Elbow scattering and inverse scattering applications to LKdV and KdV

Abstract

It is well known that many integrable partial differential equations are consistency conditions of a “Lax pair” of linear differential equations whose variables x,t can be considered as coordinates in a plane. The standard inverse scattering method, and the standard direct linearization methods, are related to the scattering problem defined on the x-axis. In a previous paper, the scattering problem defined on the t-axis was considered. Here, the scattering on an unbounded line of the plane, an “elbow,” is defined and it is shown how it can yield new reconstruction methods for the solutions of the consistency condition. The results are neither as complete or as much interesting as the standard IST, but their derivation shows that defining the “wanted” class of solutions is crucial, and it helps understanding why the standard IST is so “simple.”