Construction of minimizing travelling waves for the Gross-Pitaevskii equation on ℝ×𝕋
arxiv(2023)
摘要
As a sequel to our previous analysis in [9] arXiv:2202.09411 on the
Gross-Pitaevskii equation on the product space ℝ×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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