Freezing of Energy of a Soliton in an External Potential

In this paper we study the dynamics of a soliton in the generalized NLS with a small external potential ϵV of Schwartz class. We prove that there exists an effective mechanical system describing the dynamics of the soliton and that, for any positive integer r, the energy of such a mechanical system is almost conserved up to times of order ϵ−r. In the rotational invariant case we deduce that the true orbit of the soliton remains close to the mechanical one up to times of order ϵ−r.