Advertisement

Interaction of solitons and the formation of bound states in the generalized Lugiato-Lefever equation

  • Pedro Parra-RivasEmail author
  • Damia Gomila
  • Pere Colet
  • Lendert GelensEmail author
Regular Article
Part of the following topical collections:
  1. Topical Issue: Theory and applications of the Lugiato-Lefever Equation

Abstract

Bound states, also called soliton molecules, can form as a result of the interaction between individual solitons. This interaction is mediated through the tails of each soliton that overlap with one another. When such soliton tails have spatial oscillations, locking or pinning between two solitons can occur at fixed distances related with the wavelength of these oscillations, thus forming a bound state. In this work, we study the formation and stability of various types of bound states in the Lugiato-Lefever equation by computing their interaction potential and by analyzing the properties of the oscillatory tails. Moreover, we study the effect of higher order dispersion and noise in the pump intensity on the dynamics of bound states. In doing so, we reveal that perturbations to the Lugiato-Lefever equation that maintain reversibility, such as fourth order dispersion, lead to bound states that tend to separate from one another in time when noise is added. This separation force is determined by the shape of the envelope of the interaction potential, as well as an additional Brownian ratchet effect. In systems with broken reversibility, such as third order dispersion, this ratchet effect continues to push solitons within a bound state apart. However, the force generated by the envelope of the potential is now such that it pushes the solitons towards each other, leading to a null net drift of the solitons.

Graphical abstract

References

  1. 1.
    N. Akhmediev, in General Theory of Solitons, in Soliton-driven Photonics, edited by A.D. Boardman, A.P. Sukhorukov (Kluwer Academic Publishers, Netherlands, 2001), pp. 371–395Google Scholar
  2. 2.
    P.B. Umbanhowar, F. Melo, H.L. Swinney, Nature 382, 793 (1996)ADSCrossRefGoogle Scholar
  3. 3.
    W.J. Firth, A. Lord, A.J. Scroggie, Phys. Scr. T 12, 67 (1996)Google Scholar
  4. 4.
    W.J. Firth, G.K. Harkness, A. Lord, J. McSloy, D. Gomila, P. Colet, J. Opt. Soc. Am. B 19, 747 (2002)ADSCrossRefGoogle Scholar
  5. 5.
    V.K. Vanag, I.R. Epstein, Chaos 17, 037110 (2007)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    F. Leo, L. Gelens, P. Emplit, M. Haelterman, S. Coen, Opt. Express 21, 9180 (2013)ADSCrossRefGoogle Scholar
  7. 7.
    D. Michaelis, U. Peschel, C. Etrich, F. Lederer, IEEE J. Quantum Electron. 39, 255 (2003)ADSCrossRefGoogle Scholar
  8. 8.
    D. Gomila, M.A. Matias, P. Colet, Phys. Rev. Lett. 94, 063905 (2005)ADSCrossRefGoogle Scholar
  9. 9.
    M. Cross, P. Hohenberg, Rev. Mod. Phys. 65, 851 (1993)ADSCrossRefGoogle Scholar
  10. 10.
    J. Murray, Mathematical Biology (Springer, New York, 1989)Google Scholar
  11. 11.
    R. Hoyle, Pattern Formation: An Introduction to Methods (Cambridge University Press, 2006)Google Scholar
  12. 12.
    B.A. Malomed, Phys. Rev. A 44, 6954 (1991)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    B.A. Malomed, Phys. Rev. E 47, 2874 (1993)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    D. Cai, A.R. Bishop, N. Gronbech-Jensen, B.A. Malomed, Phys. Rev. E 49, 1677 (1994)ADSCrossRefGoogle Scholar
  15. 15.
    A.V. Buryak, N.N. Akhmediev, Phys. Rev. E 51, 3572 (1995)ADSCrossRefGoogle Scholar
  16. 16.
    N. Akhmediev, A. Ankiewicz, J.M. Soto-Crespo, Phys. Rev. Lett. 79, 4047 (1997)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    I.V. Barashenkov, Yu.S. Smirnov, N.V. Alexeeva, Phys. Rev. E 57, 2 (1998)CrossRefGoogle Scholar
  18. 18.
    B. Schäpers, M. Feldmann, T. Ackemann, W. Lange, Phys. Rev. Lett. 85, 748 (2000)ADSCrossRefGoogle Scholar
  19. 19.
    L.A. Lugiato, R. Lefever, Phys. Rev. Lett. 58, 2209 (1987)ADSCrossRefGoogle Scholar
  20. 20.
    M. Haelterman, S. Trillo, S. Wabnitz, Opt. Commun. 91, 401 (1992)ADSCrossRefGoogle Scholar
  21. 21.
    F. Leo, S. Coen, P. Kockeart, S.-P. Gorza, Ph. Emplit, M. Haelterman, Nat. Photon. 4, 471 (2010)ADSCrossRefGoogle Scholar
  22. 22.
    S. Coen, H.G. Randle, T. Sylvestre, M. Erkintalo, Opt. Lett. 38, 37 (2013)ADSCrossRefGoogle Scholar
  23. 23.
    Y.K. Chembo, C. Menyuk, Phys. Rev. A 87, 053852 (2013)ADSCrossRefGoogle Scholar
  24. 24.
    T.J. Kippenberg, R. Holzwarth, S.A. Diddams, Science 332, 555 (2011)ADSCrossRefGoogle Scholar
  25. 25.
    P. Del’Haye, T. Herr, E. Gavartin, M.L. Gorodetsky, R. Holzwarth, T.J. Kippenberg, Phys. Rev. Lett. 107, 063901 (2011)ADSCrossRefGoogle Scholar
  26. 26.
    Y. Okawachi, K. Saha, J.S. Levy, Y. Henry Wen, M. Lipson, A.L. Gaeta, Opt. Lett. 36, 3398 (2011)ADSCrossRefGoogle Scholar
  27. 27.
    T. Hansch, Rev. Mod. Phys. 78, 1297 (2006)ADSCrossRefGoogle Scholar
  28. 28.
    S.B. Papp, K. Beha, P. Del’Haye, F. Quinlan, H. Lee, K.J. Vahala, S.A. Diddams, Optica 1, 10 (2014)CrossRefGoogle Scholar
  29. 29.
    J.D. Jost, T. Herr, C. Lecaplain, V. Brasch, M.H.P. Pfeiffer, T.J. Kippenberg, Optica 2, 706 (2015)CrossRefGoogle Scholar
  30. 30.
    F.M. Mitschke, L.F. Mollenauer, Opt. Lett. 12, 355 (1987)ADSCrossRefGoogle Scholar
  31. 31.
    D.J. Kaup, Phys. Rev. A 42, 5689 (1990)ADSCrossRefGoogle Scholar
  32. 32.
    M. Tlidi, P. Mandel, R. Lefever, Phys. Rev. Lett. 73, 640 (1994)ADSCrossRefGoogle Scholar
  33. 33.
    M. Tlidi, A.G. Vladimirov, P. Mandel, J. Quantum Electron. 39, 2 (2003)CrossRefGoogle Scholar
  34. 34.
    B.A. Malomed, Phys. Rev. E 58, 7928 (1998)ADSCrossRefGoogle Scholar
  35. 35.
    J.M. Soto-Crespo, Ph. Grelu, N. Akhmediev, N. Devine, Phys. Rev. E 75, 016613 (2007)ADSCrossRefGoogle Scholar
  36. 36.
    Y. Wang, F. Leo, J. Fatome, K. Luo, J.K. Jang, M.J. Erkintalo, S.G. Murdoch, S. Coen, CLEO:QELS Fundamental Science, FF2A. 6 (2016)Google Scholar
  37. 37.
    Y. Wang, F. Leo, J. Fatome, K. Luo, J.K. Jang, M.J. Erkintalo, S.G. Murdoch, S. Coen (submitted), https://arxiv.org/abs/1703.10604
  38. 38.
    V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M.H.P. Pfeiffer, M.L. Gorodetsky, T.J. Kippenberg, Science 351, 357 (2016)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    X. Yi, Q.-F. Yang, K.Y. Yang, K. Vahala, Opt. Lett. 41, 2037 (2016)ADSCrossRefGoogle Scholar
  40. 40.
    P. Del’Haye, A. Coillet, W. Loh, K. Beha, S.B. Papp, S.A. Diddams, Nat. Commun. 6, 5668 (2015)CrossRefGoogle Scholar
  41. 41.
    P. Coullet, C. Riera, C. Tresser, Phys. Rev. Lett. 84, 3069 (2000)ADSCrossRefGoogle Scholar
  42. 42.
    P.D. Woods, A.R. Champneys, Physica D 129, 147 (1999)ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    D. Gomila, A.J. Scroggie, W.J. Firth, Physica D 227, 70 (2007)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    J. Burke, E. Knobloch, Phys. Rev. E 73, 056211 (2006)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    P. Parra-Rivas, D. Gomila, M.A. Matías, S. Coen, L. Gelens, Phys. Rev. A 89, 043813 (2014)ADSCrossRefGoogle Scholar
  46. 46.
    C. Elphick, E. Meron, E.A. Spiegel, SIAM J. Appl. Math. 50, 490 (1990)MathSciNetCrossRefGoogle Scholar
  47. 47.
    I. Aranson, K. Gorshkov, A. Lomov, M. Rabinovich, Physica D 43, 435 (1990)ADSMathSciNetCrossRefGoogle Scholar
  48. 48.
    G. Kozyreff, P. Assemat, S.J. Chapman, Phys. Rev. Lett. 103, 164501 (2009)ADSCrossRefGoogle Scholar
  49. 49.
    G. Kozyreff, L. Gelens, Phys. Rev. A 84, 023819 (2011)ADSCrossRefGoogle Scholar
  50. 50.
    L. Gelens, D. Gomila, G. Van der Sande, M.A. Matas, P. Colet, Phys. Rev. Lett. 104, 154101 (2010)ADSCrossRefGoogle Scholar
  51. 51.
    P. Colet, M.A. Matías, L. Gelens, D. Gomila, Phys. Rev. E 89, 012914 (2014)ADSCrossRefGoogle Scholar
  52. 52.
    L. Gelens, M.A. Matías, D. Gomila, T. Dorissen, P. Colet, Phys. Rev. E 89, 012915 (2014)ADSCrossRefGoogle Scholar
  53. 53.
    W.J. Firth, A. Lord, J. Mod. Opt. 43, 1071 (1996)ADSCrossRefGoogle Scholar
  54. 54.
    A.R. Champneys, Physica D 112, 158 (1998)ADSMathSciNetCrossRefGoogle Scholar
  55. 55.
    J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, New York, 1983)Google Scholar
  56. 56.
    R.L. Devaney, Trans. Am. Math. Soc. 218, 89 (1976)CrossRefGoogle Scholar
  57. 57.
    A.J. Homburg, B. Sandstede, in Handbook of Dynamical Systems, edited by B. Hasselblatt, H. Broer, F. Takens (North Holland, Amsterdam, The Netherlands, 2010), Chap. 8, pp. 379–524Google Scholar
  58. 58.
    E. Knobloch, Annu. Rev. Cond. Matter Phys. 6, 325 (2015)ADSCrossRefGoogle Scholar
  59. 59.
    J. Burke, E. Knobloch, Discrete Cont. Dyn. Syst. Suppl., September, 109 (2009)Google Scholar
  60. 60.
    M.R.E. Lamont, Y. Okawachi, A.L. Gaeta, Opt. Lett. 38, 3478 (2013)ADSCrossRefGoogle Scholar
  61. 61.
    L. Gelens, G. Van der Sande, P. Tassin, M. Tlidi, P. Kockaert, D. Gomila, I. Veretennicoff, J. Danckaert, Phys. Rev. A 75, 063812 (2007)ADSCrossRefGoogle Scholar
  62. 62.
    M. Tlidi, L. Gelens, Opt. Lett. 35, 306 (2010)ADSCrossRefGoogle Scholar
  63. 63.
    C. Milián, D.V. Skryabin, Opt. Express 22, 3732 (2014)ADSCrossRefGoogle Scholar
  64. 64.
    P. Parra-Rivas, D. Gomila, F. Leo, S. Coen, L. Gelens, Opt. Lett. 39, 2971 (2014)ADSCrossRefGoogle Scholar
  65. 65.
    C. Milián, A.V. Gorbach, M. Taki, A.V. Yulin, D.V. Skryabin, Phys. Rev. A 92, 033851 (2015)ADSCrossRefGoogle Scholar
  66. 66.
    B.A. Malomed, Europhys. Lett. 30, 507 (1995)ADSCrossRefGoogle Scholar
  67. 67.
    M.O. Magnasco, Phys. Rev. Lett. 71, 1477 (1993)ADSCrossRefGoogle Scholar
  68. 68.
    F. Gustave, C. Rimoldi, P. Walczak, L. Columbo, M. Brambilla, F. Prati, G. Tissoni, S. Barland, Eur. Phys. J. D (in press)Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Applied Physics Research Group (APHY), Vrije Universiteit BrusselBrusselsBelgium
  2. 2.Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat de les Illes BalearsPalma de MallorcaSpain
  3. 3.Laboratory of Dynamics in Biological Systems, Department of Cellular and Molecular Medicine, University of Leuven (KU Leuven)LeuvenBelgium

Personalised recommendations