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.
Nous considérons l’équation de Gross–Pitaevskii en dimension deux pour des fonctions périodiques dans une direction, soit de façon équivalente dans l’espace produit $\mathbb{R} \times \mathbb{T}_L$, où $L > 0$ et $\mathbb{T}_L = \mathbb{R} / L \mathbb{Z}$. Nous nous intéressons au problème variationnel qui consiste à minimiser l’équation de Ginzburg–Landau à moment fixé. Nous montrons qu’il existe une valeur critique pour la largeur $L$ en dessous de laquelle les minimiseurs sont les solitons sombres à une variable, et au-dessus de laquelle aucun minimiseur ne peut dépendre que d’une seule variable.
Accepté le :
Publié le :
André de Laire 1 ; Philippe Gravejat 2 ; Didier Smets 3
CC-BY 4.0
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author = {Andr\'e de Laire and Philippe Gravejat and Didier Smets},
title = {Minimizing travelling waves for the {Gross{\textendash}Pitaevskii} equation on $\mathbb{R} \times \mathbb{T}$},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {135--192},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 34},
number = {1},
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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, Série 6, Tome 34 (2025) no. 1, pp. 135-192. doi : 10.5802/afst.1808. https://afst.centre-mersenne.org/articles/10.5802/afst.1808/
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