Autoresonance excitation of a soliton of the nonlinear Schrödinger equation

  • R. N. GarifullinEmail author


The action of an external parametric perturbation with slowly changing frequency on a soliton of the nonlinear Schrödinger equation is studied. Equations for the time evolution of the parameters of the perturbed soliton are derived. Conditions for the soliton phase locking are found, which relate the rate of change of the perturbation frequency, its amplitude, the wave number, and the phase to the initial data of the soliton. The cases when the initial amplitude of the soliton is small and when the amplitude of the soliton is of the order of unity are considered.


autoresonance perturbation nonlinear Schrödinger equation soliton 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. A. Kalyakin, R. N. Garifullin, and M. A. Shamsutdinov, Comput. Math. Math. Phys. 47(7), 1158 (2007).MathSciNetCrossRefGoogle Scholar
  2. 2.
    S. V. Batalov, E. M. Maslov, and A. G. Shagalov, J. Experim. Theor. Physics 108(5), 890 (2009).CrossRefGoogle Scholar
  3. 3.
    V. I. Karpman and E. M. Maslov, Soviet Phys. JETP 46(2), 537 (1977).MathSciNetGoogle Scholar
  4. 4.
    B. I. Veksler, Dokl. Akad. Nauk SSSR 43(8), 346 (1944).Google Scholar
  5. 5.
    B. I. Veksler, Dokl. Akad. Nauk SSSR 44(9), 393 (1944).Google Scholar
  6. 6.
    E. M. McMillan, Phys. Rev. 70, 800 (1946).Google Scholar
  7. 7.
    K. S. Golovanevskii, Fizika Plazmy 11(3), 295 (1985).Google Scholar
  8. 8.
    J. Fajans and L. Friedland, Am. J. Phys. 69(10), 1096 (2001).CrossRefGoogle Scholar
  9. 9.
    B. Meerson and L. Friedland, Phys. Rev. A 41, 5233 (1990).CrossRefGoogle Scholar
  10. 10.
    L. A. Kalyakin, J. Math. Sci. 125(5), 658 (2003).MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    L. A. Kalyakin, Proc. Steklov Inst. Math., Suppl. 1, S108 (2003).Google Scholar
  12. 12.
    R. N. Garifullin, Dokl. Math. 70(2), 799 (2004).Google Scholar
  13. 13.
    L. A. Kalyakin, Russian Math. Surveys 63(5), 791 (2008).MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    A. E. El’bert and A. R. Danilin, Trudy Inst. Mat. Mekh. UrO RAN 16(2), 288 (2010).Google Scholar
  15. 15.
    S. G. Glebov, O. M. Kiselev, and V. A. Lazarev, SIAM J. Appl. Math. 65(6), 2158 (2005).MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    S. Glebov, O. Kiselev, and N. Tarkhanov, Stud. Appl. Math. 124(1), 19 (2010).MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    O. M. Kiselev, S. G. Glebov, and V. A. Lazarev;
  18. 18.
    L. A. Kalyakin, M. A. Shamsutdinov, R. N. Garifullin, and R. K. Salimov, Physics Metals Metallography 104(2), 107 (2007).CrossRefGoogle Scholar
  19. 19.
    Yu. Yu. Bagderina, Math. Notes 80(3–4), 442 (2006).MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    R. N. Garifullin, in Proc. Regional School-Conf. for Students, Graduate Students, and Young Scientists on Mathematics and Physics (Bashkirsk. Gos. Univ., Ufa, 2003), Vol. 1, pp. 189–195 [in Russian].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Institute of Mathematics with Computing CenterUfa Research Center of the Russian Academy of SciencesUfaRussia
  2. 2.Bashkir State UniversityUfaRussia

Personalised recommendations