Inverse Scattering Problems and Their Application to Nonlinear Integrable Equations

Pham Loi Vu

• Publisher: Chapman and Hall/CRC
• Year: 2023
• Print ISBN: 9781032429212
• eBook ISBN: 9781000872125
• Topic: Mathematics
• Subtopic: Arithmetic
• Index: Mathematics

Inverse Scattering Problems and Their Applications to Nonlinear Integrable Equations, Second Edition is devoted to inverse scattering problems (ISPs) for differential equations and their applications to nonlinear evolution equations (NLEEs). The book is suitable for anyone who has a mathematical background and interest in functional analysis, differential equations, and equations of mathematical physics. This book is intended for a wide community working with ISPs and their applications. There is an especially strong traditional community in mathematical physics.

In this monograph, the problems are presented step-by-step, and detailed proofs are given for considered problems to make the topics more accessible for students who are approaching them for the first time.

New to the Second Edition

• All new chapter dealing with the Bäcklund transformations between a common solution of both linear equations in the Lax pair and the solution of the associated IBVP for NLEEs on the half-line
• Updated references and concluding remarks

Features

• Solving the direct and ISP, then solving the associated initial value problem (IVP) or initial-boundary value problem (IBVP) for NLEEs are carried out step-by-step. The unknown boundary values are calculated with the help of the Lax (generalized) equations, then the time-dependent scattering data (SD) are expressed in terms of preassigned initial and boundary conditions. Thereby, the potential functions are recovered uniquely in terms of the given initial and calculated boundary conditions. The unique solvability of the ISP is proved and the SD of the scattering problem is described completely. The considered ISPs are well-solved.
• The ISPs are set up appropriately for constructing the B?ckhund transformations (BTs) for solutions of associated IBVPs or IVPs for NLEEs. The procedure for finding a BT for the IBVP for NLEEs on the half-line differs from the one used for obtaining a BT for non-linear differential equations defined in the whole space.
• The interrelations between the ISPs and the constructed BTs are established to become new powerful unified transformations (UTs) for solving IBVPs or IVPs for NLEEs, that can be used in different areas of physics and mechanics. The application of the UTs is consistent and efficiently embedded in the scheme of the associated ISP.

Table of contents

• Cover Page
• Half-Title Page
• Series Page
• Title Page
• Copyright Page
• Dedication Page
• Contents
• Acronyms
• Preface
• Preface for the second edition
• Author
• Introduction
• 1 Inverse Scattering Problems for Systems of First-Order ODEs on A Half-Line
• 2 Some Problems for A System of Non-Linear Evolution Equations on A Half-Line
• 3 Some Problems for Cubic Non-Linear Evolution Equations on A Half-Line
• 4 The Dirichlet IBVPs for sine and sinh-Gordon Equations
• 5 Inverse Scattering for Integration of The Continual System of Non-Linear Interaction Waves
• 6 Some Problems for The KdV Equation and Associated Inverse Scattering
• 7 Inverse Scattering and Its Application to The KdV Equation with Dominant Surface Tension
• 8 The Inverse Scattering Problem for The Perturbed String Equation and Its Application to Integration of The Two-Dimensional Generalization from Korteweg-de Vries Equation
• 9 Connections Between the Inverse Scattering Method and Related Methods
• 10 The Bäcklund Transformations Between A Common Solution of Both Linear Equations in The Lax Pair and The Solution of The Associated IBVP for NLEEs on The Half-Line
• Bibliography
• Index