Bambusi D. Grebert B.
Forme normale pour NLS en dimension quelconque
(20K, LaTeX)
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.