Advertisement

Theoretical and Mathematical Physics

, Volume 173, Issue 1, pp 1403–1416 | Cite as

Solution of a nonlinear Schrödinger equation in the form of two-phase freak waves

  • A. O. SmirnovEmail author
Article

Abstract

We construct a family of two-gap solutions of the focusing nonlinear Schrödinger equation and derive a condition under which the solutions behave as the so-called freak waves located at the nodes of a two-dimensional lattice. We also study how the lattice parameters depend on the parameters of the spectral curve.

Keywords

rogue wave freak wave nonlinear Schrödinger equation theta function reduction covering 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. Akhmediev, A. Ankiewicz, and M. Taki, Phys. Lett. A, 373, 675–678 (2009).ADSzbMATHCrossRefGoogle Scholar
  2. 2.
    N. Akhmediev and E. Pelinovsky, Eur. Phys. J. Spec. Top., 185, 1–4 (2010).CrossRefGoogle Scholar
  3. 3.
    A. I. Dyachenko and V. E. Zakharov, JETP Letters, 88, 307–311 (2008).ADSCrossRefGoogle Scholar
  4. 4.
    V. E. Zakharov and R. V. Shamin, JETP Letters, 91, 62–65 (2010).ADSCrossRefGoogle Scholar
  5. 5.
    N. Akhmediev and A. Ankevich, Solitons [in Russian], Fizmatlit, Moscow (2003).Google Scholar
  6. 6.
    B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, Nature Physics, 6, 790–795 (2010).ADSCrossRefGoogle Scholar
  7. 7.
    P. Dubard, P. Gaillard, C. Klein, and V. B. Matveev, Eur. Phys. J. Spec. Top., 185, 247–258 (2010).CrossRefGoogle Scholar
  8. 8.
    P. Dubard and V. B. Matveev, Nat. Hazards Earth Syst. Sci., 11, 667–672 (2011).ADSCrossRefGoogle Scholar
  9. 9.
    A. Ankiewicz, D. J. Kedzora, and N. Akhmediev, Phys. Lett. A, 375, 2782–2785 (2011).ADSzbMATHCrossRefGoogle Scholar
  10. 10.
    D. J. Kedzora, A. Ankiewicz, and N. Akhmediev, Phys. Rev. E, 84, 056611 (2011).ADSCrossRefGoogle Scholar
  11. 11.
    B. Guo, L. Ling, and Q. P. Liu, Phys. Rev. E, 85, 026607 (2012); arXiv:1108.2867v2 [nlin.SI] (2011).ADSCrossRefGoogle Scholar
  12. 12.
    A. R. Osborne, M. Onorato, and M. Serio, Phys. Lett. A, 275, 386–393 (2000).MathSciNetADSzbMATHCrossRefGoogle Scholar
  13. 13.
    A. Calini and C. M. Schober, Phys. Lett. A, 298, 335–349 (2002).MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. 14.
    C. M. Schober, Eur. J. Mech. B, 25, 602–620 (2006).MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    K. B. Dysthe, Proc. Roy. Soc. London A, 369, 105–114 (1979).ADSzbMATHCrossRefGoogle Scholar
  16. 16.
    K. Trulsen and K. B. Dysthe, Wave Motion, 24, 281–289 (1996).MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    K. Trulsen, I. Kliakhandler, K. B. Dysthe, and M. G. Velarde, Phys. Fluids, 12, 2433–2437 (2000).MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    A. Saini, V. M. Vyas, S. N. Pandey, T. S. Raju, and P. K. Panigrahi, “Traveling wave solutions to nonlinear Schröedinger equation with self-steepening and self-frequency shift,” arXiv:0911.2788v2 [nlin.PS] (2009).Google Scholar
  19. 19.
    A. L. Islas and C. M. Schober, Phys. Fluids, 17, 031701 (2005).MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    S. P. Novikov, Funct. Anal. Appl., 8, No. 3, 236–246 (1974).zbMATHCrossRefGoogle Scholar
  21. 21.
    P. D. Lax, “Periodic solutions of the KdV equations,” in: Nonlinear Wave Motion (Lect. Appl. Math., Vol. 15, A. C. Newell, ed.), Amer. Math. Soc., Providence, R. I. (1974), pp. 85–96.Google Scholar
  22. 22.
    B. A. Dubrovin and S. P. Novikov, Sov. Math. Dokl., 15, 1597–1601 (1974).zbMATHGoogle Scholar
  23. 23.
    A. R. Its and V. B. Matveev, Theor. Math. Phys., 23, 343–355 (1975).MathSciNetCrossRefGoogle Scholar
  24. 24.
    B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, Russ. Math. Surveys, 31, 59–146 (1976).MathSciNetADSzbMATHCrossRefGoogle Scholar
  25. 25.
    I. M. Krichever, Russ. Math. Surveys, 32, 185–213 (1977).ADSzbMATHCrossRefGoogle Scholar
  26. 26.
    B. A. Dubrovin, Russ. Math. Surveys, 36, 11–92 (1981).MathSciNetADSCrossRefGoogle Scholar
  27. 27.
    E. D. Belokolos, A. I. Bobenko, V. Z. Enol’skii, A. R. Its, and V. B. Matveev, Algebro-Geometrical Approach to Nonlinear Evolution Equations, Springer, Berlin (1994).Google Scholar
  28. 28.
    C. Klein and O. Richter, Ernst Equation and Riemann Surfaces: Analytical and Numerical Methods (Lect. Notes Phys., Vol. 685), Springer, Berlin (2005).Google Scholar
  29. 29.
    C. Klein, V. B. Matveev, and A. O. Smirnov, Theor. Math. Phys., 152, 1132–1145 (2007).MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    A. I. Bobenko and C. Klein, eds., Computational Approach to Riemann Surfaces (Lect. Notes Math., Vol. 2013), Springer, Berlin (2011).zbMATHGoogle Scholar
  31. 31.
    C. Kalla and C. Klein, “On the numerical evaluation of algebro-geometric solutions to integrable equations,” arXiv:1107.2108v2 [math-ph] (2011).Google Scholar
  32. 32.
    C. Kalla and C. Klein, “New construction of algebro-geometric solutions to the Camassa-Holm equation and their numerical evaluation,” arXiv:1109.5301v2 [math-ph] (2011).Google Scholar
  33. 33.
    E. G. Amosenok and A. O. Smirnov, Lett. Math. Phys., 96, 157–168 (2011).MathSciNetADSzbMATHCrossRefGoogle Scholar
  34. 34.
    A. O. Smirnov, G. M. Golovachev, and E. G. Amosenok, Nonlinear Dynamics, 7, 239–256 (2011).Google Scholar
  35. 35.
    D. W. McLaughlin and C. M. Schober, Phys. D, 57, 447–465 (1992).MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    M. J. Ablowitz, C. M. Schober, and B. M. Herbst, Phys. Rev. Lett., 71, 2683–2686 (1993).ADSCrossRefGoogle Scholar
  37. 37.
    A. Calini, N. M. Ercolani, D. W. McLaughlin, and C. M. Schober, Phys. D, 89, 227–260 (1996).MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    A. R. Its, Vestn. Leningr. Univ. Mat. Mekh. Astron., 7, No. 2, 39–46 (1976).MathSciNetGoogle Scholar
  39. 39.
    B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, Sov. Math. Dokl., 17, 947–951 (1977).Google Scholar
  40. 40.
    J. D. Fay, Theta Functions on Riemann Surfaces (Lect. Notes Math., Vol. 352), Springer, Berlin (1973).zbMATHGoogle Scholar
  41. 41.
    H. F. Baker, Abel’s Theorem and the Allied Theory Including the Theory of Theta Functions, Cambridge Univ. Press, Cambridge (1897).zbMATHGoogle Scholar
  42. 42.
    A. Krazer, Lehrbuch der Thetafunktionen, Teubner, Leipzig (1903).zbMATHGoogle Scholar
  43. 43.
    A. O. Smirnov, Math. USSR-Sb., 61, 379–388 (1988).MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    N. I. Akhiezer, Elements of the Theory of Elliptic Functions (Transl. Math. Monogr., Vol. 79), Amer. Math. Soc., Providence, R. I. (1990).zbMATHGoogle Scholar
  45. 45.
    A. O. Smirnov, Math. Notes, 58, 735–743 (1995).MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    A. O. Smirnov, Russian Acad. Sci. Sb. Math., 82, 461–470 (1995).MathSciNetGoogle Scholar
  47. 47.
    A. O. Smirnov, Theor. Math. Phys., 107, 568–578 (1996).zbMATHCrossRefGoogle Scholar
  48. 48.
    A. O. Smirnov, Sb. Math., 188, 115–135 (1997).MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    A. O. Smirnov, Theor. Math. Phys., 78, 6–13 (1989).CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.St. Petersburg State University of Aerospace InstrumentationSt. PetersburgRussia

Personalised recommendations