AVERAGING THEOREMS FOR NLS:PROBABILISTIC AND DETERMINISTIC RESULTS

In this thesis, we study the dynamics of NLS, in particular, we deal with the problem of the construction of prime integrals, either in the probabilistic or in the deterministic case. In the first part of the thesis, we consider the non linear Schr\uf6dinger equation on the one dimensional torus with a defocusing polynomial nonlinearity and we study the dynamics corresponding to initial data in a set of a large measure with respect to the Gibbs measure. We prove that along the corresponding solutions the modulus of the Fourier coefficients is approximately constant for long time. The proof is obtained by adapting to the context of Gibbs measure for PDEs some tools of Hamiltonian perturbation theory. In the second part, we consider the nonlinear Schr\uf6dinger equation on the two dimensional torus with a time-dependent nonlinearity starting with cubic terms. In this case, using perturbation theory techniques, we construct an approximate integral of motion that change slowly for initial data with small H^1-norm, this allows to ensure long time existence of solutions in H^1 on the two dimensional torus. The main difficulty is that H^1 on the two dimensional torus is not an algebra