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In this note we study stability times for a family of parameter dependent nonlinear Schrödinger equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first introduced by Bourgain), we state a rather flexible Birkho¤ Normal Form theorem, which implies, e.g., exponential and sub-exponential time estimates in the Sobolev and Gevrey class respectively. Complete proofs are given elsewhere (see [BMP18]).
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Luca Biasco, Jessica Elisa Massetti, Michela Procesi, Exponential and sub-exponential stability times for the NLS on the circle. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 30 (2019), no. 2, pp. 351–364