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

TL;DR

This paper derives and analyzes variants of the focusing nonlinear Schrödinger equation from Maxwell's equations, aiming to better model laser beam propagation and address phenomena like filamentation that standard models cannot fully explain.

Contribution

The paper introduces new variants of the focusing NLS equation derived from Maxwell's equations and provides rigorous error estimates for these models.

Findings

Derivation of NLS variants from Maxwell's equations

Error estimates for the proposed models

Discussion of open problems in modified NLS equations

Abstract

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.