Exponential Stability Estimates for the 1D NLS

We study stability times for a family of parameter dependent nonlinear Schr\"odinger 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.