Multidimensional nonlinear evolution equations and inverse scattering

Abstract

In this paper we will review some recent work done in the field of integrable nonlinear evolution equations and inverse scattering. We will concentrate on the basic underlying areas and refer interested readers to suitable references for complete details; specifically background material can be found in various texts on this subject (e.g. [1] by Ablowitz and Segur). More recent references will be given as necessary. The outline of the paper is as follows. 1. 1) Introductory remarks. 2. 2) A discussion of two separate but related issues. Namely, (a) solving certain nonlinear evolutions equations in infinite space; and (b) inverse scattering. These are important problems having many physical applications. Moreover, they are related to each other by what we refer to as the Inverse Scattering Transform (IST). 3. 3) At the end of the paper we will make some remarks on the possibility of solving nonlinear evolution equations in high dimensions (i.e. equations with more than two spatial and one time variable) by using the IST method as we now understand it.