  We consider the Gross-Petaevskii equation in 1 space dimension with a
$n$-well trapping potential. We prove, in the semiclassical limit, that the
finite dimensional eigenspace associated to the lowest n eigenvalues of the
linear operator is slightly deformed by the nonlinear term into an almost
invariant manifold M. Precisely, one has that solutions starting on M, or close
to it, will remain close to M for times exponentially long with the inverse of
the size of the nonlinearity. As heuristically expected the effective equation
on M is a perturbation of a discrete nonlinear Schroedinger equation. We deduce
that when the size of the nonlinearity is large enough then tunneling among the
wells essentially disappears: that is for almost all solutions starting close
to M their restriction to each of the wells has norm approximatively constant
over the considered time scale. In the particular case of a double well
potential we give a more precise result showing persistence or destruction of
the beating motions over exponentially long times. The proof is based on
canonical perturbation theory; surprisingly enough, due to the Gauge invariance
of the system, no non-resonance condition is required.
