Recovery of a cubic nonlinearity for the nonlinear Schr\"odinger equation

TL;DR

This paper demonstrates that in 2D and 3D, the nonlinear phase of solutions to a nonlinear Schrödinger equation with localized cubic nonlinearity encodes the X-ray transform of the nonlinearity, enabling its recovery.

Contribution

It establishes a method to recover a localized cubic nonlinearity in the Schrödinger equation from wave packet solutions, linking nonlinear phase to the X-ray transform.

Findings

Nonlinear phase encodes the X-ray transform of the nonlinearity.

Recovery of the nonlinearity is possible from small-amplitude wave packet data.

Results apply to dimensions two and three.

Abstract

We consider the problem of recovering a spatially-localized cubic nonlinearity in a nonlinear Schr\"odinger equation in dimensions two and three. We prove that solutions with data given by small-amplitude wave packets accrue a nonlinear phase that determines the X-ray transform of the nonlinear coefficient.