  In the present contribution we investigate some features of dynamical lattice
systems near periodic traveling waves. First, following the formal averaging
method of Whitham, we derive modulation systems expected to drive at main order
the time evolution of slowly modulated wavetrains. Then, for waves whose period
is commensurable to the lattice, we prove that the formally-derived first-order
averaged system must be at least weakly hyperbolic if the background waves are
to be spectrally stable, and, when weak hyperbolicity is met, the
characteristic velocities of the modulation system provide group velocities of
the original system. Historically, for dynamical evolutions obeying partial
differential equations, this has been proved, according to increasing level of
algebraic complexity, first for systems of reaction-diffusion type, then for
generic systems of balance laws, at last for Hamiltonian systems. Here, for
their semi-discrete counterparts, we give at once simultaneous proofs for all
these cases. Our main analytical tool is the discrete Bloch transform, a
discrete analogous to the continuous Bloch transform. Nevertheless , we needed
to overcome the absence of genuine space-translation invariance, a key
ingredient of continuous analyses.
