Skip to main content
SpringerLink
Log in

Spatiotemporal chaos and two-dimensional dissipative rogue waves in Lugiato-Lefever model

  • Regular Article
  • Published:
The European Physical Journal D Aims and scope Submit manuscript

Abstract

Driven nonlinear optical cavities can exhibit complex spatiotemporal dynamics. We consider the paradigmatic Lugiato-Lefever model describing driven nonlinear optical resonator. This model is one of the most-studied nonlinear equations in optics. It describes a large spectrum of nonlinear phenomena from bistability, to periodic patterns, localized structures, self-pulsating localized structures and to a complex spatiotemporal behavior. The model is considered also as prototype model to describe several optical nonlinear devices such as Kerr media, liquid crystals, left handed materials, nonlinear fiber cavity, and frequency comb generation. We focus our analysis on a spatiotemporal chaotic dynamics in one-dimension. We identify a route to spatiotemporal chaos through an extended quasiperiodicity. We have estimated the Kaplan-Yorke dimension that provides a measure of the strange attractor complexity. Likewise, we show that the Lugiato-Leferver equation supports rogues waves in two-dimensional settings. We characterize rogue-wave formation by computing the probability distribution of the pulse height.

Graphical abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  ADS  Google Scholar 

  2. Y.K. Chembo, I.S. Grudinin, N. Yu, Phys. Rev. A 92, 043818 (2015)

    Article  ADS  Google Scholar 

  3. M. Anderson, F. Leo, S. Coen, M. Erkintalo, S.G. Murdoch, Optica 3, 1071 (2016)

    Article  Google Scholar 

  4. D. Ruelle, Commun. Math. Phys. 87, 287 (1982)

    Article  ADS  Google Scholar 

  5. P. Manneville, Dissipative Structures and Weak Turbulence (Academic Press, San Diego, 1990)

  6. A. Pikovsky, A. Politi, Lyapunov Exponents: A Tool to Explore Complex Dynamics (Cambridge University Press, 2016)

  7. D.R. Solli, C. Koonath, B. Jalali, Nature 450, 1054 (2007)

    Article  ADS  Google Scholar 

  8. N. Akhmediev, J.M. Dudley, D.R. Solli, S.K. Turitsyn, J. Opt. 15, 060201 (2013)

    Article  ADS  Google Scholar 

  9. M. Onorato, S. Residori, U. Bortolozzo, A. Montina, F.T. Arecchi, Phys. Rep. 528, 47 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  10. M. Dudley, F. Dias, M. Erkintalo, G. Genty, Nat. Photon. 8, 755 (2014)

    Article  ADS  Google Scholar 

  11. N. Akhmediev et al., J. Opt. 18, 063001 (2016)

    Article  ADS  Google Scholar 

  12. Y.K. Chembo, N. Yu, Phys. Rev. A 82, 033801 (2010)

    Article  ADS  Google Scholar 

  13. J.M. Soto-Crespo, Ph. Grelu, N. Akhmediev, Phys. Rev. E 84, 016604 (2011)

    Article  ADS  Google Scholar 

  14. M. Tlidi, K. Panajotov, Chaos 27, 013119 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  15. M. Tlidi, Y. Gandica, G. Sonnino, E. Averlant, K. Panajotov, Entropy 18, 64 (2016)

    Article  ADS  Google Scholar 

  16. H. Bénard, Ph.D. thesis, Université de Paris, Thése de doctorat Sciences physiques, Faculté des sciences de l’Université de Paris, (1901)

  17. A.M. Turing, Phil. Trans. R. Soc. Lond. B: Biol. Sci. 237, 37 (1952)

    Article  ADS  Google Scholar 

  18. I. Prigogine, R. Lefever, J. Chem. Phys. 48, 1695 (1968)

    Article  ADS  Google Scholar 

  19. P. Glansdorff, I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations (Wiley Interscience, 1971)

  20. M. Haelterman, S. Trillo, S. Wabnitz, Opt. Commun. 91, 401 (1992)

    Article  ADS  Google Scholar 

  21. P. Kockaert, P. Tassin, G. Van der Sande, I. Veretennicoff, M. Tlidi, Phys. Rev. A 74, 033822 (2006)

    Article  ADS  Google Scholar 

  22. L. Gelens, G. Van der Sande, P. Tassin, M. Tlidi, P. Kockaert, D. Gomila, I. Veretennicoff, J. Danckaert, Phys. Rev. A 75, 063812 (2007)

    Article  ADS  Google Scholar 

  23. P. Tassin, G. Van der Sande, N. Veretenov, P. Kockaert, I. Veretennicoff, M. Tlidi, Opt. Express 14, 9338 (2006)

    Article  ADS  Google Scholar 

  24. V. Odent, M. Tlidi, M.G. Clerc, P. Glorieux, E. Louvergneaux, Phys. Rev. A 90, 011806 (2014)

    Article  ADS  Google Scholar 

  25. M Tlidi, M. Haelterman, P. Mandel, Quant. Semiclassical Opt. 10, (1998)

  26. L. Lugiato, F. Prati, M. Brambilla, Nonlinear Optical Systems (Cambridge University Press, 2015)

  27. W.J. Firth, A.J. Scroggie, G.S. McDonald, L. Lugiato, Phys. Rev. A 46, 3609 (1992)

    Article  ADS  Google Scholar 

  28. M. Tlidi, Ph.D. thesis, ULB Belgium, unpublished (1995)

  29. M. Tlidi, R. Lefever, P. Mandel, Quant. Semiclassical Opt. 8, 931 (1996)

    Article  ADS  Google Scholar 

  30. M. Tlidi, M. Haelterman, P. Mandel, Europhys. Lett. 42, 505 (1998)

    Article  ADS  Google Scholar 

  31. A.J. Scorggie, W.J. Firth, G.S. McDonald, M. Tlidi, R. Lefever, L.A. Lugiato, Chaos Solitons Fract. 4, 1323 (1994)

    Article  ADS  Google Scholar 

  32. Z. Liu, M. Ouali, S. Coulibaly, M.G. Clerc, M. Taki, M. Tlidi, Opt. Lett. 24, 1063 (2017)

    Article  ADS  Google Scholar 

  33. D. Gomila, P. Colet, Phys. Rev. A 68, 011801 (2003)

    Article  ADS  Google Scholar 

  34. M.G. Clerc, N. Verschueren, Phys. Rev. E 88, 052916 (2013)

    Article  ADS  Google Scholar 

  35. N. Akhmediev, J.M. Soto-Crespo, G. Town, Phys. Rev. E 63, 056602 (2001)

    Article  ADS  Google Scholar 

  36. F. Leo, L. Gelens, P. Emplit, M. Haelterman, S. Coen, Opt. Express 21, 9180 (2013)

    Article  ADS  Google Scholar 

  37. K. Panajotov, M. Tlidi, Opt. Lett. 39, 4739 (2014)

    Article  ADS  Google Scholar 

  38. E. Ott, Chaos in Dynamical Systems, 2nd edn. (Cambridge University Press, Cambridge, 2002)

  39. U. Frisch, Turbulence: the legacy of AN Kolmogorov (Cambridge, university press, 1995)

  40. A. Coillet, Y.K. Chembo, Chaos 24, 013113 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  41. F. Selmi, S. Coulibaly, Z. Loghmari, I. Sagnes, G. Beaudoin, M.G. Clerc, S. Barbay, Phys. Rev. Lett. 116, 013901 (2016)

    Article  ADS  Google Scholar 

  42. S. Coulibaly, M.G. Clerc, F. Selmi, S. Barbay, Phys. Rev. E 95, 023816 (2017)

    Article  ADS  Google Scholar 

  43. D.H. Peregrine, J. Aust. Math. Soc. Ser. B 25, 16 (1983)

    Article  MathSciNet  Google Scholar 

  44. A Mussot, E Louvergneaux, N. Akhmediev, F. Reynaud, L. Delage, M. Taki, Phys. Rev. Lett. 101, 113904 (2008)

    Article  ADS  Google Scholar 

  45. B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, J.M. Dudley, Nat. Phys. 6, 790 (2010)

    Article  Google Scholar 

  46. A. Chabchoub, N.P. Hoffmann, N. Akhmediev, Phys. Rev. Lett. 106, 204502 (2011)

    Article  ADS  Google Scholar 

  47. A. Chabchoub, N. Akhmediev, N.P. Hoffmann, Phys. Rev. E 86, 016311 (2012)

    Article  ADS  Google Scholar 

  48. N. Akhmediev, J.M. Soto-Crespo, A. Ankiewicz, Phys. Lett. A 373, 2137 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  49. V. Odent, M. Taki, E. Louvergneaux, New J. Phys. 13, 113026 (2011)

    Article  ADS  Google Scholar 

  50. M. Tlidi, M. Taki, T. Kolokolnikov, Chaos 17, 037101 (2007)

    Article  ADS  Google Scholar 

  51. N. Akhmediev, A. Ankiewicz (eds.), Dissipative Solitons: from Optics to Biology and Medicine, Lecture Notes in Physics (Springer, Heidelberg, 2008), Vol. 751

  52. H. Leblond, D. Mihalache, Phys. Rep. 523, 61 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  53. O. Descalzi, M.G. Clerc, S. Residori, G. Assanto, Localized States in Physics: Solitons and Patterns: Solitons and Patterns (Springer, 2011)

  54. M. Tlidi, K. Staliunas, K. Panajotov, A.G. Vladimiorv, M. Clerc, Phil. Trans. R. Soc. A 372, 20140101 (2014)

    Article  ADS  Google Scholar 

  55. M. Tlidi, M.G. Clerc (eds.), Nonlinear Dynamics: Materials, Theory and Experiments (Springer Proceedings in Physics, 2016), Vol. 173

  56. D. Turaev, A.G. Vladimirov, S. Zelik, Phys. Rev. Lett. 108, 263906 (2012)

    Article  ADS  Google Scholar 

  57. W.J. Firth, G.K. Harkness, A. Lord, J.M. McSloy, D. Gomila, P. Colet, J. Opt. Soc. Am. B 19, 747 (2002)

    Article  ADS  Google Scholar 

  58. P. Parra-Rivas, D. Gomila, M.A. Matas, S. Coen, L. Gelens Phys. Rev. A 89, 043813 (2014)

    Article  ADS  Google Scholar 

  59. C. Rimoldi, S. Barland, F. Prati, G. Tissoni, Phys. Rev. A 95, 023841 (2017)

    Article  ADS  Google Scholar 

  60. K. Panajotov, D. Puzyrev, A.G. Vladimirov, S.V. Gurevich, M. Tlidi, Phys. Rev. A 93, 043835 (2016)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Physics and Photonics (IR-TONA), Vrije Universiteit Brussels, Pleinlaan 2, 1050, Brussels, Belgium

    Krassimir Panajotov

  2. Institute of Solid State Physics, 72 Tzarigradsko Chaussee Blvd., 1784, Sofia, Bulgaria

    Krassimir Panajotov

  3. Departamento de Física, FCFM, Universidad de Chile, Casilla 487-3, Santiago, Chile

    Marcel G. Clerc

  4. Faculté des Sciences, Optique Nonlinéaire Thérorique, Université Libre de Bruxelles (U.L.B.), C.P. 231, Campus Plaine, 1050, Bruxelles, Belgium

    Mustapha Tlidi

Authors
  1. Krassimir Panajotov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcel G. Clerc
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mustapha Tlidi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Krassimir Panajotov.

Additional information

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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Panajotov, K., Clerc, M.G. & Tlidi, M. Spatiotemporal chaos and two-dimensional dissipative rogue waves in Lugiato-Lefever model. Eur. Phys. J. D 71, 176 (2017). https://doi.org/10.1140/epjd/e2017-80068-y

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1140/epjd/e2017-80068-y

Search

Navigation