  We investigate the long-time evolution of weakly perturbed single-site
breathers (localized stationary states) in the discrete nonlinear Schroedinger
(DNLS) equation. The perturbations we consider correspond to time-periodic
solutions of the linearized equations around the breather, and can be either
(i) spatially localized, or (ii) spatially extended. For case (i), which
corresponds to the excitation of an internal mode of the breather, we find that
the nonlinear interaction between the breather and its internal mode always
leads to a slow growth of the breather amplitude and frequency. In case (ii),
corresponding to interaction between the breather and a standing-wave phonon,
the breather will grow provided that the wave vector of the phonon is such that
the generation of radiating higher-harmonics at the breather is possible. In
other cases, breather decay is observed. This condition yields a limit value
for the breather frequency above which no further growth is possible. We also
discuss another mechanism for breather growth and destruction which becomes
important when the amplitude of the perturbation is non-negligible, and which
originates from the oscillatory instabilities of the nonlinear standing-wave
phonons.
