The question of the linear stability of spatially periodic waves for the Boussinesq equation (in the cases p = 2, 3) and the Klein-Gordon-Zakharov system is considered. For a wide class of solutions, we completely and explicitly characterize their linear stability (instability) when the perturbations are taken with the same period T. In particular, our results allow us to completely recover the linear stability results, in the limit T → ∞, for the whole-line case. Copyright © Royal Society of Edinburgh 2014.
