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

Fabrice Béthuel; Philippe Gravejat; Didier Smets

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arXiv · 1212.5027

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

Fabrice Béthuel·Philippe Gravejat·Didier Smets

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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, Schrödinger equations do not.

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Fabrice Béthuel, Philippe Gravejat, Didier Smets. 2013-07-10. Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation. https://arxiv.org/abs/1212.5027

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