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In this paper we prove that ground states of the NLS which satisfy thesufficient conditions for orbital stability of M.Weinstein, are alsoasymptotically stable, for seemingly generic equations. Here we assume that theNLS has a smooth short range nonlinearity. We assume also the presence of avery short range and smooth linear potential, to avoid translation invariance.The basic idea is to perform a Birkhoff normal form argument on thehamiltonian, as in a paper by Bambusi and Cuccagna on the stability of the 0solution for NLKG. But in our case, the natural coordinates arising from thelinearization are not canonical. So we need also to apply the Darboux Theorem.With some care though, in order not to destroy some nice features of theinitial hamiltonian.

The Hamiltonian structure of the nonlinear Schr\"odinger equation and the asymptotic stability of its ground states