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 $\mathbb{R}\times \mathbb{T}$, 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.

References in corpus (2)

• Travelling waves for the Gross-Pitaevskii equation II
• Orbital stability of the black soliton to the Gross-Pitaevskii equation

Cited by in corpus (2)

• Periodic traveling waves for nonlinear Schrödinger equations with non-zero conditions at infinity in ${\mathbb{R}}^2$
• Exotic traveling waves for a quasilinear Schrödinger equation with nonzero background