[Une nouvelle preuve d’un théorème de Liouville pour l’équation de Gross–Pitaevskii en dimension un]
The asymptotic stability of the black and dark solitons of the one-dimensional Gross–Pitaevskii equation was proved by Béthuel, Gravejat and Smets (Ann. Sci. Éc. Norm. Supér., 2015), and Gravejat and Smets (Proc. Lond. Math. Soc., 2015), using a rigidity property in the vicinity of solitons. We provide an alternate proof of the Liouville theorems in those two articles, using a factorization identity for the linearized operator which trivializes the spectral analysis.
La stabilité asymptotique des solitons de l’équation de Gross–Pitaevskii en dimension un a été démontrée par Béthuel, Gravejat et Smets (Ann. Sci. Éc. Norm. Supér., 2015), et Gravejat et Smets (Proc. Lond. Math. Soc., 2015), à l’aide d’une propriété de rigidité dans le voisinage d’un soliton. On donne une nouvelle démonstration des théorèmes de Liouville contenus dans ces articles, utilisant une identité de factorisation pour l’opérateur linéarisé qui rend triviale l’analyse spectrale du problème.
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Keywords: Solitons, Gross–Pitaevskii equation, asymptotic stability
Mots-clés : Solitons, équation de Gross–Pitaevskii, stabilité asymptotique
Michał Kowalczyk 1 ; Yvan Martel 2
CC-BY 4.0
@article{CRMATH_2025__363_G12_1339_0,
author = {Micha{\l} Kowalczyk and Yvan Martel},
title = {A new proof of a {Liouville} theorem for the one dimensional {Gross{\textendash}Pitaevskii} equation},
journal = {Comptes Rendus. Math\'ematique},
pages = {1339--1350},
year = {2025},
publisher = {Acad\'emie des sciences, Paris},
volume = {363},
doi = {10.5802/crmath.799},
language = {en},
}
TY - JOUR AU - Michał Kowalczyk AU - Yvan Martel TI - A new proof of a Liouville theorem for the one dimensional Gross–Pitaevskii equation JO - Comptes Rendus. Mathématique PY - 2025 SP - 1339 EP - 1350 VL - 363 PB - Académie des sciences, Paris DO - 10.5802/crmath.799 LA - en ID - CRMATH_2025__363_G12_1339_0 ER -
Michał Kowalczyk; Yvan Martel. A new proof of a Liouville theorem for the one dimensional Gross–Pitaevskii equation. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1339-1350. doi: 10.5802/crmath.799
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