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Oscillatory standing wave instabilities in Hamiltonian lattices
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Magnus Johansson **^{(1,2)},
Anna Maria Morgante ^{(2)},
Serge Aubry ^{(2)},
George Kopidakis ^{(2,3)}

^{(1)}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.