  Starting from the three-dimensional (3D) time-dependent nonlinear
Gross-Pitaevskii equation for a Bose-Einstein condensate (BEC) and density
functional (DF) equation for a Fermi superfluid at the unitarity and
Bardeen-Cooper-Schrieffer (BCS) limits, we derive effective one- (1D) and
two-dimensional (2D) mean-field equations, respectively, for the dynamics of a
trapped cigar- and disk-shaped BEC and Fermi superfluid by using the adiabatic
approximation. The reduced 1D and 2D equations for a cigar- and disk-shaped
Fermi superfluid have simple analytic nonlinear terms and at unitarity produce
results for stationary properties and non-stationary breathing oscillation and
free expansion in excellent agreement with the solution of the full 3D
equation.
