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

Variants of the focusing NLS equation. Derivation, justification and open problems related to filamentation