  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.
