Existence and nonexistence of traveling waves for the Gross-Pitaevskii equation in tori

TL;DR

This paper investigates the existence of traveling wave solutions to the Gross-Pitaevskii equation on tori, establishing conditions for existence, nonexistence, and variational characterizations based on the size of the torus.

Contribution

It proves the existence of traveling waves as minimizers and mountain-pass solutions for large T, and shows only constant solutions exist for small T.

Findings

Existence of solutions for large T as minimizers.

Existence of mountain-pass solutions in the subsonic case.

Only constant solutions for small T.

Abstract

In this paper we consider traveling waves for the Gross-Pitaevskii equation which are T-periodic in each variable. We prove that if T is large enough, there exists a solution as a global minimizer of the corresponding action functional. In the subsonic case, we can use variational methods to prove the existence of a mountain-pass solution. Moreover, we show that for small T the problem admits only constant solutions.