Nonlinear Waves And Inverse Scattering Transform

Nonlinear waves are essential phenomena in scientific and engineering disciplines. The features of nonlinear waves are usually described by solutions to nonlinear partial differential equations, which are fundamental to students and researchers.This book provides students, who are familiar with nonlinear waves, methods for solving nonlinear partial differential equations, enabling them to expand their studies into other related areas. The selection of topics and the focus given to each provide essential materials for a lecturer to cover the bases in a nonlinear wave course.Chapter 1 introduces 'Mode' types in terms of the nonlinear wave forms as well as the generic nonlinear partial differential equations (NLPDEs). B cklund transform is introduced to solve different types of NLPDEs for stationary solutions. Chapters 2 and 3 are devoted to the derivation and solution characterization of three generic nonlinear equations: nonlinear Schrodinger equation, Korteweg-de Vries (KdV) equation, and Burgers equation. Chapter 4 is devoted to the inverse scattering transform (IST) which addresses the initial value problems of a group NLPDEs. Lax and AKNS equations, which represent a NLPDE, subject to a specific pair of operators are introduced. K-dV equation is adopted to illustrate methods to determine the pair of operators, which lead to two auxiliary equations, one is in the form of linear Schrodinger equation to define the scattering by the solution function of the NLPDE; the second one is a rate equation, which updates the scattering data having the initial condition of the NLPDE as the scatterer. IST then reconstructs the potential function (scatterer) of the Schrodinger equation via the scattering data. In Chapter 5, derivations and proofs of the IST formulas are presented. Steps of applying IST to solve NLPDE for solitary solutions are illustrated in Chapter 6.