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A bifurcation analysis for the Lugiato-Lefever equation

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Abstract

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.

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References

  1. C. Godey, I.V. Balakireva, A. Coillet, Y.K. Chembo, Phys. Rev. A 89, 063814 (2014)

    Article  ADS  Google Scholar 

  2. Y.K. Chembo, C.R. Menyuk, Phys. Rev. A 87, 053852 (2013)

    Article  ADS  Google Scholar 

  3. L.A. Lugiato, R. Lefever, Phys. Rev. Lett. 58, 2209 (1987)

    Article  ADS  Google Scholar 

  4. T. Miyaji, I. Ohnishi, Y. Tsutsumi, Phys. D: Nonlinear Phenom. 239, 2066 (2010)

    Article  ADS  Google Scholar 

  5. R. Mandel, W. Reichel, SIAM J. Appl. Math. 77, 315 (2017)

    Article  MathSciNet  Google Scholar 

  6. T. Miyaji, I. Ohnishi, Y. Tsutsumi, Tohoku Math. J. 63, 651 (2011)

    Article  MathSciNet  Google Scholar 

  7. K. Kirchgässner, J. Differ. Equ. 45, 113 (1982)

    Article  ADS  Google Scholar 

  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)

  9. G. Iooss, M.C. Pérouème, J. Differ. Equ. 102, 68 (1993)

    Article  ADS  Google Scholar 

  10. E. Lombardi, Oscillatory integrals and phenomena beyond all algebraic orders, vol. 1741 of Lecture Notes in Mathematics (Springer-Verlag, Berlin, 2000)

  11. C. Godey, C.R. Math. Acad. Sci. Paris 354, 175 (2016)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Mathématiques de Besançon, Univ. Bourgogne Franche-Comté, 16 route de Gray, 25030, Besançon Cedex, France

    Cyril Godey

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  1. Cyril Godey
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Correspondence to Cyril Godey.

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Contribution to the Topical Issue: “Theory and Applications of the Lugiato-Lefever Equation”, edited by Yanne K. Chembo, Damia Gomila, Mustapha Tlidi, Curtis R. Menyuk.

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Godey, C. A bifurcation analysis for the Lugiato-Lefever equation. Eur. Phys. J. D 71, 131 (2017). https://doi.org/10.1140/epjd/e2017-80057-2

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  • DOI: https://doi.org/10.1140/epjd/e2017-80057-2

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