Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation
[Stabilité asymptotique dans l'espace d'énergie pour les solitons sombres de l'équation de Gross-Pitaevskii]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 6, pp. 1327-1381.

Nous poursuivons notre analyse [5] 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.

We pursue our work [5] 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.

Publié le :
DOI : 10.24033/asens.2271
Classification : 35Q55, 35B35, 35C08.
Keywords: Gross-Pitaevskii equation, soliton, asymptotic stability.
Mot clés : Équation de Gross-Pitaevskii, soliton, stabilité asymptotique. 
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     title = {Asymptotic stability  in the energy space for dark solitons  of the {Gross-Pitaevskii} equation},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1327--1381},
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Béthuel, Fabrice; Gravejat, Philippe; Smets, Didier. Asymptotic stability  in the energy space for dark solitons  of the Gross-Pitaevskii equation. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 6, pp. 1327-1381. doi : 10.24033/asens.2271. http://www.numdam.org/articles/10.24033/asens.2271/

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