  We consider existence and stability properties of nonlinear spatially
periodic or quasiperiodic standing waves (SWs) in one-dimensional lattices of
coupled anharmonic oscillators. Specifically, we consider Klein-Gordon (KG)
chains with either soft (e.g., Morse) or hard (e.g., quartic) on-site
potentials, as well as discrete nonlinear Schroedinger (DNLS) chains
approximating the small-amplitude dynamics of KG chains with weak inter-site
coupling. The SWs are constructed as exact time-periodic multibreather
solutions from the anticontinuous limit of uncoupled oscillators. In the
validity regime of the DNLS approximation these solutions can be continued into
the linear phonon band, where they merge into standard harmonic SWs. For SWs
with incommensurate wave vectors, this continuation is associated with an
inverse transition by breaking of analyticity. When the DNLS approximation is
not valid, the continuation may be interrupted by bifurcations associated with
resonances with higher harmonics of the SW. Concerning the stability, we
identify one class of SWs which are always linearly stable close to the
anticontinuous limit. However, approaching the linear limit all SWs with
nontrivial wave vectors become unstable through oscillatory instabilities,
persisting for arbitrarily small amplitudes in infinite lattices. Investigating
the dynamics resulting from these instabilities, we find two qualitatively
different regimes for wave vectors smaller than or larger than pi/2,
respectively. In one regime persisting breathers are found, while in the other
regime the system rapidly thermalizes.
