Some variants of the focusing NLS equations
Dumas, Éric1
1 Université Grenoble Alpes France
Journées équations aux dérivées partielles (2018), Exposé no. 1, 15 p.

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. We derive from Maxwell’s equations some known variants of NLS and propose some new ones, providing rigorous error estimates for all the models considered. These notes result from the work [9], in collaboration with D. Lannes and J. Szeftel.

Publié le :
DOI : 10.5802/jedp.661
Dumas, Éric 1

1 Université Grenoble Alpes France 
@article{JEDP_2018____A1_0,
     author = {Dumas, \'Eric},
     title = {Some variants of the focusing {NLS} equations},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     note = {talk:1},
     pages = {1--15},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2018},
     doi = {10.5802/jedp.661},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jedp.661/}
}
TY  - JOUR
AU  - Dumas, Éric
TI  - Some variants of the focusing NLS equations
JO  - Journées équations aux dérivées partielles
N1  - talk:1
PY  - 2018
SP  - 1
EP  - 15
PB  - Groupement de recherche 2434 du CNRS
UR  - http://archive.numdam.org/articles/10.5802/jedp.661/
DO  - 10.5802/jedp.661
LA  - en
ID  - JEDP_2018____A1_0
ER  - 
%0 Journal Article
%A Dumas, Éric
%T Some variants of the focusing NLS equations
%J Journées équations aux dérivées partielles
%Z talk:1
%D 2018
%P 1-15
%I Groupement de recherche 2434 du CNRS
%U http://archive.numdam.org/articles/10.5802/jedp.661/
%R 10.5802/jedp.661
%G en
%F JEDP_2018____A1_0
Dumas, Éric. Some variants of the focusing NLS equations. Journées équations aux dérivées partielles (2018), Exposé no. 1, 15 p. doi : 10.5802/jedp.661. http://archive.numdam.org/articles/10.5802/jedp.661/

[1] Antonelli, Paolo; Arbunich, Jack; Sparber, Christof Regularizing nonlinear Schrödinger equations through partial off-axis variations, SIAM J. Math. Anal., Volume 51 (2019) no. 1, pp. 110-130 | Zbl

[2] Arbunich, Jack; Klein, Christian; Sparber, Christof On a class of derivative Nonlinear Schrödinger-type equations in two spatial dimensions (2018) (https://arxiv.org/abs/1805.12351, to appear in Math. Model. Anal.)

[3] Bergé, Luc; Skupin, Stefan Modeling ultrashort filaments of light, Discrete Contin. Dyn. Syst., Volume 23 (2009) no. 4, pp. 1099-1139 | Zbl

[4] Bergé, Luc; Skupin, Stefan; Nuter, Rachel; Kasparian, Jérôme; Wolf, Jean-Pierre Ultrashort filaments of light in weakly ionized, optically transparent media, Rep. Prog. Phys., Volume 70 (2007) no. 10, pp. 1633-1713

[5] Bloembergen, Nicolaas Nonlinear optics, Frontiers in Physics, 21, Addison-Wesley, 1977

[6] Colin, Mathieu; Lannes, David Short pulses approximations in dispersive media, SIAM J. Math. Anal., Volume 41 (2009) no. 2, pp. 708-732 | Zbl

[7] Colin, Thierry; Gallice, Gérard; Laurioux, Karen Intermediate models in nonlinear optics, SIAM J. Math. Anal., Volume 36 (2005) no. 5, pp. 1664-1688 | Zbl

[8] Donnat, Phillipe; Joly, Jean-Luc; Métivier, Guy; Rauch, Jeffrey Diffractive nonlinear geometric optics, Sémin. Équ. Dériv. Partielles, Éc. Polytech., Volume 1995-1996 (1996) (Exp. no. 17, 23 pages) | Zbl

[9] Dumas, Éric; Lannes, David; Szeftel, Jeremie Variants of the focusing NLS equation. Derivation, justification and open problems related to filamentation, Laser filamentation. Mathematical methods and models (CRM Series in Mathematical Physics), Springer, 2016, pp. 19-75

[10] Ginibre, Jean; Velo, Giorgio On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal. (1979), pp. 1-32 | Zbl

[11] Joly, Jean-Luc; Metivier, Guy; Rauch, Jeffrey Diffractive nonlinear geometric optics with rectification, Indiana Univ. Math. J., Volume 47 (1998) no. 4, pp. 1167-1241 | Zbl

[12] Katznelson, Yitzhak An introduction to harmonic analysis, Cambridge Mathematical Library, Cambridge University Press, 2004 | Zbl

[13] Kolesik, Miroslav; Moloney, Jerome V. Nonlinear optical pulse propagation simulation: From maxwell’s to unidirectional equations, Phys. Rev. E, Volume 70 (2004), 036604 | DOI

[14] Lannes, David Dispersive effects for nonlinear geometrical optics with rectification, Asymptotic Anal., Volume 18 (1998) no. 1-2, pp. 111-146

[15] Lannes, David High-frequency nonlinear optics: from the nonlinear Schrödinger approximation to ultrashort-pulses equations, Proc. R. Soc. Edinb., Sect. A, Math., Volume 141 (2011) no. 2, pp. 253-286 | Zbl

[16] Lorentz, Hendrik A. The Theory of Electrons, Dover, 1952

[17] Owyoung, Adelbert The origins of the nonlinear refractive indices of liquids and gases, California Institute of Technology (USA) (1971) (Ph. D. Thesis)

[18] Sulem, Catherine; Sulem, Pierre-Louis The nonlinear Schrödinger equation. Self-focusing and wave collapse, Applied Mathematical Sciences, 139, Springer, 1999 | Zbl

[19] Tao, Terence; Visan, Monica; Zhang, Xiaoyi The nonlinear Schrödinger equation with combined power-type nonlinearities, Commun. Partial Differ. Equations, Volume 32 (2007) no. 8, pp. 1281-1343 | Zbl

Cité par Sources :