Asymptotic stability of solitary waves in dispersive equations

We consider the subcritical Hamiltonian NLS in R ; it is well known that under suitable assumptions on the nonlinearity it admitts a family of travelling solitary waves. We prove that generically they are asymptotically stable. The result was known when the Floquet spectrum of the soliton has no non trivial eigenvalues. It is here extended to the general case. The proof (which is developped in an abstract framework) is based on the combination of Hamiltonian and dispersive techniques. The main technical difficulties one has to face are related to the fact that the generators of the symmetry are unbounded operators. This oblidges to develop Marsden-Weinstein reduction theory when the group action is only continuous and normal form theory when the generating vector field is not smooth. This also causes some difficulties for dispersive estimates. Such difficulties are solved using recent results by Parelman and Beceanu on Strichartz estimates for time dependent potentials.