An Abstract Birkhoff Normal Form Theorem and Exponential Type Stability of the 1d NLS
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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 prove a rather flexible Birkhoff Normal Form theorem, which implies, e.g., exponential and sub-exponential time estimates in the Sobolev and Gevrey class respectively.
The three authors have been supported by the ERC grant HamPDEs under FP7 n. 306414 and the PRIN Variational Methods in Analysis, Geometry and Physics. J.E. Massetti also acknowledges Centro di Ricerca Matematica Ennio de Giorgi and UniCredit Bank R&D group for financial support through the “Dynamics and Information Theory Institute” at the Scuola Normale Superiore. The authors would like to thank D. Bambusi, M. Berti, B. Grebert, Z. Hani and A. Maspero for helpful suggestions and fruitful discussions.
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