  We consider a family of regularized defocusing nonlinear Schrodinger (NLS)
equations proposed in the context of the cubic NLS equation with a bounded
dispersion relation. The time evolution is well-posed if the black soliton is
perturbed by a small perturbation in the Sobolev space $H^s(\R)$ with s > 1/2.
We prove that the black soliton is spectrally stable (unstable) if the
regularization parameter is below (above) some explicitly specified threshold.
We illustrate the stable and unstable dynamics of the perturbed black solitons
by using the numerical finite-difference method. The question of orbital
stability of the black soliton is left open due to the mismatch of the function
spaces for the energy and momentum conservation.
