Nekhoroshev theorem for small amplitude solutions in nonlinear Schrodingerequations

We prove a Nekhoroshev type result [26,27] for the nonlinear Schrodinger eq uation iu(t) = -u(xx) - mu - u phi(\u\(2)), (0.1) with vanishing or periodic boundary conditions on [0, pi]; here m is an ele ment of IR is a parameter and phi : IR --> IR is a function analytic in a n eighborhood of the origin and such that phi(0) = 0, '(0) not equal 0. More precisely, we consider the Cauchy problem for (0.1) with initial data which extend to analytic entire functions of finite order, and prove that all th e actions of the Linearized system are approximate constants of motion up t o times growing faster than any negative power of the size of the initial d atum. The proof is obtained by a method which applies to Hamiltonian pertur bations of linear PDE's with the following properties: (i) the linear dynam ics is periodic (ii) there exists a finite order Birkhoff normal form which is integrable and quasi convex as a function of the action variables. Eq. (0.1) satisfies (i) and (ii) when restricted to a level surface of \\u\\(L2 ), which is an integral of motion. The main technical tool used in the proo f is a normal form lemma for systems with symmetry which is also proved her e.