Inverse fixed energy scattering problem for the two-dimensional nonlinear Schroedinger operator

TL;DR

This paper addresses the inverse scattering problem for a 2D nonlinear Schrödinger equation, demonstrating unique potential recovery from fixed energy data using new estimates for the Faddeev-Green's function.

Contribution

It introduces a method to uniquely recover compactly supported potentials in 2D nonlinear Schrödinger equations using fixed energy scattering data and new analytical estimates.

Findings

Unique potential reconstruction from fixed energy data

New estimates for Faddeev-Green's function in L-infinity space

Applicability to general nonlinear index of refraction

Abstract

This work studies the direct and inverse fixed energy scattering problem for two-dimensional Schroedinger equation with rather general nonlinear index of refraction. In particular, using the Born approximation we prove that all singularities of the unknown compactly supported potential from $L^2$-space can be obtained uniquely by the scattering data with fixed positive energy. The proof is based on the new estimates for the Faddeev-Green's function in $L^\infty$-space.