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Annales scientifiques de l'ENS - Parutions - série 4, 48 (2015)

Parutions < série 4, 48

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 48, fascicule 6 (2015)

Fabrice Béthuel, Philippe Gravejat, Didier Smets
Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation
Annales scientifiques de l'ENS 48, fascicule 6 (2015), 1327-1381

Télécharger cet article : Fichier PDF

Résumé :
Stabilité asymptotique dans l'espace d'énergie pour les solitons sombres de l'équation de Gross-Pitaevskii
Nous poursuivons notre analyse [Ann. Inst. Fourier 64 (2014)] de la stabilité dynamique des solitons sombres pour l'équation de Gross-Pitaevskii en dimension un. Dans cet article, nous démontrons leur stabilité asymptotique par rapport à de petites perturbations dans l'espace d'énergie. En particulier, nos résultats ne requièrent aucune condition de petitesse dans des espaces à poids, aussi bien qu'aucune hypothèse spectrale a priori. Notre stratégie s'appuie sur celle développée par Martel et Merle dans plusieurs articles au sujet des équations de Korteweg-de Vries généralisées. Notre contribution principale réside dans le fait que les équations de Korteweg-de Vries possèdent une dispersion unidirectionnelle, ce qui n'est plus le cas des équations de Schrödinger.

Mots-clefs : Équation de Gross-Pitaevskii, soliton, stabilité asymptotique.

Abstract:
We pursue our work [Ann. Inst. Fourier 64 (2014)] 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.

Keywords: Gross-Pitaevskii equation, soliton, asymptotic stability.

Class. math. : 35Q55, 35B35, 35C08.


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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