Minimizing travelling waves for the one-dimensional nonlinear Schr\"odinger equation with non-zero condition at infinity

TL;DR

This paper establishes the existence and stability of traveling wave solutions for a one-dimensional nonlinear Schrödinger equation with non-zero boundary conditions at infinity, using energy minimization under fixed momentum.

Contribution

It introduces a novel variational approach to construct and prove the stability of traveling waves for this class of equations.

Findings

Existence of traveling wave solutions via energy minimization.

Proof of stability for the family of minimizers.

Application of concentration-compactness to ensure convergence.

Abstract

This paper deals with the existence of travelling wave solutions for a general one-dimensional nonlinear Schr\"odinger equation. We construct these solutions by minimizing the energy under the constraint of fixed momentum. We also prove that the family of minimizers is stable. Our method is based on recent articles about the orbital stability for the classical and non-local Gross-Pitaevskii equations [3, 10]. It relies on a concentration-compactness theorem, which provides some compactness for the minimizing sequences and thus the convergence (up to a subsequence) towards a travelling wave solution.