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Variants of the Focusing NLS Equation: Derivation, Justification, and Open Problems Related to Filamentation

  • Éric DumasEmail author
  • David Lannes
  • Jérémie Szeftel
Chapter
Part of the CRM Series in Mathematical Physics book series (CRM)

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.

Keywords

Strichartz Estimates Slowly Varying Envelope Approximation (SVEA) Local Well-posedness Optical Shock Frequency-dependent Polarizability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

David Lannes acknowledges support from the ANR-13-BS01-0003-01 DYFICOLTI, the ANR BOND.

The authors warmly thank Christof Sparber for pointing [28] out.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institut FourierUniversité Joseph FourierSaint Martin d’HèresFrance
  2. 2.Institut de Mathématiques de BordeauxUniversité de Bordeaux & CNRScours de la LibérationFrance
  3. 3.Laboratoire Jacques-Louis LionsUniversité Pierre et Marie CurieParisFrance

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