On a class of derivative Nonlinear Schr\"odinger-type equations in two spatial dimensions

Abstract

We present analytical results and numerical simulations for a class of nonlinear dispersive equations in two spatial dimensions. These equations are of (derivative) nonlinear Schr\"odinger type and have recently been obtained in \cite{DLS} in the context of nonlinear optics. In contrast to the usual nonlinear Schr\"odinger 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, extending earlier results of \cite{AAS}. 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.