Nonlinear Schr\"odinger equation for a superfluid Fermi gas from BCS to Bose crossover

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 $k_F\mathrm{\mid }a\mathrm{\mid }\ll 1$ with $a$ the s-wave scattering length and $k_F$ the Fermi momentum, through the unitarity limit, $k_{F}a\to \pm \mathrm{\infty }$, to the Bose regime where $k_{F}a>0$. 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.