Abstract
Starting from the three-dimensional (3D) time-dependent nonlinear Gross–Pitaevskii equation for a Bose–Einstein condensate (BEC) and the 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 disc-shaped BEC and Fermi superfluid by using the adiabatic approximation. The reduced 1D and 2D equations for a cigar- and disc-shaped Fermi superfluid have simple analytic non-linear 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.
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