Advertisement

Localized structures and spatiotemporal chaos: comparison between the driven damped sine-Gordon and the Lugiato-Lefever model

  • Michel A. Ferré
  • Marcel G. ClercEmail author
  • Saliya Coulibally
  • René G. Rojas
  • Mustapha Tlidi
Regular Article
Part of the following topical collections:
  1. Topical Issue: Theory and applications of the Lugiato-Lefever Equation

Abstract

Driven damped coupled oscillators exhibit complex spatiotemporal dynamics. An archetype model is the driven damped sine-Gordon equation, which can describe several physical systems such as coupled pendula, extended Josephson junction, optical systems and driven magnetic wires. Close to resonance an enveloped model in the form Lugiato-Lefever equation can be derived from the driven damped sine-Gordon equation. We compare the dynamics obtained from both models. Unexpectedly, qualitatively similar dynamical behaviors are obtained for both models including homogeneous steady states, localized structures, and pattern waves. For large forcing, both systems share similar spatiotemporal chaos.

Graphical abstract

References

  1. 1.
    L.A. Lugiato, R. Lefever, Phys. Rev. Lett. 58, 2209 (1987)ADSCrossRefGoogle Scholar
  2. 2.
    D.W. McLaughlin, J.V. Moloney, A.C. Newell, Phys. Rev. Lett. 51, 75 (1983)ADSCrossRefGoogle Scholar
  3. 3.
    D.W. McLaughlin, J.V. Moloney, A.C. Newell, Phys. Rev. Lett. 54, 681 (1985)ADSCrossRefGoogle Scholar
  4. 4.
    A.M. Turing, Phil. Trans. R. Soc. Lond. B: Biol. Sci. 237, 37 (1952)ADSCrossRefGoogle Scholar
  5. 5.
    I. Prigogine, R. Lefever, J. Chem. Phys. 48, 1695 (1968)ADSCrossRefGoogle Scholar
  6. 6.
    P. Glansdorff, I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations (NY Interscience, New York, 1971)Google Scholar
  7. 7.
    P. Kockaert, P. Tassin, G. Van der Sande, I. Veretennicoff, M. Tlidi, Phys. Rev. A 74, 033822 (2006)ADSCrossRefGoogle Scholar
  8. 8.
    U. Peschel, O. Egorov, F. Lederer, Opt. Lett. 29, 1909 (2004)ADSCrossRefGoogle Scholar
  9. 9.
    M. Haelterman, S. Trillo, S. Wabnitz, Opt. Commun. 91, 401 (1992)ADSCrossRefGoogle Scholar
  10. 10.
    Y.K. Chembo, C.R. Menyuk, Phys. Rev. A 87, 053852 (2013)ADSCrossRefGoogle Scholar
  11. 11.
    T. Hansson, D. Modotto, S. Wabnitz, Phys. Rev. A 88, 023819 (2013)ADSCrossRefGoogle Scholar
  12. 12.
    A. Coillet, J. Dudley, G. Genty, L. Larger, Y.K. Chembo, Phys. Rev. A 89, 013835 (2014)ADSCrossRefGoogle Scholar
  13. 13.
    M. Tlidi, M. Haelterman, P. Mandel, Europhys. Lett. 42, 505 (1998)ADSCrossRefGoogle Scholar
  14. 14.
    M. Tlidi, M. Haelterman, P. Mandel, Quant. Semiclass. Opt.: J. Eur. Opt. Soc. B 10, 869 (1998)ADSCrossRefGoogle Scholar
  15. 15.
    P. Tassin, G. Van der Sande, N. Veretenov, P. Kockaert, I. Veretennicoff, M. Tlidi, Opt. Express 14, 9338 (2006)ADSCrossRefGoogle Scholar
  16. 16.
    G.J. Morales, Y.C. Lee, Phys. Rev. Lett. 33, 1016 (1974)ADSCrossRefGoogle Scholar
  17. 17.
    K. Nozaki, N. Bekki, Phys. Lett. A 102, 383 (1984)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    D.J. Kaup, A.C. Newell, Phys. Rev. B 18, 5162 (1978)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    G. Terrones, D.W. McLaughlin, E.A. Overman, A.J. Pearlstein, SIAM J. Appl. Math. 50, 791 (1990)MathSciNetCrossRefGoogle Scholar
  20. 20.
    L.D. Landau, E.M. Lifshitz, Mechanics, Course of Theoretical Physics (Pergamon Press, 1976), Vol. 1Google Scholar
  21. 21.
    A. Frova, M. Marenzana, Thus spoke Galileo: the great scientist’s ideas and their relevance to the present day (Oxford University Press, 2006)Google Scholar
  22. 22.
    G.L. Baker, J.A. Blackburn, The pendulum: a case study in physics (Oxford University Press, 2005)Google Scholar
  23. 23.
    Y.S. Kivshar, B.A. Malomed, Rev. Mod. Phys. 61, 763 (1989)ADSCrossRefGoogle Scholar
  24. 24.
    N.N. Bogoliubov, Y.A. Mitropolski, Asymptotic methods in the theory of non-linear oscillations (Gordon and Breach, New York, 1961)Google Scholar
  25. 25.
    J. Cuevas-Maraver, P.G. Kevrekidis, F. Williams, The Sine-Gordon Model and its Applications (Springer, 2014)Google Scholar
  26. 26.
    D. Bennett, A. Bishop, S. Trullinger, Zeitschrift für Physik B Condensed Matter 47, 265 (1982)ADSCrossRefGoogle Scholar
  27. 27.
    L.P. Gor’kov, G. Grüner, Charge density waves in solids (Elsevier, 2012), Vol. 25Google Scholar
  28. 28.
    O.M. Braun, Y. Kivshar, The Frenkel-Kontorova model: concepts, methods, and applications (Springer Science & Business Media, 2013)Google Scholar
  29. 29.
    E. Berrios-Caro, M.G. Clerc, A.O. Leon, Phys. Rev. E 94, 052217 (2016)ADSCrossRefGoogle Scholar
  30. 30.
    M. Clerc, P. Coullet, E. Tirapegui, Phys. Rev. Lett. 83, 3820 (1999)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    M. Clerc, P. Coullet, E. Tirapegui, Opt. Commun. 167, 159 (1999)ADSCrossRefGoogle Scholar
  32. 32.
    M. Clerc, P. Coullet, E. Tirapegui, Int. J. Bifurc. Chaos 11, 591 (2001)CrossRefGoogle Scholar
  33. 33.
    I.V. Barashenkov, Y.S. Smirnov, Phys. Rev. E 54, 5707 (1996)ADSCrossRefGoogle Scholar
  34. 34.
    M. Tlidi, R. Lefever, P. Mandel, Quant. Semiclass. Opt.: J. Eur. Opt. Soc. B 8, 931 (1996)ADSCrossRefGoogle Scholar
  35. 35.
    A.J. Scroggie, W.J. Firth, G.S. McDonald, M. Tlidi, R. Lefever, L.A. Lugiato, Chaos Solitons Fract. 4, 1323 (1994)ADSCrossRefGoogle Scholar
  36. 36.
    I.V. Barashenkov, Y.S. Smirnov, N.V. Alexeeva, Phys. Rev. E 57, 2350 (1998)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    S. Coen, M. Tlidi, P. Emplit, M. Haelterman, Phys. Rev. Lett. 83, 2328 (1999)ADSCrossRefGoogle Scholar
  38. 38.
    S. Coulibaly, M. Taki, M. Tlidi, Opt. Express 22, 483 (2014)ADSCrossRefGoogle Scholar
  39. 39.
    Z. Liu, M. Ouali, S. Coulibaly, M. Clerc, M. Taki, M. Tlidi, Opt. Lett. 42, 1063 (2017)ADSCrossRefGoogle Scholar
  40. 40.
    N. Akhmediev, B. Kibler, F. Baronio, M. Beli, W.P. Zhong, Y. Zhang, W. Chang, J.M. Soto-Crespo, P. Vouzas, P. Grelu et al., J. Opt. 18, 063001 (2016)ADSCrossRefGoogle Scholar
  41. 41.
    M. Tlidi, K. Panajotov, Chaos: Interdiscip. J. Nonlinear Sci. 27, 013119 (2017)CrossRefGoogle Scholar
  42. 42.
    F. Palmero, J. Han, L.Q. English, T. Alexander, P. Kevrekidis, Phys. Lett. A 380, 402 (2016)ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    Y. Xu, T.J. Alexander, H. Sidhu, P.G. Kevrekidis, Phys. Rev. E 90, 042921 (2014)ADSCrossRefGoogle Scholar
  44. 44.
    M. Tlidi, M. Taki, T. Kolokolnikov, Chaos: Interdiscip. J. Nonlinear Sci. 17, 037101 (2007)CrossRefGoogle Scholar
  45. 45.
    O. Descalzi, M.G. Clerc, S. Residori, G. Assanto, Localized states in physics: solitons and patterns (Springer-Verlag Berlin Heidelberg, 2011)Google Scholar
  46. 46.
    H.G. Purwins, H. Bdeker, S. Amiranashvili, Adv. Phys. 59, 485 (2010)ADSCrossRefGoogle Scholar
  47. 47.
    M. Tlidi, K. Staliunas, K. Panajotov, A.G. Vladimirov, M.G. Clerc, Phil. Trans. R. Soc. Lond. A: Math. Phys. Eng. Sci. (2014)Google Scholar
  48. 48.
    L. Lugiato, P. Franco, M. Brambilla, Nonlinear Optical Systems (Cambridge University Press, 2015)Google Scholar
  49. 49.
    D. Mihalache, Rom. Rep. Phys. 67, 1383 (2015)Google Scholar
  50. 50.
    Y. He, X. Zhu, D. Mihalache, Rom. J. Phys. 61, 595 (2016)Google Scholar
  51. 51.
    D. Mihalache, Rom. Rep. Phys. 69, 403 (2017)Google Scholar
  52. 52.
    M.G. Clerc, S. Coulibaly, D. Laroze, EPL 97, 30006 (2012)ADSCrossRefGoogle Scholar
  53. 53.
    I. Barashenkov, E. Zemlyanaya, Phys. D: Nonlinear Phenom. 132, 363 (1999)ADSCrossRefGoogle Scholar
  54. 54.
    J. Cuevas, L.Q. English, P.G. Kevrekidis, M. Anderson, Phys. Rev. Lett. 102, 224101 (2009)ADSCrossRefGoogle Scholar
  55. 55.
    F. Leo, L. Gelens, P. Emplit, M. Haelterman, S. Coen, Opt. Express 21, 9180 (2013)ADSCrossRefGoogle Scholar
  56. 56.
    M.G. Clerc, C. Falcón, M.A. García-Ñustes, V. Odent, I. Ortega, Chaos: Interdiscip. J. Nonlinear Sci. 24, 023133 (2014)CrossRefGoogle Scholar
  57. 57.
    A.R. Bishop, K. Fesser, P.S. Lomdahl, W.C. Kerr, M.B. Williams, S.E. Trullinger, Phys. Rev. Lett. 50, 1095 (1983)ADSMathSciNetCrossRefGoogle Scholar
  58. 58.
    A.R. Bishop, M.G. Forest, D.W. McLaughlin, E. Overman, Phys. D: Nonlinear Phenom. 23, 293 (1986)ADSCrossRefGoogle Scholar
  59. 59.
    E. Overman, D.W. McLaughlin, A.R. Bishop, Phys. D: Nonlinear Phenom. 19, 1 (1986)ADSCrossRefGoogle Scholar
  60. 60.
    A.R. Bishop, D. McLaughlin, M.G. Forest, E.A. Overman, Phys. Lett. A 127, 335 (1988)ADSMathSciNetCrossRefGoogle Scholar
  61. 61.
    A. Bishop, M.G. Forest, D. McLaughlin, E. Overman, Phys. Lett. A 144, 17 (1990)ADSCrossRefGoogle Scholar
  62. 62.
    A.R. Bishop, R. Flesch, M.G. Forest, D.W. McLaughlin, E.A. Overman, II, SIAM J. Math. Anal. 21, 1511 (1990)MathSciNetCrossRefGoogle Scholar
  63. 63.
    A. Gavrielides, T. Kottos, V. Kovanis, G.P. Tsironis, Phys. Rev. E 58, 5529 (1998)ADSCrossRefGoogle Scholar
  64. 64.
    A. Pikovsky, A. Politi, Lyapunov Exponents: A Tool to Explore Complex Dynamics (Cambridge University Press, 2016)Google Scholar
  65. 65.
    H. Chaté, Nonlinearity 7, 185 (1994)ADSMathSciNetCrossRefGoogle Scholar
  66. 66.
    J.F. Claerbout, Fundamentals of Geophysical Data Processing (McGraw-Hill, New York, 1976)Google Scholar
  67. 67.
    M.G. Clerc, N. Verschueren, Phys. Rev. E 88, 052916 (2013)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Michel A. Ferré
    • 1
  • Marcel G. Clerc
    • 1
    Email author
  • Saliya Coulibally
    • 2
  • René G. Rojas
    • 3
  • Mustapha Tlidi
    • 4
  1. 1.Departamento de Física, FCFM, Universidad de ChileSantiagoChile
  2. 2.Université de Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes et MoléculesLilleFrance
  3. 3.Intituto de Física, Pontificia Universidad Católica de ValparaísoValparaísoChile
  4. 4.Faculté des Sciences, Université Libre de Bruxelles (U.L.B.)BruxellesBelgium

Personalised recommendations