  We present theoretical calculations of collective modes of the one-band
attractive Hubbard model which is widely used to study the s-wave superfluid
phases of atomic Fermi gases of two-hyperfine states loaded in a deep optical
lattice. To make our theory applicable for both superconductivity and
superfluidity, we assume the more general $t-U-J$ Hamiltonian.
  Using the functional differentiation we derive Schwinger-Dyson equations for
the single-particle Green's functions.
  The method of Legendre transform is used to give a systematic derivation of
the Bethe-Salpeter (BS) equation for the two-particle Green's function and the
associated collective modes. The numerical solution of the BS equation in the
limit $J\rightarrow 0$ shows the existence of two rotonlike collective modes
with different low-energy Goldstone dispersions and different positions of the
rotonlike minima. The two rotonlike modes lie outside of the region determined
by the lower boundary of the particle-hole continuum, and therefore, the two
modes are not damped and they should be experimentally observable.
  In the presence of superfluid flow at a certain critical flow momentum, the
minimum of the first rotonlike mode reaches zero energy, but this occurs before
the minimum of the second mode and the lower boundary of the particle-hole
continuum do, i.e. there are two critical flow momenta related to the existence
of two rotonlike excitations.
