  In this paper we investigate the emergence of time-periodic and and
time-quasiperiodic (sometimes infinitely long lived and sometimes very long
lived or metastable) solutions of discrete nonlinear wave equations: discrete
sine Gordon, discrete $\phi^4$ and discrete nonlinear Schr\"odinger. The
solutions we consider are periodic oscillations on a kink or standing wave
breather background. The origin of these oscillations is the presence of
internal modes, associated with the static ground state. Some of these modes
are associated with the breaking of translational invariance, in going from a
spatially continuous to a spatially discrete system. Others are associated with
discrete modes which bifurcate from the continuous spectrum. It is also
possible that such modes exist in the continuum limit and persist in the
discrete case. The regimes of existence, stability and metastability of states
as the lattice spacing is varied are investigated analytically and numerically.
A consequence of our analysis is a class of spatially localized, time
quasiperiodic solutions of the discrete nonlinear Schr\"odinger equation. We
demonstrate, however, that this class of quasiperiodic solution is rather
special and that its natural generalizations yield only metastable
quasiperiodic solutions.
