Existence and properties of travelling waves for the Gross-Pitaevskii equation

TL;DR

This paper reviews recent rigorous results on the existence and properties of travelling wave solutions to the Gross-Pitaevskii equation in two and three dimensions, highlighting differences from classical nonlinear Schrödinger equations.

Contribution

It provides a comprehensive overview of recent mathematical advances in understanding travelling waves for the Gross-Pitaevskii equation, especially in higher dimensions.

Findings

Existence of travelling wave solutions in 2D and 3D

Qualitative properties of these solutions

Connection to classical results in 1D

Abstract

This paper presents recent results concerning the existence and qualitative properties of travelling wave solutions to the Gross-Pitaevskii equation posed on the whole space R^N. Unlike the defocusing nonlinear Schr\"odinger equations with null condition at infinity, the presence of non-zero conditions at infinity yields a rather rich and delicate dynamics. We focus on the case N = 2 and N = 3, and also briefly review some classical results on the one-dimensional case. The works we survey provide rigorous justifications to the impressive series of results which Jones, Putterman and Roberts established by formal and numerical arguments.