  We present the study of the dark soliton dynamics in an inhomogenous fiber by
means of a variable coefficient modified nonlinear Schr\"{o}dinger equation
(Vc-MNLSE) with distributed dispersion, self-phase modulation, self-steepening
and linear gain/loss. The ultrashort dark soliton pulse evolution and
interaction is studied by using the Hirota bilinear (HB) method. In particular,
we give much insight into the effect of self-steepening (SS) on the dark
soliton dynamics. The study reveals a shock wave formation, as a major effect
of SS. Numerically, we study the dark soliton propagation in the continuous
wave background, and the stability of the soliton solution is tested in the
presence of photon noise. The elastic collision behaviors of the dark solitons
are discussed by the asymptotic analysis. On the other hand, considering the
nonlinear tunneling of dark soliton through barrier/well, we find that the
tunneling of the dark soliton depends on the height of the barrier and the
amplitude of the soliton. The intensity of the tunneling soliton either forms a
peak or valley and retains its shape after the tunneling. For the case of
exponential background, the soliton tends to compress after tunneling through
the barrier/well.
