Families of dark solitons exist in superfluid Fermi gases. The
energy-velocity dispersion and number of depleted particles completely
determines the dynamics of dark solitons on a slowly-varying background
density. For the unitary Fermi gas we determine these relations from general
scaling arguments and conservation of local particle number. We find solitons
to oscillate sinusoidally at the trap frequency reduced by a factor of
1/3​. Numerical integration of the time-dependent Bogoliubov-de Gennes
equation determines spatial profiles and soliton dispersion relations across
the BEC-BCS crossover and proves consistent with the scaling relations at
unitarity.Comment: Small changes in response to referee's comments; fig 1 revised and
refs updated. Cross listed to nucl-th due to interest in the unitary Fermi
ga