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

André de Laire (1, 2) , Philippe Gravejat (3) , Didier Smets (4)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 Paradyse - Systèmes de particules et systèmes dynamiques
3 AGM - UMR 8088 - Analyse, Géométrie et Modélisation
4 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions

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.

Mots clés: Gross-Pitaevksii equation; Travelling waves

Domaines: Equations aux dérivées partielles [math.AP]

André de Laire, Philippe Gravejat, Didier Smets. Minimizing travelling waves for the Gross-Pitaevskii equation on $\mathbb{R} \times \mathbb{T}$. Annales de la Faculté des Sciences de Toulouse. Mathématiques., In press. ⟨hal-03586112⟩