Advertisement

Soliton interactions and degenerate soliton complexes for the focusing nonlinear Schrödinger equation with nonzero background

  • Sitai Li
  • Gino Biondini
Regular Article
  • 29 Downloads

Abstract.

We characterize soliton interactions in focusing media described by the nonlinear Schrödinger equation in the presenze of a nonzero background field, including the cases of bound states (degenerate soliton trains) and interactions between solitons and Akhmediev breathers. We first characterize bound states, which, as in the case of zero background, are obtained when several solitons travel with the same velocity. We then turn to the case when the soliton velocities are distinct, and we compute the long-time asymptotic behavior of soliton interactions by calculating the position shift for each soliton as \(t\rightarrow\pm\infty\). We also identify conditions that give rise to large position shifts. Moreover, we characterize the asymptotic phase of the nonzero background in each sector of the xt -plane that is separated by individual solitons or breathers, and we show that the asymptotic phase can be easily determined from whether the region is on the left or on the right of a soliton or an Akhmediev breather.

References

  1. 1.
    N.N. Akhmediev, Nature 413, 267 (2001)ADSCrossRefGoogle Scholar
  2. 2.
    D.R. Solli, C. Ropers, P. Koonath, B. Jalali, Nature 450, 1054 (2007)ADSCrossRefGoogle Scholar
  3. 3.
    B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, J.M. Dudley, Nat. Phys. 6, 790 (2010)CrossRefGoogle Scholar
  4. 4.
    M.J. Ablowitz, J. Hammack, D. Henderson, C.M. Schober, Phys. Rev. Lett. 84, 887 (2000)ADSCrossRefGoogle Scholar
  5. 5.
    M. Onorato, A.R. Osborne, M. Serio, Phys. Rev. Lett. 96, 014503 (2006)ADSCrossRefGoogle Scholar
  6. 6.
    V.E. Zakharov, L.A. Ostrovsky, Physica D 238, 540 (2009)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    V.E. Zakharov, Stud. Appl. Math. 122, 219 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    V.E. Zakharov, A. Gelash, Phys. Rev. Lett. 111, 054101 (2013)ADSCrossRefGoogle Scholar
  9. 9.
    D.S. Agafontsev, V.E. Zakharov, Nonlinearity 28, 2791 (2015)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    S. Randoux, P. Walczak, M. Onorato, P. Suret, Phys. Rev. Lett. 113, 113902 (2014)ADSCrossRefGoogle Scholar
  11. 11.
    G. Van Simaeys, Ph. Emplit, M. Haelterman, Phys. Rev. Lett. 87, 033902 (2001)ADSCrossRefGoogle Scholar
  12. 12.
    O. Kimmoun, H.C. Hsu, H. Branger, M.S. Li, Y.Y. Chen, C. Kharif, M. Onorato, E.J.R. Kelleher, B. Kibler, N. Akhmediev, A. Chabchoub, Sci. Rep. 6, 28516 (2016)ADSCrossRefGoogle Scholar
  13. 13.
    G. Biondini, D. Mantzavinos, Phys. Rev. Lett. 116, 043902 (2016)ADSCrossRefGoogle Scholar
  14. 14.
    G. Biondini, S. Li, D. Mantzavinos, Phys. Rev. E 94, 060201R (2016)ADSCrossRefGoogle Scholar
  15. 15.
    G. Biondini, S. Li, D. Mantzavinos, S. Trillo, arXiv:1710.05068 (2017) to be published in SIAM Rev. (2018)Google Scholar
  16. 16.
    G. Biondini, E. Fagerstrom, SIAM J. Appl. Math. 75, 136 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    M. Pichler, G. Biondini, IMA J. Appl. Math. 82, 131 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    B. Kibler, A. Chabchoub, A. Gelash, N. Akhmediev, V.E. Zakharov, Phys. Rev. X 5, 041026 (2015)Google Scholar
  19. 19.
    A.E. Kraych, P. Suret, G. El, S. Randoux, arXiv:1805.05074 (2018)Google Scholar
  20. 20.
    D. Bilman, P.D. Miller, arXiv:1710.06568 (2017)Google Scholar
  21. 21.
    M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981)Google Scholar
  22. 22.
    C. Sulem, P.-L. Sulem, The Nonlinear Schrödinger Equation -- Self-focusing and Wave Collapse (Springer, 1999)Google Scholar
  23. 23.
    E. Infeld, G. Rowlands, Nonlinear Waves, Solitons and Chaos (Cambridge University Press, Cambridge, 2000)Google Scholar
  24. 24.
    G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, New York, 2001)Google Scholar
  25. 25.
    L.P. Pitaevskii, S. Stringari, Bose-Einstein Condensation (Clarendon Press, Oxford, 2003)Google Scholar
  26. 26.
    R.R. Alfano, The Supercontinuum Laser Source: Fundamentals with Updated References (Springer, 2006)Google Scholar
  27. 27.
    V.E. Zakharov, A.B. Shabat, Sov. Phys. JETP 34, 62 (1972)ADSGoogle Scholar
  28. 28.
    S.P. Novikov, S.V. Manakov, L.P. Pitaevskii, V.E. Zakharov, Theory of Solitons: The Inverse Scattering Transform (Plenum, 1984)Google Scholar
  29. 29.
    M.J. Ablowitz, B. Prinari, A.D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems (Cambridge University Press, 2004)Google Scholar
  30. 30.
    S. Li, G. Biondini, C. Schiebold, J. Math. Phys. 58, 033507 (2017)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    T. Aktosun, F. Demontis, C. van der Mee, Inverse Probl. 23, 2171 (2007)ADSCrossRefGoogle Scholar
  32. 32.
    C. Schiebold, Nonlinearity 30, 2930 (2017)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    S.V. Manakov, Sov. Phys. JETP 38, 248 (1974)ADSGoogle Scholar
  34. 34.
    M.J. Ablowitz, B. Prinari, A.D. Trubatch, Inverse Probl. 20, 1217 (2004)ADSCrossRefGoogle Scholar
  35. 35.
    B. Prinari, M.J. Ablowitz, G. Biondini, J. Math. Phys. 47, 063508 (2006)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    B. Prinari, G. Biondini, A.D. Trubatch, Stud. Appl. Math. 126, 245 (2011)MathSciNetCrossRefGoogle Scholar
  37. 37.
    G. Dean, T. Klotz, B. Prinari, F. Vitale, Appl. Anal. 92, 379 (2013)MathSciNetCrossRefGoogle Scholar
  38. 38.
    G. Biondini, G. Kovačič, J. Math. Phys. 55, 031506 (2014)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    G. Biondini, B. Prinari, Stud. Appl. Math. 132, 138 (2014)MathSciNetCrossRefGoogle Scholar
  40. 40.
    G. Biondini, D.K. Kraus, SIAM J. Math. Anal. 47, 706 (2015)MathSciNetCrossRefGoogle Scholar
  41. 41.
    B. Prinari, F. Vitale, G. Biondini, J. Math. Phys. 56, 071505 (2015)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    G. Biondini, D.K. Kraus, B. Prinari, F. Vitale, J. Phys. A 48, 395202 (2015)MathSciNetCrossRefGoogle Scholar
  43. 43.
    G. Biondini, D.K. Kraus, B. Prinari, Commun. Math. Phys. 348, 475 (2016)ADSCrossRefGoogle Scholar
  44. 44.
    A.A. Gelash, Phys. Rev. E 97, 022208 (2018)ADSCrossRefGoogle Scholar
  45. 45.
    B. Prinari, F. Demontis, S. Li, T.P. Horikis, Physica D 368, 22 (2018)ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    S. Li, B. Prinari, G. Biondini, Phys. Rev. E 97, 022221 (2018)ADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    G.C. Katsimiga, P.G. Kevrekidis, B. Prinari, G. Biondini, P. Schmelcher, Phys. Rev. A 97, 043623 (2018)ADSCrossRefGoogle Scholar
  48. 48.
    V.E. Zakharov, A.B. Shabat, Sov. Phys. JETP 37, 823 (1973)ADSGoogle Scholar
  49. 49.
    Y.S. Kivshar, S.K. Turitsyn, Opt. Lett. 18, 337 (1993)ADSCrossRefGoogle Scholar
  50. 50.
    R. Radhakrishnan, M. Lakshmanan, J. Phys. A 28, 2683 (1995)ADSMathSciNetCrossRefGoogle Scholar
  51. 51.
    A.P. Sheppard, Y.S. Kivshar, Phys. Rev. E 55, 4773 (1997)ADSMathSciNetCrossRefGoogle Scholar
  52. 52.
    Q.H. Park, H.J. Shin, Phys. Rev. E 61, 3093 (2000)ADSMathSciNetCrossRefGoogle Scholar
  53. 53.
    M. Tajiri, Y. Watanabe, Phys. Rev. E 57, 3510 (1998)ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    E.A. Kuznetsov, Sov. Phys. Dokl. 22, 507 (1977)ADSGoogle Scholar
  55. 55.
    Y.-C. Ma, Stud. Appl. Math. 60, 43 (1979)ADSMathSciNetCrossRefGoogle Scholar
  56. 56.
    N.N. Akhmediev, V.I. Korneev, Theor. Math. Phys. 69, 1089 (1986)CrossRefGoogle Scholar
  57. 57.
    D.H. Peregrine, Austral. Math. Soc. Ser. B 25, 16 (1983)MathSciNetCrossRefGoogle Scholar
  58. 58.
    S. Cuccagna, R. Jenkins, Commun. Math. Phys. 343, 921 (2016)ADSCrossRefGoogle Scholar
  59. 59.
    M. Borghese, R. Jenkins, K. McLaughlin, Ann. Inst. H. Poincaré C 35, 887 (2018)CrossRefGoogle Scholar
  60. 60.
    K. Grunnert, G. Teschl, Math. Phys. Anal. Geom. 12, 287 (2009)MathSciNetCrossRefGoogle Scholar
  61. 61.
    P. Giavedoni, Nonlinearity 30, 1165 (2017)ADSMathSciNetCrossRefGoogle Scholar
  62. 62.
    J. Liu, P. Perry, C. Sulem, Ann. Inst. H. Poincaré C 35, 217 (2018)CrossRefGoogle Scholar
  63. 63.
    L.D. Faddeev, L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer, Berlin, 1987)Google Scholar
  64. 64.
    S. Schechter, Math. Comput. 13, 73 (1959)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsState University of New York at BuffaloBuffaloUSA
  2. 2.Department of PhysicsState University of New York at BuffaloBuffaloUSA

Personalised recommendations