Exponential times in the one-dimensional Gross-Pitaevskii equation with multiple well potential

We consider the Gross-Pitaevskii 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 script M sign . Precisely, one has that solutions starting on script M sign , or close to it, will remain close to script M sign for times exponentially long with the inverse of the size of the nonlinearity. As heuristically expected the effective equation on script M sign is a perturbation of a discrete nonlinear Schrödinger 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 script M sign 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.