Dark solitons dynamics and snake instability in superfluid Fermi gases trapped by an anisotropic harmonic potential

We present an investigation of generation, dynamics and stability of dark solitons in anisotropic Fermi gases for a range of particle numbers and trap aspect ratios within the framework of the order-parameter equation. We calculate the periods of dark solitons oscillating in a trap, and find a good agreement with the results based on the Bogoliubov-de Gennes equations. By studying the stability of initially off-center dark solitons under various tight transverse confinements in the unitarity limit, we not only give the criterion of dynamical stability, but also find that the soliton and a hybrid of solitons and vortex rings can be characterized by different oscillation period. The stability criterion is not fulfilled by the parameters of the recent experiment [Nature {\bf 499}, 426 (2013)]. Therefore, instead of a very slow oscillation as observed experimentally, we find that the created dark soliton undergoes a transverse snake instability with collapsing into vortex rings, which propagate in soliton-like manner with a nearly two times larger period.