A bifurcation analysis for the Lugiato-Lefever equation

  • Cyril GodeyEmail author
Regular Article
Part of the following topical collections:
  1. Topical Issue: Theory and applications of the Lugiato-Lefever Equation


The Lugiato-Lefever equation is a cubic nonlinear Schrödinger equation, including damping, detuning and driving, which arises as a model in nonlinear optics. We study the existence of stationary waves which are found as solutions of a four-dimensional reversible dynamical system in which the evolutionary variable is the space variable. Relying upon tools from bifurcation theory and normal forms theory, we discuss the codimension 1 bifurcations. We prove the existence of various types of steady solutions, including spatially localized, periodic, or quasi-periodic solutions.

Graphical abstract


  1. 1.
    C. Godey, I.V. Balakireva, A. Coillet, Y.K. Chembo, Phys. Rev. A 89, 063814 (2014)ADSCrossRefGoogle Scholar
  2. 2.
    Y.K. Chembo, C.R. Menyuk, Phys. Rev. A 87, 053852 (2013)ADSCrossRefGoogle Scholar
  3. 3.
    L.A. Lugiato, R. Lefever, Phys. Rev. Lett. 58, 2209 (1987)ADSCrossRefGoogle Scholar
  4. 4.
    T. Miyaji, I. Ohnishi, Y. Tsutsumi, Phys. D: Nonlinear Phenom. 239, 2066 (2010)ADSCrossRefGoogle Scholar
  5. 5.
    R. Mandel, W. Reichel, SIAM J. Appl. Math. 77, 315 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    T. Miyaji, I. Ohnishi, Y. Tsutsumi, Tohoku Math. J. 63, 651 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    K. Kirchgässner, J. Differ. Equ. 45, 113 (1982)ADSCrossRefGoogle Scholar
  8. 8.
    M. Haragus, G. Iooss, Local bifurcations, center manifolds, and normal forms in infinite-dimensional dynamical systems, Universitext (Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2011)Google Scholar
  9. 9.
    G. Iooss, M.C. Pérouème, J. Differ. Equ. 102, 68 (1993)ADSCrossRefGoogle Scholar
  10. 10.
    E. Lombardi, Oscillatory integrals and phenomena beyond all algebraic orders, vol. 1741 of Lecture Notes in Mathematics (Springer-Verlag, Berlin, 2000)Google Scholar
  11. 11.
    C. Godey, C.R. Math. Acad. Sci. Paris 354, 175 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques de BesançonBesançon CedexFrance

Personalised recommendations