Long time stability for cubic nonlinear Schrödinger equations on non-rectangular flat tori

Joackim Bernier; Nicolas Camps

doi:10.5802/jep.300

Long time stability for cubic nonlinear Schrödinger equations on non-rectangular flat tori
[Stabilité en temps long des équations de Schrödinger non linéaires cubiques sur des tores plats non rectangulaires]

Joackim Bernier ¹ ; Nicolas Camps ²

¹ Nantes Université, CNRS, Laboratoire de Mathématiques Jean Leray, LMJL, F-44000 Nantes, France
² Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 713-800.

Abstract (VO)
Résumé

We consider nonlinear Schrödinger equations on flat tori satisfying a simple and explicit Diophantine non-degeneracy condition. Provided that the nonlinearity contains a cubic term, we prove the almost global existence and stability of most of the small solutions in high regularity Sobolev spaces. To this end, we develop a normal form approach designed to handle general resonant Hamiltonian partial differential equations for which it is possible to modulate the frequencies by using the initial data.

Nous considérons des équations de Schrödinger non linéaires sur des tores plats satisfaisant à une condition simple et explicite de non-dégénérescence diophantienne. Sous la condition que la non-linéarité contienne un terme cubique, nous prouvons l’existence et la stabilité presque globales de la plupart des petites solutions dans des espaces de Sobolev de forte régularité. À cette fin, nous développons une approche de forme normale conçue pour traiter des équations aux dérivées partielles hamiltoniennes résonnantes générales, pour lesquelles il est possible de moduler les fréquences en utilisant les données initiales.

Reçu le : 2024-04-04
Accepté le : 2025-05-13
Publié le : 2025-05-20

DOI : 10.5802/jep.300

Classification : 35B34, 35B35, 35Q55, 37K45, 37K55
Keywords: Almost global existence, flat tori, nonlinear Schrödinger equations, normal forms, internal parameters
Mots-clés : Existence presque globale, tores plats, équations de Schrödinger non linéaires, formes normales, paramètres internes

Affiliations des auteurs :

Joackim Bernier ¹ ; Nicolas Camps ²

¹ Nantes Université, CNRS, Laboratoire de Mathématiques Jean Leray, LMJL, F-44000 Nantes, France
² Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Joackim Bernier and Nicolas Camps},
     title = {Long time stability for {cubic~nonlinear~Schr\"odinger} equations on non-rectangular flat tori},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {713--800},
     publisher = {\'Ecole polytechnique},
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     year = {2025},
     doi = {10.5802/jep.300},
     language = {en},
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Joackim Bernier; Nicolas Camps. Long time stability for cubic nonlinear Schrödinger equations on non-rectangular flat tori. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 713-800. doi : 10.5802/jep.300. https://jep.centre-mersenne.org/articles/10.5802/jep.300/

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