Instabilities of a matter wave in a matter grating

We investigate the stability of Bloch waves for a Bose-Einstein condensate moving through a periodic lattice created by another condensate modulated by an optical lattice. We show that the coupling of phonon-antiphonon modes of the two species give rise to a very rich structure of the regimes for dynamical instability, with significant differences with respect to the case of a single condensate in an optical lattice. We characterize the relative weight of each condensate in the mixing and discuss an analytic limit that accounts for the bare structure of the instability diagrams.