Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation

TL;DR

This paper proves the asymptotic stability of dark solitons in the one-dimensional Gross-Pitaevskii equation within the energy space, without requiring weighted space smallness or spectral assumptions.

Contribution

It introduces a novel approach to establish asymptotic stability for dark solitons in Schrödinger-type equations, extending techniques from KdV equations.

Findings

Dark solitons are asymptotically stable under small energy space perturbations.

The method does not rely on weighted space smallness or spectral assumptions.

Results apply to the one-dimensional Gross-Pitaevskii equation.

Abstract

We pursue our work on the dynamical stability of dark solitons for the one-dimensional Gross-Pitaevskii equation. In this paper, we prove their asymptotic stability under small perturbations in the energy space. In particular, our results do not require smallness in some weighted spaces or a priori spectral assumptions. Our strategy is reminiscent of the one used by Martel and Merle in various works regarding generalized Korteweg-de Vries equations. The important feature of our contribution is related to the fact that while Korteweg-de Vries equations possess unidirectional dispersion, Schr\"odinger equations do not.