Dynamics of the black soliton in a regularized nonlinear Schrödinger equation

We consider a family of regularized defocusing nonlinear Schrödinger (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 $H^s(\mathbb{R})$ with $s>\frac{1}{2}$ . 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.