03-105 Bambusi D. Grebert B.
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Abstract. We consider the non linear Sch\"odinger equation $$-iu_{t} = -\Delta u + V\ast u +g(u,\bar u )$$ with periodic boundary conditions on $[-\pi, \pi]^d$, $d\geq 1$; $g$ 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 $\varepsilon$, the solution remains smaller than $2\varepsilon$ for all times $t$ smaller than $\varepsilon^{-(M-1)}$. Moreover, for the same times, the solution is close to an infinite dimensional torus.

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