Stability of bound states for regularized nonlinear Schrödinger equations

We consider the stability of bound-state solutions of a family of regularized nonlinear Schr & ouml;dinger equations which were introduced by Dumas et al. as models for the propagation of laser beams. Among these bound-state solutions are ground states, which are defined as solutions of a variational problem. We give a sufficient condition for existence and orbital stability of ground states, and use it to verify that ground states exist and are stable over a wider range of nonlinearities than for the nonregularized nonlinear Schr & ouml;dinger equation. We also give another sufficient and almost necessary condition for stability of general bound states, and show that some stable bound states exist which are not ground states.