  Black solitons are identical in the nonlinear Schr\"{o}dinger (NLS) equation
with intensity-dependent dispersion and the cubic defocusing NLS equation. We
prove that the intensity-dependent dispersion introduces new properties in the
stability analysis of the black soliton. First, the spectral stability problem
possesses only isolated eigenvalues on the imaginary axis. Second, the
energetic stability argument holds in Sobolev spaces with exponential weights.
Third, the black soliton persists with respect to addition of a small decaying
potential and remains spectrally stable when it is pinned to the minimum points
of the effective potential. The same model exhibits a family of traveling dark
solitons for every wave speed and we incorporate properties of these dark
solitons for small wave speeds in the analysis of orbital stability of the
black soliton.
