Inverse Scattering Problems and Their Application to Nonlinear Integrable Equations is devoted to inverse scattering problems (ISPs) for differential equations and their application to nonlinear evolution equations (NLEEs). The book is suitable for anyone who has a mathematical background and interest in functional analysis, partial differential equations, equations of mathematical physics, and functions of a complex variable. This book is intended for a wide community working with inverse scattering problems and their applications; in particular, there is a traditional community in mathematical physics.
In this monograph, the problems are solved step-by-step, and detailed proofs are given for the problems to make the topics more accessible for students who are approaching them for the first time.
• The unique solvability of ISPs are proved. The scattering data of the considered inverse scattering problems (ISPs) are described completely.
• Solving the associated initial value problem or initial-boundary value problem for the nonlinear evolution equations (NLEEs) is carried out step-by-step. Namely, the NLEE can be written as the compatibility condition of two linear equations. The unknown boundary values are calculated with the help of the Lax (generalized) equation, and then the time-dependent scattering data (SD) are constructed from the initial and boundary conditions.
• The potentials are recovered uniquely in terms of time-dependent SD, and the solution of the NLEEs is expressed uniquely in terms of the found solutions of the ISP.
• Since the considered ISPs are solved well, then the SPs generated by two linear equations constitute the inverse scattering method (ISM). The application of the ISM to solving the NLEEs is consistent and is effectively embedded in the schema of the ISM.
Table of Contents
Chapter 1: Inverse scattering problems for systems of rst-order ODEs on a half-line
Chapter 2: Some problems for a system of nonlinear evolution equations.on a half-line
Chapter 3: Some problems for cubic nonlinear evolution equations on a half-line
Chapter 4: The Dirichlet IBVPs for sine and sinh-Gordon equations
Chapter 5: Inverse scattering for integration of the continual system of nonlinear interaction waves
Chapter 6: Some problems for the KdV equation and associated inverse scattering
Chapter 7: Inverse scattering and its application to the KdV equation with dominant surface tension
Chapter 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
Chapter 9: Connections between the inverse scattering method and related methods
Pham Loi Vu is a professor, doctor of mathematics, and a leading Vietnamese expert in inverse problems and their applications. He has authored 50 papers published in prominent journals, including Inverse Problems (Q1), Acta Applicandae Mathematicae (Q2), Journal of Nonlinear Mathematical Physics (Q2), and others.
His previous research focused on seismic waves in seismic prospecting for petroleum-gas complexes and on determining coordinates of epicenter in near earthquakes at the Institute of Geophysics of the Vietnam Academy. He is now a researcher at VAST's Institute of Mechanics and the National Foundation for Science and Technology Development (NAFOSTED).