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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.
Berniér et al. (Tue,) studied this question.