A new proof of a Liouville theorem for the one dimensional Gross–Pitaevskii equation

Michał Kowalczyk; Yvan Martel

doi:10.5802/crmath.799

Research article - Partial differential equations

A new proof of a Liouville theorem for the one dimensional Gross–Pitaevskii equation

Michał Kowalczyk ¹; Yvan Martel ²

¹ Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
² Laboratoire de mathématiques de Versailles, UVSQ, Université Paris-Saclay, CNRS, and Institut Universitaire de France, 45 avenue des États-Unis, 78035 Versailles Cedex, France

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1339-1350

This article is part of the thematic issue From Generation to Generation: The Mathematical Legacy of Haïm Brezis edited by Henri Berestycki et al..

Abstract (Original language)
Résumé

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.

Received: 2025-05-29
Revised: 2025-09-26
Accepted: 2025-09-29
Published online: 2025-10-20

DOI: 10.5802/crmath.799

Classification: 35L71, 35B40, 37K40
Keywords: Solitons, Gross–Pitaevskii equation, asymptotic stability
Mots-clés : Solitons, équation de Gross–Pitaevskii, stabilité asymptotique

Author's affiliations:

Michał Kowalczyk ¹; Yvan Martel ²

¹ Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
² Laboratoire de mathématiques de Versailles, UVSQ, Université Paris-Saclay, CNRS, and Institut Universitaire de France, 45 avenue des États-Unis, 78035 Versailles Cedex, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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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

References
Cited by

[1] Fabrice Bethuel; Philippe Gravejat; Didier Smets Asymptotic stability in the energy space for dark solitons of the Gross–Pitaevskii equation, Ann. Sci. Éc. Norm. Supér. (4), Volume 48 (2015) no. 6, pp. 1327-1381 | DOI | MR | Zbl | Numdam

[2] Patrick Gérard The Gross–Pitaevskii equation in the energy space, Stationary and time dependent Gross–Pitaevskii equations (Alberto Farina; Jean-Claude Saut, eds.) (Contemporary Mathematics), Volume 473, American Mathematical Society, 2008, pp. 129-148 | DOI | MR | Zbl

[3] Philippe Gravejat; Didier Smets Asymptotic stability of the black soliton for the Gross–Pitaevskii equation, Proc. Lond. Math. Soc. (3), Volume 111 (2015) no. 2, pp. 305-353 | DOI | MR | Zbl

[4] Michał Kowalczyk; Yvan Martel Kink dynamics under odd perturbations for $(1 + 1)$ -scalar field models with one internal mode, Math. Res. Lett., Volume 31 (2024) no. 3, pp. 795-832 | DOI | MR | Zbl

[5] Michał Kowalczyk; Yvan Martel; Claudio Muñoz Soliton dynamics for the 1D NLKG equation with symmetry and in the absence of internal modes, J. Eur. Math. Soc., Volume 24 (2022) no. 6, pp. 2133-2167 | DOI | MR | Zbl

[6] Yvan Martel Asymptotic stability of solitary waves for the 1D cubic-quintic Schrödinger equation with no internal mode, Probab. Math. Phys., Volume 3 (2022) no. 4, pp. 839-867 | DOI | MR | Zbl

[7] Guillaume Rialland Asymptotic stability of solitary waves for the 1D near-cubic non-linear Schrödinger equation in the absence of internal modes, Nonlinear Anal., Theory Methods Appl., Volume 241 (2024), 113474, 30 pages | DOI | MR | Zbl

[8] Peter E. Zhidkov Korteweg–de Vries and nonlinear Schrödinger equations: qualitative theory, Lecture Notes in Mathematics, 1756, Springer, 2001 | MR | Zbl

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