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

Fabrice Bethuel (1) , Philippe Gravejat (2) , Didier Smets (1)

1 LJLL - Laboratoire Jacques-Louis Lions
2 CMLS - Centre de Mathématiques Laurent Schwartz

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.

Mots clés: Asymptotic stability; soliton; Gross-Pitaevskii equation

Domaines: Equations aux dérivées partielles [math.AP]

Fabrice Bethuel, Philippe Gravejat, Didier Smets. Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation. 2012. ⟨hal-00767588v1⟩