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We consider a modulated discrete nonlinear Schrödinger (DNLS) model with
alternating
on-site potential, having a linear spectrum
with two branches separated by a `forbidden' gap.
Nonlinear localized
time-periodic solutions with frequencies in
the gap and near the gap — discrete gap and out-gap
breathers (DGBs and DOGBs) — are investigated. Their linear stability
is studied varying the system parameters from the continuous to
the anti-continuous limit, and different types of oscillatory and real
instabilities are revealed. It is shown, that generally DGBs in infinite
modulated
DNLS chains with hard (soft) nonlinearity do not possess any oscillatory
instabilities
for breather frequencies in the lower (upper) half of the gap.
Regimes of
`exchange of stability' between symmetric and antisymmetric DGBs are observed,
where an increased breather mobility is expected.
The transformation
from DGBs to DOGBs when the breather frequency enters the linear
spectrum is studied, and
the general bifurcation picture for DOGBs
with tails of different wave numbers is described.
Close to the anti-continuous limit, the localized linear
eigenmodes and their corresponding eigenfrequencies are calculated analytically
for several gap/out-gap breather configurations, yielding explicit
proof of their linear stability or instability close to this limit
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