Long Time Stability of Some Small Amplitude Solutions in Nonlinear Schr Odinger Equations

We consider small perturbations of the Zakharov-Shabat nonlinear Schrr odinger equation on 0; ] with vanishing or periodic boundary conditions; we prove a Nekhoroshev type result for solutions starting in the neighbourhood (in the H 1 topology) of the majority of small amplitude nite dimensional invariant tori of the linearized system. More precisely we will prove that along the considered solutions all the actions of the linearized system are approximatively constant up to times growing exponentially with the inverse of a suitable small parameter.