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

Eric Dumas (1) , David Lannes (2) , Jeremie Szeftel (3)

1 IF - Institut Fourier
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 LJLL - Laboratoire Jacques-Louis Lions

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.

Domaines: Equations aux dérivées partielles [math.AP]

Eric Dumas, David Lannes, Jeremie Szeftel. Variants of the focusing NLS equation. Derivation, justification and open problems related to filamentation. CRM Series in Mathematical Physics, Springer International Publishing, pp.19-75, 2016, Laser Filamentation. ⟨hal-00997611v2⟩