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arXiv · 2307.06015

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

André de Laire·Philippe Gravejat·Didier Smets
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Abstract

As a sequel to our previous analysis in [9] arXiv:2202.09411 on the Gross-Pitaevskii equation on the product space $\mathbb{R} \times \mathbb{T}$, we construct a branch of finite energy travelling waves as minimizers of the Ginzburg-Landau energy at fixed momentum. We deduce that minimizers are precisely the planar dark solitons when the length of the transverse direction is less than a critical value, and that they are genuinely two-dimensional solutions otherwise. The proof of the existence of minimizers is based on the compactness of minimizing sequences, relying on a new symmetrization argument that is well-suited to the periodic setting.

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André de Laire, Philippe Gravejat, Didier Smets. 2023-07-12. Construction of minimizing travelling waves for the Gross-Pitaevskii equation on $\mathbb{R} \times \mathbb{T}$. https://doi.org/10.2140/tunis.2024.6.157

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