  The focusing cubic NLS is a canonical model for the propagation of laser
beams. In dimensions 2 and 3, it is known that a large class of initial data
leads to finite time blow-up. Now, physical experiments suggest that this
blow-up does not always occur. This might be explained by the fact that some
physical phenomena neglected by the standard NLS model become relevant at large
intensities of the beam. Many ad hoc variants of the focusing NLS equation have
been proposed to capture such effects. In this paper, we derive some of these
variants from Maxwell's equations and propose some new ones. We also provide
rigorous error estimates for all the models considered. Finally, we discuss
some open problems related to these modified NLS equations.
