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

André de Laire, Philippe Gravejat, Didier Smets

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.

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

Abstract

As a sequel to our previous analysis in [9] arXiv:2202.09411 on the Gross-Pitaevskii equation on the product space , 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.
Paper Structure (17 sections, 17 theorems, 211 equations)

This paper contains 17 sections, 17 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Sketch of the proof of Theorem \ref{['thm:exist-min']}
  3. Properties of the minimizing energy
  4. The concentration-compactness argument
  5. The analysis of the minimizing energy
  6. Proof of Lemma \ref{['lem:prop-I-l']}
  7. Proof of Lemma \ref{['lem:presymmetry']}
  8. Proof of Lemma \ref{['lem:deriv-0']}
  9. Proof of Proposition \ref{['prop:prop-I-l']}
  10. The compactness of the minimizing sequences
  11. Proof of Lemma \ref{['lem:conv-const']}
  12. Proof of Lemma \ref{['lem:lions']}
  13. Proof of Lemma \ref{['lem:farfromone']}
  14. Proof of Lemma \ref{['lem:vaud']}
  15. Proof of Proposition \ref{['prop:tessin']}
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\ell > 0$ and $\mathfrak{p} \in \mathbb{R} / \pi \mathbb{Z}$. The minimization problem $\mathcal{I}_{\textup{2d}}(\mathfrak{p})$ is achieved by some function $U_{\mathfrak{p}, \ell} \in X(\mathbb{R} \times \mathbb{T}_\ell)$. Moreover, $U_{\mathfrak{p}, \ell}$ is smooth on $\mathbb{R} \times \ma

Theorems & Definitions (23)

  • Theorem 1
  • Lemma 1
  • Proposition 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Definition
  • ...and 13 more