various classes of nonlinear partial differential equations. More exactly the main purpose of this book is to give a complete and rigorous description of the emergence of solitons from various classes of equations on coordinate regions where the dispersive component of the solution is sufficiently small. The book contains 8 chapters. Chapter 1 presents a rigorous demonstration of the emergence of solitons from the Korteweg-de Vries initial value problem with arbitrary real initial functions; in chapter 2 existence and uniqueness results for Korteweg-de Vries equation are discussed and asymptotic analysis from chapter 1 is extended onto more complicated and interesting cases. Chapter 3 deals with multisoliton solutions to Korteweg-de Vries equations. Chapter 4 deals with Zakharov-Shabat systems and the question how well a real potential in such a system is approximated by its reflectionless part; the corresponding results are applied to the Korteweg-de Vries equation in chapter 5 and 6; in particular the remarkably similar and important general phase shift formula is presented. Chapter 7 deals with the asymptotic analysis of the sine-Gordon equation. The last chapter 8 is devoted to Zakharov-Shabat systems with complex potential. In a small appendix the author explains that his method cannot apply in the case when the associated group velocity is not of constant sign.
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35R30
zbMATH DE Number: 4048171
zbMATH Keywords: inverse scattering technique
zbMATH Keywords: large time
zbMATH Keywords: soliton
zbMATH Keywords: dispersive component
zbMATH Keywords: Korteweg-de Vries
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: multisoliton
zbMATH Keywords: Zakharov-Shabat systems
zbMATH Keywords: phase shift formula
zbMATH Keywords: sine-Gordon equation
zbMATH Keywords: complex potential
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0073054
OpenAlex ID: W276754816
DOI: 10.1007/BFB0073054
description: scientific article; zbMATH DE number 4048171