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Construction of minimizing traveling waves for the Gross–Pitaevskii equation on $\mathbb{R} \times \mathbb{T}$

André de Laire, Philippe Gravejat and Didier Smets

Vol. 6 (2024), No. 1, 157–188

As a sequel to our previous analysis of the Gross–Pitaevskii equation on the product space × 𝕋, we construct finite energy traveling waves as minimizers of the Ginzburg–Landau energy at fixed momentum. The proof of the compactness of minimizing sequences relies on a new symmetrization argument, well-suited to the periodic setting.

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Gross–Pitaevskii equation, traveling waves
Mathematical Subject Classification
Primary: 35C07, 35C08, 35J20, 35Q55, 37K40
Received: 11 July 2023
Accepted: 3 October 2023
Published: 20 January 2024
André de Laire
Laboratoire Paul Painlevé
Université de Lille
Cité Scientifique
Villeneuve d’Ascq
Philippe Gravejat
Laboratoire Analyse Géométrie Modélisation
CY Cergy Paris Université
Didier Smets
Laboratoire Jacques-Louis Lions
Sorbonne Université