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.