  We derive the Bethe-Salpeter (BS) equations for the collective-mode spectrum
of superfluid Fermi gases of equal mixture of atomic Fermi gas of two hyperfine
states loaded into a moving optical lattice. In a moving lattice the superfluid
state is unstable due to spontaneous emission of the short-wavelength rotonlike
excitations which appear in the spectrum of the collective modes. It is shown
that the spectrum obtained by the BS equations is in an excellent agreement
with the collective-mode dispersion calculated by the perturbation approach,
while there are some differences between the results obtained by density
response function method and by the BS approach. The difference increases with
increasing the lattice velocity, which can be seen in the analytical
approximations for the dispersion relation in the long-wavelength limit in a
weak-coupling regime.
