Long time stability in perturbations of completely resonant PDE's

In this paper we will present some results concerning long time stability in nonlinear perturbations of resonant linear PDE's with discrete spectrum. In particular we will prove that if the perturbation satisfies a suitable nondegeneracy condition then there exists a periodic like trajectory, i.e. a closed curve in the phase space with the property that solutions starting close to it remain close to it for very long times. Secondly, in the special case where the average of the main part of the perturbation is integrable we will prove that if the energy is initially essentially concentrated on finitely many modes, then along the corresponding solutions all the actions are approximatively constant for very long times. Applications to nonlinear wave and Schrodinger equations on a segment will also be given.