Normal form for NLS in arbitrary dimension

Normal form for NLS in arbitrary dimension. We consider the nonlinear Schrodinger equation -iu(t) = - Deltau + V * u + g(u, (u) over bar) with periodic boundary conditions on [-pi, pi](d), d greater than or equal to 1; g is analytic and g(0, 0) = Dg(0,0) = 0; V is a potential in L-2. Under a nonresonance condition which is fulfilled for most V s 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 epsilon, the solution remains smaller than 2epsilon for all times t smaller than epsilon(-(M- 1)). Moreover, for the same times, the solution is close to an infinite dimensional torus. (C) 2003 Academie des sciences. Publie par Editions scientifiques et medicales Elsevier SAS. Tous droits reserves.