Orbitally but not asymptotically stable ground states for the discrete NLS

We consider examples of discrete nonlinear Schr\"odingerequations in Z admitting ground states which are orbitally butnot asymptotically stable in l ^2(Z ). The ground statescontain internal modes which decouple from the continuous modes.The absence of leaking of energy from discrete to continues modesleads to an almost conservation and perpetual oscillation of thediscrete modes. This is quite different from what is known fornonlinear Schr\"odinger equations in R ^d. To achieve our goal weprove a Siegel normal form theorem, prove dispersive estimates forthe linearized operators and prove some nonlinear estimates