The black solitons of one-dimensional NLS equations

In this paper, we prove a criterion to determine if a black soliton solution (which is an odd solution that does not vanish at infinity) to a one-dimensional nonlinear Schr?dinger equation is linearly stable or not. This criterion handles the sign of the limit at 0 of the Vakhitov?Kolokolov function. For some nonlinearities, we numerically compute the black soliton and the Vakhitov?Kolokolov function in order to investigate linear stability of black solitons. We then show that linearly unstable black solitons are also orbitally unstable. In the Gross?Pitaevskii case, we rigorously prove the linear stability of the black soliton. Finally, we numerically study the dynamical stability of these solutions solving both linearized and fully nonlinear equations with a finite differences algorithm.