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We consider the stability of bound-state solutions of a family of regularized nonlinear Schr\"odinger equations which were introduced by Dumas, Lannes and Szeftel 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\"odinger 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., Comment: 39 pages

We prove maximal speed estimates for nonlinear quantum propagation in the context of the Hartree equation. More precisely, under some regularity and integrability assumptions on the pair (convolution) potential, we construct a set of energy and space localized initial conditions such that, up to time-decaying tails, solutions starting in this set stay within the light cone of the corresponding initial datum. We quantify precisely the light cone speed, and hence the speed of nonlinear propagation, in terms of the momentum of the initial state., Comment: 38 pages. Appendices D and E are adapted from arXiv:2011.04570. To appear in Communications in Partial Differential Equations