JETP Letters

, Volume 105, Issue 2, pp 125–129 | Cite as

Fermi–Pasta–Ulam recurrence and modulation instability

  • E. A. Kuznetsov
Nonlinear Phenomena


We give a qualitative conceptual explanation of the Fermi–Pasta–Ulam (FPU) like recurrence in the onedimensional focusing nonlinear Schrodinger equation (NLSE). The recurrence can be considered as a result of the nonlinear development of the modulation instability. All known exact localized solitary wave solutions describing propagation on the background of the modulationally unstable condensate show the recurrence to the condensate state after its interaction with solitons. The condensate state locally recovers its original form with the same amplitude but a different phase after soliton leave its initial region. Based on the integrability of the NLSE, we demonstrate that the FPU recurrence takes place not only for condensate, but also for a more general solution in the form of the cnoidal wave. This solution is periodic in space and can be represented as a solitonic lattice. That lattice reduces to isolated soliton solution in the limit of large distance between solitons. The lattice transforms into the condensate in the opposite limit of dense soliton packing. The cnoidal wave is also modulationally unstable due to soliton overlapping. The recurrence happens at the nonlinear stage of the modulation instability. Due to generic nature of the underlying mathematical model, the proposed concept can be applied across disciplines and nonlinear systems, ranging from optical communications to hydrodynamics.


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  1. 1.
    E. Fermi, J. Pasta, and S. Ulam, Los Alamos Report LA-1940 (Los Alamos, 1955), p.978.Google Scholar
  2. 2.
    N. J. Zabusky, J. Math. Phys. 3, 1028 (1962).ADSCrossRefGoogle Scholar
  3. 3.
    N. J. Zabusky and M. D. Kruskal, Phys. Rev. Lett. 15, 240 (1965).ADSCrossRefGoogle Scholar
  4. 4.
    N. J. Zabusky and G. S. Deem, J. Comp. Phys. 2, 126 (1967).ADSCrossRefGoogle Scholar
  5. 5.
    C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Phys. Rev. Lett. 19, 1095 (1967).ADSCrossRefGoogle Scholar
  6. 6.
    V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).ADSGoogle Scholar
  7. 7.
    V. E. Zakharov and L. D. Faddeev, Funct. Anal. Appl. 5, 280 (1971).CrossRefGoogle Scholar
  8. 8.
    V. E. Zakharov, Sov. Phys. JETP 38, 108 (1974).ADSGoogle Scholar
  9. 9.
    J. Ford, Phys. Rep. 213, 271 (1992)ADSMathSciNetCrossRefGoogle Scholar
  10. 9a.
    N. J. Zabusky, Chaos 15, 015102 (2005)ADSMathSciNetCrossRefGoogle Scholar
  11. 9b.
    M. A. Porter, N. J. Zabusky, B. Hu, and D. K. Campbell, Am. Sci. 97, 214 (2009).CrossRefGoogle Scholar
  12. 10.
    G. van Simaeys, P. Emplit, and M. Haelterman, Phys. Rev. Lett. 87, 033902 (2001); J. Opt. Soc. Am. B 19, 477 (2002).ADSCrossRefGoogle Scholar
  13. 11.
    A. Mussot, A. Kudlinski, M. Droques, P. Szriftgiser, and N. Akhmediev, Phys. Rev. X 4, 011054 (2014)Google Scholar
  14. 11a.
    O. Kimmoun et al., Sci. Rep. 6, 28516 (2016).ADSCrossRefGoogle Scholar
  15. 12.
    D. K. Campbell, S. Flash, and Yu. S. Kivshar, Phys. Today 43 (2004)Google Scholar
  16. 12a.
    S. Flash, M. V. Ivanchenko, and O. I. Kanakov, Phys. Rev. Lett. 95, 064102 (2005); Phys. Rev. E 73, 036618 (2006)ADSCrossRefGoogle Scholar
  17. 12b.
    M. Onorato, L. Vozellaa, D. Proment, and Yu. V. Lvov, Proc. Natl. Acad. Sci. 112, 4208 (2015).ADSCrossRefGoogle Scholar
  18. 13.
    V. E. Zakharov and E. A. Kuznetsov, Phys. Usp. 40, 1087 (1997).ADSCrossRefGoogle Scholar
  19. 14.
    A. Hasegawa and F. Tappet, Appl. Phys. Lett. 23, 142 (1973).ADSCrossRefGoogle Scholar
  20. 15.
    T. B. Benjamin and J. E. Feir, J. Fluid Mech. 27, 417 (1967).ADSCrossRefGoogle Scholar
  21. 16.
    V. E. Zakharov and L. A. Ostrovsky, Physica D 238, 540 (2009).ADSMathSciNetCrossRefGoogle Scholar
  22. 17.
    E. A. Kuznetsov and M. D. Spector, Theor. Math. Phys. 120, 997 (1999).CrossRefGoogle Scholar
  23. 18.
    E. A. Kuznetsov, M. D. Spector, and G. E. Falkovich, Physica D 100, 379 (1984).ADSCrossRefGoogle Scholar
  24. 19.
    E. A. Kuznetsov, Sov. Phys. Dokl. 22, 507 (1977).ADSGoogle Scholar
  25. 20.
    D. H. Peregrine, J. Austral. Math. Soc., Ser. B 25, 16 (1983)MathSciNetCrossRefGoogle Scholar
  26. 20a.
    N. Akhmediev, V. Eleonsky, and N. Kulagin, Sov. Phys. JETP 62, 894 (1985)Google Scholar
  27. 20b.
    V. E. Zakharov and A. A. Gelash, Phys. Rev. Lett. 111, 054101 (2013).ADSCrossRefGoogle Scholar
  28. 21.
    D. S. Agafontsev and V. E. Zakharov, Nonlinearity 28, 2791 (2015). arXiv:1512.06332 (2016); Nonlinearity (in press).ADSMathSciNetCrossRefGoogle Scholar
  29. 22.
    E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge Univ. Press, Cambridge, 1996), Vol.2.Google Scholar
  30. 23.
    E. A. Kuznetsov and A. V. Mikhailov, Sov. Phys. JETP 40, 855 (1974).ADSGoogle Scholar
  31. 24.
    V. E. Zakharov and A. B. Shabat, Funct. Anal. Appl. 8, 226 (1974), Funct. Anal. Appl. 13, 166 (1979).CrossRefGoogle Scholar
  32. 25.
    S. P. Novikov, S. V. Manakov, L. P. Pitaevsky, and V. E. Zakharov, Theory of Solitons (Consultants Bureau, New York, 1984).Google Scholar

Copyright information

© Pleiades Publishing, Inc. 2017

Authors and Affiliations

  1. 1.Lebedev Physical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.Landau Institute for Theoretical PhysicsRussian Academy of SciencesMoscowRussia
  3. 3.Novosibirsk State UniversityNovosibirskRussia

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