J. Cuevas-Maraver, P.G. Kevrekidis, and D.E. Pelinovsky
Nonlinear instabilities of multi-site breathers in Klein-Gordon lattices
Studies in Applied Mathematics 137 (2016), 214-237
In the present work, we explore the possibility of excited breather states
in a nonlinear Klein--Gordon lattice to become nonlinearly unstable, even
if they are found to be spectrally stable. The mechanism for this
fundamentally nonlinear instability is through the resonance
with the wave continuum of a multiple
of an internal mode eigenfrequency in the linearization of excited
breather states. For the nonlinear instability, the internal mode must have
its Krein signature opposite to that of the wave continuum.
This mechanism is not only theoretically proposed,
but also numerically corroborated through two concrete examples of the Klein-Gordon lattice
with a soft (Morse) and a hard (phi-four) potential.
Compared to the case of the nonlinear Schrodinger lattice, the Krein
signature of the internal mode relative to that of the wave continuum may change depending on
the period of the excited breather state. For the periods
for which the Krein signatures of the internal mode and the wave continuum coincide,
excited breather states are observed to be nonlinearly stable.
discrete Klein-Gordon equation, multi-site breathers,
instability, Krein signature, internal modes.