The scattering map determines the nonlinearity

TL;DR

This paper introduces a method to uniquely recover the nonlinearity in the nonlinear Schrödinger equation from scattering data, using inverse convolution techniques and the Beurling--Lax Theorem.

Contribution

It establishes that the scattering map uniquely determines the nonlinearity for a broad class of nonlinear dispersive equations, providing a new inverse problem solution.

Findings

The wave operator uniquely determines the nonlinearity.

The scattering map can be used to recover the nonlinearity from initial data.

Inverse convolution approach effectively solves the problem.

Abstract

Using the two-dimensional nonlinear Schr\"odinger equation (NLS) as a model example, we present a general method for recovering the nonlinearity of a nonlinear dispersive equation from its small-data scattering behavior. We prove that under very mild assumptions on the nonlinearity, the wave operator uniquely determines the nonlinearity, as does the scattering map. Evaluating the scattering map on well-chosen initial data, we reduce the problem to an inverse convolution problem, which we solve by means of an application of the Beurling--Lax Theorem.