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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.
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Dates and versions

hal-03586112 , version 1 (23-02-2022)

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André de Laire, Philippe Gravejat, Didier Smets. Minimizing travelling waves for the Gross-Pitaevskii equation on $\mathbb{R} \times \mathbb{T}$. 2022. ⟨hal-03586112⟩
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