Minimizing travelling waves for the Gross-Pitaevskii equation on $\mathbb{R} \times \mathbb{T}$

TL;DR

This paper investigates the minimization of the Ginzburg-Landau energy for the Gross-Pitaevskii equation on a mixed domain, identifying a threshold in the domain size that determines the structure of energy minimizers.

Contribution

It establishes a threshold for the domain size L that dictates whether minimizers are one-dimensional dark solitons or not, advancing understanding of wave minimization in mixed geometries.

Findings

Existence of a threshold L for minimizer structure.

Below threshold, minimizers are one-dimensional dark solitons.

Above threshold, minimizers are not one-dimensional.

Abstract

We study the Gross-Pitaevskii equation in dimension two with periodic conditions in one direction, or equivalently on the product space $\mathbb{R} \times \mathbb{T}_L$ where $L > 0$ and $\mathbb{T}_L = \mathbb{R} / L \mathbb{Z}.$ We focus on the variational problem consisting in minimizing the Ginzburg-Landau energy under a fixed momentum constraint. We prove that there exists a threshold value for $L$ below which minimizers are the one-dimensional dark solitons, and above which no minimizer can be one-dimensional.