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

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.

References in corpus (2)

• Travelling waves for the Gross-Pitaevskii equation II
• Orbital stability of the black soliton to the Gross-Pitaevskii equation