Skip to main content
SpringerLink
Log in

Periodic two-phase “Rogue waves”

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

A family of periodic (in x and z) two-gap solutions of the focusing nonlinear Schrödinger equation is constructed. A condition under which the two-gap solutions exhibit the behavior of periodic “rogue waves” is obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. Ruban et al., “Rogue Waves-towards a Unifying Concept?: Discussions & Debates,” Eur. Phys. J. Special Topics 185(1), 5–15 (2010).

    Article  Google Scholar 

  2. N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373(6), 675–678 (2009).

    Article  MATH  Google Scholar 

  3. A. Ankiewicz, D. J. Kedzora, and N. Akhmediev, “Rogue waves triplets,” Phys. Lett. A 375(28–29), 2782–2785 (2011).

    Article  MATH  Google Scholar 

  4. D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Circular rogue wave clusters,” Phys. Rev. A 84(056611) (2011).

    Google Scholar 

  5. P. Dubard, P. Gaillard, C. Klein, and V. B. Matveev, “On multi-rogue waves solutions of the focusing NLS equation and positon solutions of the KdV equation,” Eur. Phys. J. Spec. Top. 185(1), 247–261 (2010).

    Article  Google Scholar 

  6. P. Dubard and V. B. Matveev, “Multi-rogue waves solutions of to the focusing NLS equation and the KP-I equation,” Nat. Hazards Earth Syst. Sci. 11, 667–672 (2011).

    Article  Google Scholar 

  7. B. Guo, L. Ling, and Q. P. Liu, “Nonlinear Schrödinger equation: Generalized Darboux transformation and rogue wave solutions,” Phys. Rev. E 85(026607) (2012).

    Google Scholar 

  8. Y. Ohta and J. Yang, “General higher order rogue waves and their dynamics in the nonlinear Schrödinger equation,” Proc. R. Soc. A 468(2142), 1716–1740 (2012).

    Article  MathSciNet  Google Scholar 

  9. A. I. Dyachenko and V. E. Zakharov, “On the formation freak waves on the surface of deep water,” Pis’ma Zh. Éksper. Teoret. Fiz. 88(5), 356–359 (2008) [JETP Lett. 88 (5), 307–311 (2008)].

    Google Scholar 

  10. V. E. Zakharov and R. V. Shamin, “On the probability of the occurrence of the rogue waves,” Pis’ma Zh. Éksper. Teoret. Fiz. 91(2), 68–71 (2010) [JETP Lett. 91 (2), 62–65 (2010)].

    Google Scholar 

  11. N. N. Akhmediev and A. Ankiewicz, Solitons (Fizmatlit, Moscow, 2003; Chapman and Hall, 1997).

    Google Scholar 

  12. K. B. Dysthe, “Note on a modification to the nonliear Shrödinger equation for application to deep water waves,” Proc. R. Soc. Lond. A 369(1736), 105–114 (1979).

    Article  MATH  Google Scholar 

  13. K. Trulsen and K. B. Dysthe, “Amodified nonlinear Shrödinger equation for broader bandwidth gravity waves on deep water,” Wave Motion 24(3), 281–289 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  14. K. Trulsen, I. Kliakhandler, K. B. Dysthe, and M. G. Velarde, “On weakly nonlinear modulation waves on deep water,” Phys. Fluids 12(10), 2432–2437 (2000).

    Article  MathSciNet  Google Scholar 

  15. A. Saini et al., Travelling Wave Solutions to Nonlinear Schrodinger Equation with Self-Steepening and Self-Frequency Shift, arXiv: nlin. PS/0911.2788 (2009).

    Google Scholar 

  16. S. P. Novikov, “The periodic problem for the Korteweg-de Vries equation,” Funct. Anal. Appl. 8, 236–246 (1974) Funktsional. Anal. Prilozhen. 8 (3), 54–66 (1974) [Functional Anal. Appl. 8, 236–246 (1974)].

    Article  MATH  Google Scholar 

  17. P. D. Lax, “Periodic solutions of the KdV equations,” in Nonlinear Wave Motion, Lectures in Appl. Math. (Amer.Math. Soc., Providence, RI, 1974), Vol. 15, pp. 85–96.

    Google Scholar 

  18. B. A. Dubrovin and S. P. Novikov, “A periodic problem for the Korteweg-de Vries and Sturm-Liouville equations. Their connection with algebraic geometry,” Dokl. Akad. Nauk SSSR 219(3), 19–22 (1974).

    MathSciNet  Google Scholar 

  19. A. R. Its and V. B. Matveev, “Schrodinger operators with finite-gap spectrum and N-soliton solutions of the Korteweg-de Vries equation,” Teoret. Mat. Fiz. 23(1), 51–68 (1975) [Theoret. and Math. Phys. 23 (1), 343–355 (1975)].

    MathSciNet  Google Scholar 

  20. B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, “Non-linear equations of Korteweg-de Vries type, finitezone linear operators, and Abelian varieties,” Uspekhi Mat. Nauk 31(1), 55–136 (1976) [Russian Math. Surveys 31 (1), 59–146 (1976)].

    MATH  MathSciNet  Google Scholar 

  21. I. M. Krichever, “Methods of algebraic geometry in the theory of nonlinear equations,” Uspekhi Mat. Nauk 32(6), 183–208 (1977) [RussianMath. Surveys 32 (6), 185–213 (1977)].

    MATH  Google Scholar 

  22. B. A. Dubrovin, “Theta functions and non-linear equations,” Uspekhi Mat. Nauk 36(2), 11–80 (1981) [RussianMath. Surveys 36 (2), 11–92 (1981)].

    MathSciNet  Google Scholar 

  23. V. B. Matveev, “30 years of finite-gap integration theory,” Phil. Trans. R. Soc. A 366(1867), 837–875 (2008).

    Article  MATH  Google Scholar 

  24. E. D. Belokolos, A. I. Bobenko, V. Z. Enol’skii, A. R. Its, and V. B. Matveev, Algebro-Geometrical Approach to Nonlinear Evolution Equations, in Springer Ser. Nonlinear Dynam. (Springer-Verlag, Berlin, 1994).

    Google Scholar 

  25. A. O. Smirnov, “A matrix analogue of Appell’s theorem and reductions of multidimensional Riemann thetafunctions,” Mat. Sb. 133(3), 382–391 (1987) [Math. USSR-Sb. 61 (2), 379–388 (1987)].

    Google Scholar 

  26. A. R. Its, “Inversion of hyperelliptic integrals and integration of nonlinear differential equations,” Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 7(2), 39–46 (1976).

    MathSciNet  Google Scholar 

  27. B. A. Dubrovin, I.M. Krichever, and S. P. Novikov, “The Schrödinger equation in a periodic field and Riemann surfaces,” Dokl. Akad. Nauk SSSR 229(1), 15–18 (1976) [Soviet Math. Dokl. 17, 947–951 (1977)].

    MathSciNet  Google Scholar 

  28. J. D. Fay, Theta-Functions on Riemann Surfaces, in Lecture Notes in Math. (Springer-Verlag, Berlin, 1973), Vol. 352.

    Google Scholar 

  29. H. F. Baker, Abel’s Theorem and the Allied Theory Including the Theory of the Theta Functions (Cambridge Univ. Press, Cambridge, 1897).

    MATH  Google Scholar 

  30. A. Krazer, Lehrbuch der Thetafunktionen (B. G. Teubner, Leipzig, 1903).

    MATH  Google Scholar 

  31. N. I. Akhiezer, Elements of the Theory of Elliptic Functions (Nauka, Moscow, 1970; Amer. Math. Soc., Providence, RI, 1990).

    Google Scholar 

  32. A. O. Smirnov, “Two-gap elliptic solutions to integrable nonlinear equations,” Mat. Zametki 58(1), 86–97 (1995) [Math. Notes 58 (1), 735–743 (1995)].

    MathSciNet  Google Scholar 

  33. E. G. Amosenok and A. O. Smirnov, “Two-gap 2-elliptic solution Boussinesq equation,” Lett. Math. Phys. 96(1–3), 157–168 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  34. A. O. Smirnov, G. M. Golovachev, and E. G. Amosenok, “Two-gap 3-elliptic solutions of the Boussinesq and Korteweg-de Vries equations,” Nonlinear Dynamics 7(2), 239–256 (2011).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St. Petersburg State University of Aerospace Instrumentation, St. Petersburg, Russia

    A. O. Smirnov

Authors
  1. A. O. Smirnov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. O. Smirnov.

Additional information

Original Russian Text © A. O. Smirnov, 2013, published in Matematicheskie Zametki, 2013, Vol. 94, No. 6, pp. 871–883.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Smirnov, A.O. Periodic two-phase “Rogue waves”. Math Notes 94, 897–907 (2013). https://doi.org/10.1134/S0001434613110266

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434613110266

Keywords

Search

Navigation