  We study the superfluid properties of a system of interacting bosons on a
lattice which, moreover, are coupled to the vibrational modes of this lattice,
treated here in terms of Einstein phonon model. The ground state corresponds to
two correlated condensates: that of the bosons and that of the phonons. Two
competing effects determine the common collective soundwave-like mode with
sound velocity $v$, arising from gauge symmetry breaking: i) The sound velocity
$v_0$ (corresponding to a weakly interacting Bose system on a rigid lattice) in
the lowest order approximation is reduced due to reduction of the repulsive
boson-boson interaction, arising from the attractive part of phonon mediated
interaction in the static limit. ii) the second order correction to the sound
velocity is enhanced as compared to the one of bosons on a rigid lattice when
the the boson-phonon interaction is switched on due to the retarded nature of
phonon mediated interaction. The overall effect is that the sound velocity is
practically unaffected by the coupling with phonons, indicating the robustness
of the superfluid state. The induction of a coherent state in the phonon
system, driven by the condensation of the bosons could be of experimental
significance, permitting spectroscopic detections of superfluid properties of
the bosons. Our results are based on an extension of the Beliaev - Popov
formalism for a weakly interacting Bose gas on a rigid lattice to that on a
deformable lattice with which it interacts.
