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

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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.

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References

  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. 371–395

  2. P.B. Umbanhowar, F. Melo, H.L. Swinney, Nature 382, 793 (1996)

    Article  ADS  Google Scholar 

  3. W.J. Firth, A. Lord, A.J. Scroggie, Phys. Scr. T 12, 67 (1996)

    Google Scholar 

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

    Article  ADS  Google Scholar 

  5. V.K. Vanag, I.R. Epstein, Chaos 17, 037110 (2007)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  7. D. Michaelis, U. Peschel, C. Etrich, F. Lederer, IEEE J. Quantum Electron. 39, 255 (2003)

    Article  ADS  Google Scholar 

  8. D. Gomila, M.A. Matias, P. Colet, Phys. Rev. Lett. 94, 063905 (2005)

    Article  ADS  Google Scholar 

  9. M. Cross, P. Hohenberg, Rev. Mod. Phys. 65, 851 (1993)

    Article  ADS  Google Scholar 

  10. J. Murray, Mathematical Biology (Springer, New York, 1989)

  11. R. Hoyle, Pattern Formation: An Introduction to Methods (Cambridge University Press, 2006)

  12. B.A. Malomed, Phys. Rev. A 44, 6954 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  13. B.A. Malomed, Phys. Rev. E 47, 2874 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  14. D. Cai, A.R. Bishop, N. Gronbech-Jensen, B.A. Malomed, Phys. Rev. E 49, 1677 (1994)

    Article  ADS  Google Scholar 

  15. A.V. Buryak, N.N. Akhmediev, Phys. Rev. E 51, 3572 (1995)

    Article  ADS  Google Scholar 

  16. N. Akhmediev, A. Ankiewicz, J.M. Soto-Crespo, Phys. Rev. Lett. 79, 4047 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  17. I.V. Barashenkov, Yu.S. Smirnov, N.V. Alexeeva, Phys. Rev. E 57, 2 (1998)

    Article  Google Scholar 

  18. B. Schäpers, M. Feldmann, T. Ackemann, W. Lange, Phys. Rev. Lett. 85, 748 (2000)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  21. F. Leo, S. Coen, P. Kockeart, S.-P. Gorza, Ph. Emplit, M. Haelterman, Nat. Photon. 4, 471 (2010)

    Article  ADS  Google Scholar 

  22. S. Coen, H.G. Randle, T. Sylvestre, M. Erkintalo, Opt. Lett. 38, 37 (2013)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  24. T.J. Kippenberg, R. Holzwarth, S.A. Diddams, Science 332, 555 (2011)

    Article  ADS  Google Scholar 

  25. P. Del’Haye, T. Herr, E. Gavartin, M.L. Gorodetsky, R. Holzwarth, T.J. Kippenberg, Phys. Rev. Lett. 107, 063901 (2011)

    Article  ADS  Google Scholar 

  26. Y. Okawachi, K. Saha, J.S. Levy, Y. Henry Wen, M. Lipson, A.L. Gaeta, Opt. Lett. 36, 3398 (2011)

    Article  ADS  Google Scholar 

  27. T. Hansch, Rev. Mod. Phys. 78, 1297 (2006)

    Article  ADS  Google Scholar 

  28. S.B. Papp, K. Beha, P. Del’Haye, F. Quinlan, H. Lee, K.J. Vahala, S.A. Diddams, Optica 1, 10 (2014)

    Article  Google Scholar 

  29. J.D. Jost, T. Herr, C. Lecaplain, V. Brasch, M.H.P. Pfeiffer, T.J. Kippenberg, Optica 2, 706 (2015)

    Article  Google Scholar 

  30. F.M. Mitschke, L.F. Mollenauer, Opt. Lett. 12, 355 (1987)

    Article  ADS  Google Scholar 

  31. D.J. Kaup, Phys. Rev. A 42, 5689 (1990)

    Article  ADS  Google Scholar 

  32. M. Tlidi, P. Mandel, R. Lefever, Phys. Rev. Lett. 73, 640 (1994)

    Article  ADS  Google Scholar 

  33. M. Tlidi, A.G. Vladimirov, P. Mandel, J. Quantum Electron. 39, 2 (2003)

    Article  Google Scholar 

  34. B.A. Malomed, Phys. Rev. E 58, 7928 (1998)

    Article  ADS  Google Scholar 

  35. J.M. Soto-Crespo, Ph. Grelu, N. Akhmediev, N. Devine, Phys. Rev. E 75, 016613 (2007)

    Article  ADS  Google Scholar 

  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)

  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. V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M.H.P. Pfeiffer, M.L. Gorodetsky, T.J. Kippenberg, Science 351, 357 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  39. X. Yi, Q.-F. Yang, K.Y. Yang, K. Vahala, Opt. Lett. 41, 2037 (2016)

    Article  ADS  Google Scholar 

  40. P. Del’Haye, A. Coillet, W. Loh, K. Beha, S.B. Papp, S.A. Diddams, Nat. Commun. 6, 5668 (2015)

    Article  Google Scholar 

  41. P. Coullet, C. Riera, C. Tresser, Phys. Rev. Lett. 84, 3069 (2000)

    Article  ADS  Google Scholar 

  42. P.D. Woods, A.R. Champneys, Physica D 129, 147 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  43. D. Gomila, A.J. Scroggie, W.J. Firth, Physica D 227, 70 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  44. J. Burke, E. Knobloch, Phys. Rev. E 73, 056211 (2006)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  46. C. Elphick, E. Meron, E.A. Spiegel, SIAM J. Appl. Math. 50, 490 (1990)

    Article  MathSciNet  Google Scholar 

  47. I. Aranson, K. Gorshkov, A. Lomov, M. Rabinovich, Physica D 43, 435 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  48. G. Kozyreff, P. Assemat, S.J. Chapman, Phys. Rev. Lett. 103, 164501 (2009)

    Article  ADS  Google Scholar 

  49. G. Kozyreff, L. Gelens, Phys. Rev. A 84, 023819 (2011)

    Article  ADS  Google Scholar 

  50. L. Gelens, D. Gomila, G. Van der Sande, M.A. Matas, P. Colet, Phys. Rev. Lett. 104, 154101 (2010)

    Article  ADS  Google Scholar 

  51. P. Colet, M.A. Matías, L. Gelens, D. Gomila, Phys. Rev. E 89, 012914 (2014)

    Article  ADS  Google Scholar 

  52. L. Gelens, M.A. Matías, D. Gomila, T. Dorissen, P. Colet, Phys. Rev. E 89, 012915 (2014)

    Article  ADS  Google Scholar 

  53. W.J. Firth, A. Lord, J. Mod. Opt. 43, 1071 (1996)

    Article  ADS  Google Scholar 

  54. A.R. Champneys, Physica D 112, 158 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  55. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, New York, 1983)

  56. R.L. Devaney, Trans. Am. Math. Soc. 218, 89 (1976)

    Article  Google Scholar 

  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. 379–524

  58. E. Knobloch, Annu. Rev. Cond. Matter Phys. 6, 325 (2015)

    Article  ADS  Google Scholar 

  59. J. Burke, E. Knobloch, Discrete Cont. Dyn. Syst. Suppl., September, 109 (2009)

  60. M.R.E. Lamont, Y. Okawachi, A.L. Gaeta, Opt. Lett. 38, 3478 (2013)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  62. M. Tlidi, L. Gelens, Opt. Lett. 35, 306 (2010)

    Article  ADS  Google Scholar 

  63. C. Milián, D.V. Skryabin, Opt. Express 22, 3732 (2014)

    Article  ADS  Google Scholar 

  64. P. Parra-Rivas, D. Gomila, F. Leo, S. Coen, L. Gelens, Opt. Lett. 39, 2971 (2014)

    Article  ADS  Google Scholar 

  65. C. Milián, A.V. Gorbach, M. Taki, A.V. Yulin, D.V. Skryabin, Phys. Rev. A 92, 033851 (2015)

    Article  ADS  Google Scholar 

  66. B.A. Malomed, Europhys. Lett. 30, 507 (1995)

    Article  ADS  Google Scholar 

  67. M.O. Magnasco, Phys. Rev. Lett. 71, 1477 (1993)

    Article  ADS  Google Scholar 

  68. F. Gustave, C. Rimoldi, P. Walczak, L. Columbo, M. Brambilla, F. Prati, G. Tissoni, S. Barland, Eur. Phys. J. D (in press)

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Authors and Affiliations

  1. Applied Physics Research Group (APHY), Vrije Universiteit Brussel, 1050, Brussels, Belgium

    Pedro Parra-Rivas & Lendert Gelens

  2. Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat de les Illes Balears, 07122, Palma de Mallorca, Spain

    Pedro Parra-Rivas, Damia Gomila & Pere Colet

  3. Laboratory of Dynamics in Biological Systems, Department of Cellular and Molecular Medicine, University of Leuven (KU Leuven), 3000, Leuven, Belgium

    Lendert Gelens

Authors
  1. Pedro Parra-Rivas
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  2. Damia Gomila
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  4. Lendert Gelens
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Correspondence to Pedro Parra-Rivas or Lendert Gelens.

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.

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Parra-Rivas, P., Gomila, D., Colet, P. et al. Interaction of solitons and the formation of bound states in the generalized Lugiato-Lefever equation. Eur. Phys. J. D 71, 198 (2017). https://doi.org/10.1140/epjd/e2017-80127-5

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  • DOI: https://doi.org/10.1140/epjd/e2017-80127-5

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