The direct and inverse scattering problem for the semilinear Schr\"odinger equation

TL;DR

This paper investigates the direct and inverse scattering problems for a semilinear Schrödinger equation, establishing well-posedness for small solutions and proving the unique determination of the nonlinear term from scattering data.

Contribution

It introduces a linearization approach using multiple parameters to uniquely identify the nonlinear function in the inverse problem.

Findings

Well-posedness of the direct problem for small solutions.

Unique determination of the nonlinear function from scattering data.

Use of linearization with multiple parameters to solve the inverse problem.

Abstract

We study the direct and inverse scattering problem for the semilinear Schr\"{o}dinger equation $\Delta u+a(x,u)+k^2u=0$ in $\mathbb{R}^d$. We show well-posedness in the direct problem for small solutions based on the Banach fixed point theorem, and the solution has the certain asymptotic behavior at infinity. We also show the inverse problem that the semilinear function $a(x,z)$ is uniquely determined from the scattering data. The idea is certain linearization that by using sources with several parameters we differentiate the nonlinear equation with respect to these parameter in order to get the linear one.