  We present numerical calculations of the snake instability in a Fermi
superfluid within the Bogoliubov-de Gennes theory of the BEC to BCS crossover
using the random phase approximation complemented by time-dependent
simulations. We examine the snaking behaviour across the crossover and quantify
the timescale and lengthscale of the instability. While the dynamic shows
extensive snaking before eventually producing vortices and sound on the BEC
side of the crossover, the snaking dynamics is preempted by decay into sound
due to pair breaking in the deep BCS regime. At the unitarity limit,
hydrodynamic arguments allow us to link the rate of snaking to the
experimentally observable ratio of inertial to physical mass of the soliton. In
this limit we witness an unresolved discrepancy between our numerical estimates
for the critical wavenumber of suppression of the snake instability and recent
experimental observations with an ultra-cold Fermi gas.
