Oscillatory standing wave instabilities in Hamiltonian lattices

Magnus Johansson ^(1,2), Anna Maria Morgante ⁽²⁾, Serge Aubry ⁽²⁾, George Kopidakis <sup>(2,3)

(1)</sup>Department of Physics and Measurement Technology, Linköping University, S-581 83 Linköping, Sweden
⁽²⁾Laboratoire Léon Brillouin (CEA-CNRS), CEA Saclay, F-91191 Gif-sur-Yvette Cedex, France
⁽³⁾Department of Physics, University of Crete, P.O. Box 2208, GR-71003, Heraklion, Crete, Greece

Abstract: Modulational (Benjamin-Feir) instability (MI) of travelling plane waves is a wellknown mechanism leading e.g. to self-focusing in nonlinear optics and hydrodynamics. For discrete systems (e.g. nonlinear optical waveguide arrays or coupled anharmonic oscillators) MI typically occurs for a certain range of wavenumbers only, and is often considered as the first step in the generation of Intrinsically Localized Modes (ILM) ('discrete breathers'). Here, we consider Standing Waves (SWs), which naturally appear e.g. from counterpropagating waves with equal amplitude and frequency, in a general class of nonlinear Hamiltonian lattices. We show how such waves can be uniquely continued from the linear limit to the uncoupled limit as multi-site discrete breathers. Moreover, we show that even for small amplitudes, these SWs are generically unstable through oscillatory instabilities, which appear also for wave numbers where the propagating waves are stable. We analyse the dynamics resulting from these new instabilities, and find qualitatively different scenarios for wave vectors smaller than or larger than \pi/2: persisting localized structures are created in one regime but not in the other.