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

2024/02/06 by Bernier, Joackim, Camps, Nicolas
#35B34 #35B35 #35Q55 #37K45 #37K55 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2402.04122

Abstract

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.

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