Forme normale pour NLS en dimension quelconque.

We consider the nonlinear Schödinger equation with periodic boundary conditions on [−π,π]d, d1; g is analytic and g(0,0)=Dg(0,0)=0; V is a potential in L2. Under a nonresonance condition which is fulfilled for most Vs we prove that, for any integer M there exists a canonical transformation that puts the Hamiltonian in Birkhoff normal form up to a reminder of order M. The canonical tranformation is well defined in a neighbourhood of the origin of any Sobolev space of sufficiently high order. From the dynamical point of view this means in particular that if the initial data is smaller than , the solution remains smaller than 2 for all times t smaller than −(M−1). Moreover, for the same times, the solution is close to an infinite dimensional torus