Abstract
We review some recent results concerning the existence and stability of spatially localized and temporally quasiperiodic (non-stationary) excitations in discrete nonlinear Schrödinger (DNLS) models. In two dimensions, we show the existence of linearly stable, stationary and non-stationary localized vortex-like solutions. We also show that stationary on-site localized excitations can have internal ‘breathing’ modes which are spatially localized and symmetric. The excitation of these modes leads to slowly decaying, quasiperiodic oscillations. Finally, we show that for some generalizations of the DNLS equation where bistability occurs, a controlled switching between stable states is possible by exciting an internal breathing mode above a threshold value.
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