Inverse scattering for the nonlinear Schr\"{o}dinger equation with the Yukawa potential

TL;DR

This paper addresses the inverse scattering problem for a 3D nonlinear Schrödinger equation with Yukawa potential, reconstructing potential and nonlinearity from scattering data, with applications to semi-relativistic Hartree equations.

Contribution

It introduces a method to reconstruct both the potential and nonlinearity in a nonlinear Schrödinger equation with Yukawa potential from scattering states.

Findings

Successfully reconstructs potential and nonlinearity from scattering data.

Applicable to semi-relativistic Hartree equations.

Provides a framework for inverse problems with nonlocal nonlinearities.

Abstract

We study the inverse scattering problem for the three dimensional nonlinear Schroedinger equation with the Yukawa potential. The nonlinearity of the equation is nonlocal. We reconstruct the potential and the nonlinearity by the knowledge of the scattering states. Our result is applicable to reconstructing the nonlinearity of the semi-relativistic Hartree equation.