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January  2010, 26(1): 105-134. doi: 10.3934/dcds.2010.26.105

Orbitally but not asymptotically stable ground states for the discrete NLS

Scipio Cuccagna 1,

1. 

Dipartimento Metodi e Scienze dell’Ingegneria, Università di Modena e Reggio Emilia, via Amendola 2, Padiglione Morselli, Reggio Emilia 42122, Italy

Received  January 2009 Revised  July 2009 Published  October 2009

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.
Keywords: normal forms, dispersive estimates..
Mathematics Subject Classification: Primary: 35Q55; Secondary: 81Q0.
Citation: Scipio Cuccagna. Orbitally but not asymptotically stable ground states for the discrete NLS. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 105-134. doi: 10.3934/dcds.2010.26.105
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