  In this paper we elucidate the physics underlying the fact that both bright
and dark solitary waves can arise in a one-dimensional spin-polarized gas of
fermionic atoms with attractive three-dimensional p-wave interactions in a
hard-wall trap. This is possible since the one-dimensional fermion system can
be mapped to a system of bosons described by the Lieb-Liniger model with either
repulsive or attractive delta-function interactions which can support solitary
waves.
