Orbitally but not asymptotically stable ground states for the discrete NLS.

We consider examples of discrete nonlinear Schr\"odinger equations in Z admitting ground states which are orbitally but not asymptotically stable in l ^2(Z ). The ground states contain internal modes which decouple from the continuous modes. The absence of leaking of energy from discrete to continues modes leads to an almost conservation and perpetual oscillation of the discrete modes. This is quite different from what is known for nonlinear Schr\"odinger equations in R ^d. To achieve our goal we prove a Siegel normal form theorem, prove dispersive estimates for the linearized operators and prove some nonlinear estimates