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

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


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.


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.



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.


  1. 1.
    J. Ginibre, G. Velo, J. Funct. Anal. 32(1), 1 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    V.E. Zakharov, A.B. Shabat, Sov. Phys. JETP 34(1), 62 (1972)MathSciNetADSGoogle Scholar
  3. 3.
    C. Sulem, P.L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Applied Mathematical Sciences, vol. 139 (Springer, New York, 1999)Google Scholar
  4. 4.
    F. Merle, P. Raphaël, Ann. Math. 161(1), 157 (2005)CrossRefzbMATHGoogle Scholar
  5. 5.
    F. Merle, P. Raphaël, Geom. Funct. Anal. 13(3), 591 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    F. Merle, P. Raphaël, Invent. Math. 156(3), 565 (2004)MathSciNetCrossRefADSzbMATHGoogle Scholar
  7. 7.
    F. Merle, P. Raphaël, J. Am. Math. Soc. 19(1), 37 (2006)CrossRefzbMATHGoogle Scholar
  8. 8.
    F. Merle, P. Raphaël, Commun. Math. Phys. 253(3), 675 (2005)CrossRefADSzbMATHGoogle Scholar
  9. 9.
    F. Merle, P. Raphaël, J. Hyperbol. Differ. Eq. 2(4), 919 (2005)CrossRefzbMATHGoogle Scholar
  10. 10.
    F. Merle, P. Raphaël, J. Szeftel, Geom. Funct. Anal. 20(4), 1028 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    J.E. Rothenberg, Opt. Lett. 17(19), 1340 (1992)CrossRefADSGoogle Scholar
  12. 12.
    H.A. Lorentz, Theory of Electrons (Teubner, New York, 1909). Reprint (Dover, Englewood Cliffs, NJ, 1952)Google Scholar
  13. 13.
    N. Bloembergen, Nonlinear Optics. Frontiers in Physics, vol. 21 (W. A. Benjamin, Reading, MA, 1977)Google Scholar
  14. 14.
    A. Owyoung, The origins of the nonlinear refractive indices of liquids and gases. Ph.D. thesis, California Institute of Technology (1971)Google Scholar
  15. 15.
    P. Donnat, J.L. Joly, G. Métivier, J. Rauch, Équations aux dérivées partielles 1995–1996 (École Polytechnique, Palaiseau, 1996). Exp. No. XVIIGoogle Scholar
  16. 16.
    J.L. Joly, G. Métivier, J. Rauch, Indiana U. Math. J. 47(4), 1167 (1998)CrossRefzbMATHGoogle Scholar
  17. 17.
    D. Lannes, Asymptot. Anal. 18(1–2), 111 (1998)MathSciNetzbMATHGoogle Scholar
  18. 18.
    L. Bergé, S. Skupin, Discrete Contin. Dyn. Syst. 23(4), 1099 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    L. Bergé, S. Skupin, R. Nuter, J. Kasparian, J.P. Wolf, Rep. Prog. Phys. 70(10), 1633 (2007)CrossRefADSGoogle Scholar
  20. 20.
    M. Colin, D. Lannes, SIAM J. Math. Anal. 41(2), 708 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    D. Lannes, Proc. R. Soc. Edinb. A Math. 141(2), 253 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    L.A. Kalyakin, Math. USSR Sb. 60(2), 457 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    G. Schneider, Nonlinear Differ. Equ. Appl. 5(1), 69 (1998)CrossRefzbMATHGoogle Scholar
  24. 24.
    T. Colin, G. Gallice, K. Laurioux, SIAM J. Math. Anal. 36(5), 1664 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    M. Kolesik, J.V. Moloney, Phys. Rev. E 70(3), 036604 (2004)CrossRefADSGoogle Scholar
  26. 26.
    D. Lannes, The Water Waves Problem: Mathematical Analysis and Asymptotics. Mathematical Surveys and Monographs, vol. 188 (American Mathematical Society, Providence, RI, 2013)Google Scholar
  27. 27.
    T. Cazenave, F.B. Weissler, Nonlinear Anal. 14(10), 807 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    T. Tao, M. Visan, X. Zhang, Commun. Partial Differ. Equ. 32(8), 1281 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    G. Fibich, G. Papanicolaou, SIAM J. Appl. Math. 60(1), 183 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    F. Merle, C. R. Acad. Sci. Paris Sér. I Math. 304(16), 479 (1987)MathSciNetzbMATHGoogle Scholar
  31. 31.
    F. Merle, Commun. Math. Phys. 149(2), 377 (1992)MathSciNetCrossRefADSzbMATHGoogle Scholar
  32. 32.
    P. Antonelli, R. Carles, C. Sparber, Int. Math. Res. Not. 2015(3), 740 (2015)MathSciNetzbMATHGoogle Scholar
  33. 33.
    M. Tsutsumi, SIAM J. Math. Anal. 15(2), 357 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    G. Fibich, SIAM J. Appl. Math. 61(5), 1680 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    M. Ohta, G. Todorova, Discrete Contin. Dyn. Syst. 23(4), 1313 (2009)MathSciNetzbMATHGoogle Scholar
  36. 36.
    D. Anderson, M. Lisak, Phys. Rev. A 27(3), 1393 (1983)CrossRefADSGoogle Scholar
  37. 37.
    R.S. Strichartz, Duke Math. J. 44(3), 705 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    J.M. Ghidaglia, J.C. Saut, J. Nonlinear Sci. 6(2), 139 (1996)MathSciNetCrossRefADSzbMATHGoogle Scholar
  39. 39.
    P. Kevrekidis, A.R. Nahmod, C. Zeng, Nonlinearity 24(5), 1523 (2011)MathSciNetCrossRefADSzbMATHGoogle Scholar
  40. 40.
    P. Donnat, Quelques contributions mathématiques en optique non linéaire. Ph.D. thesis, École Polytechnique, Paris (1994)Google Scholar

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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