Linear stability analysis for periodic traveling waves of the Boussinesq   equation and the KGZ system
  The question for linear stability of spatially periodic waves for the Boussinesq equation (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 respectively), 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 $T\to \infty$, for the whole line case. 
Sevdzhan Hakkaev, Milena Stanislavova, Atanas Stefanov
math.AP