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.

We present analytical results and numerical simulations for a class of nonlinear dispersive equations in two spatial dimensions. These equations are of (derivative) nonlinear Schrödinger type and have recently been obtained by Dumas et al. in the context of nonlinear optics. In contrast to the usual nonlinear Schrödinger equation, this new model incorporates the additional effects of self-steepening and partial off-axis variations of the group velocity of the laser pulse. We prove global-in-time existence of the corresponding solution for various choices of parameters. In addition, we present a series of careful numerical simulations concerning the (in-)stability of stationary states and the possibility of finite-time blow-up.

We study a class of focusing nonlinear Schrödinger-type equations derived recently by Dumas, Lannes, and Szeftelwithin the mathematical description of high intensity laser beams.These equations incorporate the possibility of a (partial) off-axis variation of the group velocity of such laser beams through a second orderpartial differential operator acting in some, but not necessarily all, spatial directions.We investigate the initial value problem for such models and obtain global well-posedness in $L^2$-supercritical situations,even in the case of only partial off-axis dependence. This provides an answer to an open problem posed by Dumas, Lannes, and Szeftel.

We consider the mean-field dynamics of Bose–Einstein condensates in rotating harmonic traps and establish several stability and instability properties for the corresponding solution. We particularly emphasize the difference between the situation in which the trap is symmetric with respect to the rotation axis and the one where this is not the case.

We consider a nonlinear Schroedinger equation in two spatial dimensions subject to a periodic honeycomb lattice potential. Using a multi-scale expansion together with rigorous error estimates, we derive an effective model of nonlinear Dirac type. The latter describes the propagation of slowly modulated, weakly nonlinear waves spectrally localized near a Dirac point.