  We study the phase diagram for a dilute Bardeen-Cooper-Schrieffer superfluid
Fermi-Fermi mixture (of distinct mass) at zero temperature using energy
densities for the superfluid fermions in one (1D), two (2D), and three (3D)
dimensions. We also derive the dynamical time-dependent nonlinear
Euler-Lagrange equation satisfied by the mixture in one dimension using this
energy density. We obtain the linear stability conditions for the mixture in
terms of fermion densities of the components and the interspecies Fermi-Fermi
interaction. In equilibrium there are two possibilities. The first is that of a
uniform mixture of the two components, the second is that of two pure phases of
two components without any overlap between them. In addition, a mixed and a
pure phase, impossible in 1D and 2D, can be created in 3D. We also obtain the
conditions under which the uniform mixture is stable from an energetic
consideration. The same conditions are obtained from a modulational instability
analysis of the dynamical equations in 1D. Finally, the 1D dynamical equations
for the system are solved numerically and by variational approximation (VA) to
study the bright solitons of the system for attractive interspecies Fermi-Fermi
interaction in 1D. The VA is found to yield good agreement to the numerical
result for the density profile and chemical potential of the bright solitons.
The bright solitons are demonstrated to be dynamically stable. The experimental
realization of these Fermi-Fermi bright solitons seems possible with present
setups.
