The black solitons of one-dimensional NLS equations

In this paper, we prove a criterion determining if a black soliton solution to a one-dimensional nonlinear Schrödinger equation is linearly stable or not. This criterion handles with the sign of the limit at 0 of the Vakhitov-Kolokolov function. For some nonlinearities, we compute numerically the black soliton and the Vakhitov-Kolokolov function. We then show that linearly un-stable black solitons are also orbitally unstable. In the Gross-Pitaevskii case, we rigorously prove the linear stability of the black soliton. Finally, we nu-merically investigate the dynamical stability of these solutions solving both linearized and fully nonlinear equation with a finite differences algorithm