## Journals

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DCDS

We consider examples of discrete nonlinear Schrödinger
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 Schrödinger 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.
DCDS

We continue our study initiated in [4] of the interaction of a ground state with
a potential considering here a class of trapping potentials. We track the precise asymptotic behavior of the solution if the interaction is weak, either because
the ground state moves away from the potential or is very fast.

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