Direct and inverse scattering for nonlinear Schrödinger equation in 2D

Abstract

We discuss the scattering theory for the nonlinear Schrödinger equation \documentclass[12pt]{minimal}\begin{document}$-\Delta u(x)\break + h(x,|u(x)|)u(x) = k^{2}u(x), x \in \mathbb {R}^2,$\end{document}−Δu(x)+h(x,|u(x)|)u(x)=k2u(x),x∈R2, where h is a very general and possibly singular combination of potentials. We prove that the direct scattering problem has a unique bounded solution. We establish also the asymptotic behaviour of scattering solutions. A uniqueness result and a representation formula is proved for the inverse scattering problem with general scattering data. The method of Born approximation is applied for the recovery of local singularities and jumps.