The scattering problem for a noncommutative nonlinear Schr\"odinger equation

TL;DR

This paper studies the scattering behavior of a noncommutative nonlinear Schr"odinger equation, establishing global solutions, solitons, and a scattering framework in a Moyal-deformed setting across multiple dimensions.

Contribution

It introduces a scattering theory for a Moyal-deformed nonlinear Schr"odinger equation, including decay estimates and wave operator construction for small and arbitrary data.

Findings

Global well-posedness for general initial data

Existence of soliton solutions under certain potentials

Development of a scattering framework with decay estimates

Abstract

We investigate scattering properties of a Moyal deformed version of the nonlinear Schr\"odinger equation in an even number of space dimensions. With rather weak conditions on the degree of nonlinearity, the Cauchy problem for general initial data has a unique globally defined solution, and also has soliton solutions if the interaction potential is suitably chosen. We demonstrate how to set up a scattering framework for equations of this type, including appropriate decay estimates of the free time evolution and the construction of wave operators defined for small scattering data in the general case and for arbitrary scattering data in the rotationally symmetric case.