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 ${ϵ}^{-r}$.