  We introduce a quasi-analytic nonlinear Schr\"odinger equation with beyond
mean-field corrections to describe the dynamics of a zero-temperature dilute
superfluid Fermi gas in the crossover from the weak-coupling
Bardeen-Cooper-Schrieffer (BCS) regime, where $k_F|a| \ll 1$ with $a$ the
s-wave scattering length and $k_F$ the Fermi momentum, through the unitarity
limit, $k_Fa \to \pm \infty$, to the Bose regime where $k_Fa >0$. The energy of
our model is parametrized using the known asymptotic behavior in the BCS, Bose,
and the unitarity limits and is in excellent agreement with accurate Green
function Monte Carlo calculations. The model generates good results for
frequencies of collective breathing oscillation of a trapped Fermi superfluid.
