Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 032, 14 pages      arXiv:1412.1562      https://doi.org/10.3842/SIGMA.2015.032

Three-Phase Freak Waves

Aleksandr O. Smirnov, Sergei G. Matveenko, Sergei K. Semenov and Elena G. Semenova
St.-Petersburg State University of Aerospace Instrumentation (SUAI), 67 Bolshaya Morskaya Str., St.-Petersburg, 190000, Russia

Received December 05, 2014, in final form April 11, 2015; Published online April 21, 2015

Abstract
In the article, we describe three-phase finite-gap solutions of the focusing nonlinear Schrödinger equation and Kadomtsev-Petviashvili and Hirota equations that exhibit the behavior of almost-periodic ''freak waves''. We also study the dependency of the solution parameters on the spectral curves.

Key words: nonlinear Schrödinger equation; Hirota equation; freak waves; theta function; reduction; covering; spectral curve.

pdf (1530 kb)   tex (1018 kb)

References

  1. Akhiezer N.I., Elements of the theory of elliptic functions, Translations of Mathematical Monographs, Vol. 79, Amer. Math. Soc., Providence, RI, 1990.
  2. Akhmediev N.N., Ankiewicz A., Solitons, nonlinear pulses and beams, Chapman & Hall, London, 1997.
  3. Akhmediev N., Pelinovsky E. (Editors), Discussion & debate: rogue waves - towards a unifying concept?, Eur. Phys. J. Special Topics 185 (2010), 266 pages.
  4. Ankiewicz A., Soto-Crespo J.M., Akhmediev N., Rogue waves and rational solutions of the Hirota equation, Phys. Rev. E 81 (2010), 046602, 8 pages.
  5. Baker H.F., Abel's theorem and the allied theory including the theory of theta functions, Cambridge University Press, Cambridge, 1897.
  6. Belokolos E.D., Bobenko A.I., Enol'skii V.Z., Its A.R., Matveev V.B., Algebro-geometric approach to nonlinear integrable equations, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1994.
  7. Chiao R.Y., Garmire E., Townes C.H., Self-trapping of optical beams, Phys. Rev. Lett. 13 (1964), 479-482.
  8. Dai C.Q., Zhang J.F., New solitons for the Hirota equation and generalized higher-order nonlinear Schrödinger equation with variable coefficients, J. Phys. A: Math. Gen. 39 (2006), 723-737.
  9. Dubrovin B.A., Inverse problem for periodic finite-zoned potentials in the theory of scattering, Funct. Anal. Appl. 9 (1975), 61-62.
  10. Dubrovin B.A., Periodic problems for the Korteweg-de Vries equation in the class of finite band potentials, Funct. Anal. Appl. 9 (1975), 215-223.
  11. Dubrovin B.A., Theta functions and non-linear equations, Russian Math. Surveys 36 (1981), no. 2, 11-92.
  12. Dubrovin B.A., Matveev V.B., Novikov S.P., Nonlinear equations of Korteweg-de Vries type, finite-band linear operators and Abelian varieties, Russian Math. Surveys 31 (1976), no. 1, 59-146.
  13. Dubrovin B.A., Novikov S.P., A periodicity problem for the Korteweg-de Vries and Sturm-Liouville equations. Their connection with algebraic geometry, Sov. Math. Dokl. 15 (1974), 1597-1601.
  14. Fay J.D., Theta functions on Riemann surfaces, Lecture Notes in Math., Vol. 352, Springer-Verlag, Berlin - New York, 1973.
  15. Gesztesy F., Holden H., Soliton equations and their algebro-geometric solutions. Vol. I$.~(1+1)$-dimensional continuous models, Cambridge Studies in Advanced Mathematics, Vol. 79, Cambridge University Press, Cambridge, 2003.
  16. Gesztesy F., Holden H., Michor J., Teschl G., Soliton equations and their algebro-geometric solutions. Vol. II. $(1+1)$-dimensional discrete models, Cambridge Studies in Advanced Mathematics, Vol. 114, Cambridge University Press, Cambridge, 2008.
  17. Guo B., Ling L., Liu Q.P., Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions, Phys. Rev. E 85 (2012), 026607, 9 pages, arXiv:1108.2867.
  18. Its A.R., Inversion of hyperelliptic integrals, and integration of nonlinear differential equations, Vestnik Leningrad. Univ. (1976), no. 7, 39-46.
  19. Its A.R., ''Isomonodromy'' solutions of equations of zero curvature, Math. USSR-Izv. 26 (1986), 497-529.
  20. Its A.R., Kotlyarov V.P., On a class of solutions of the nonlinear Schrödinger equation, Dokl. Akad. Nauk USSR Ser. A 11 (1976), 965-968.
  21. Its A.R., Matveev V.B., Hill's operator with finitely many gaps, Funct. Anal. Appl. 9 (1975), 65-66.
  22. Its A.R., Matveev V.B., Schrödinger operators with the finite-band spectrum and $N$-soliton solutions of the Korteweg-de Vries equation, Theoret. and Math. Phys. 23 (1975), 343-355.
  23. Kalla C., Klein C., New construction of algebro-geometric solutions to the Camassa-Holm equation and their numerical evaluation, Proc. R. Soc. Lond. Ser. A 468 (2012), 1371-1390, arXiv:1109.5301.
  24. Kalla C., Klein C., On the numerical evaluation of algebro-geometric solutions to integrable equations, Nonlinearity 25 (2012), 569-596, arXiv:1107.2108.
  25. Krazer A., Lehrbuch der Thetafunktionen, Teubner, Leipzig, 1903.
  26. Krichever I.M., Methods of algebraic geometry in the theory of non-linear equations, Russian Math. Surveys 32 (1977), no. 6, 185-213.
  27. Kundu A., Mukherjee A., Naskar T., Modelling rogue waves through exact dynamical lump soliton controlled by ocean currents, Proc. R. Soc. Lond. Ser. A 470 (2014), 20130576, 20 pages, arXiv:1204.0916.
  28. Kuznetsov E.A., Solitons in a parametrically unstable plasma, Sov. Phys. Dokl. 22 (1977), 507-508.
  29. Lax P.D., Periodic solutions of the KdV equations, in Nonlinear Wave Motion (Proc. AMS-SIAM Summer Sem., Clarkson Coll. Tech., Potsdam, N.Y., 1972), Lectures in Appl. Math., Vol. 15, Amer. Math. Soc., Providence, R.I., 1974, 85-96.
  30. Li C.Z., He J.S., Darboux transformation and positons of the inhomogeneous Hirota and the Maxwell-Bloch equation, Sci. China Phys. Mech. Astronomy 57 (2014), 898-907, arXiv:1210.2501.
  31. Marchenko V.A., The periodic Korteweg-de Vries problem, Math. USSR Sb. 24 (1974), 319-344.
  32. Matveev V.B., 30 years of finite-gap integration theory, Philos. Trans. R. Soc. Lond. Ser. A 366 (2008), 837-875.
  33. McKean H.P., van Moerbeke P., The spectrum of Hill's equation, Invent. Math. 30 (1975), 217-274.
  34. Mumford D., Tata lectures on theta. II. Jacobian theta functions and differential equations, Progress in Mathematics, Vol. 43, Birkhäuser Boston, Inc., Boston, MA, 1984.
  35. Novikov S.P., The periodic problem for the Korteweg-de Vries equation, Funct. Anal. Appl. 8 (1974), 236-246.
  36. Previato E., Hyperelliptic quasiperiodic and soliton solutions of the nonlinear Schrödinger equation, Duke Math. J. 52 (1985), 329-377.
  37. Smirnov A.O., A matrix analogue of a theorem of Appell and reductions of multidimensional Riemann theta-functions, Math. USSR Sb. 61 (1988), 379-388.
  38. Smirnov A.O., Elliptic solutions of the nonlinear Schrödinger equation and a modified Korteweg-de Vries equation, Mat. Sb. 185 (1994), 103-114.
  39. Smirnov A.O., Solution of a nonlinear Schrödinger equation in the form of two-phase freak waves, Theoret. and Math. Phys. 173 (2012), 1403-1416.
  40. Smirnov A.O., Periodic two-phase ''rogue waves'', Math. Notes 94 (2013), 897-907.
  41. Smirnov A.O., Semenova E.G., Zinger V., Zinger N., On a periodic solution of the focusing nonlinear Schrödinger equation, arXiv:1407.7974.
  42. Wang L.H., Porsezian K., He J.S., Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation, Phys. Rev. E 87 (2013), 053202, 10 pages, arXiv:1304.8085.
  43. Yan Z., Vector financial rogue waves, Phys. Lett. A 375 (2011), 4274-4279, arXiv:1101.3107.
  44. Zakharov V.E., Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys. 9 (1968), 190-194.

Previous article  Next article   Contents of Volume 11 (2015)