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Current issue   Ukr. J. Phys. 2014, Vol. 59, N 6, p.640-649
https://doi.org/10.15407/ujpe59.06.0640    Paper

Vakhnenko O.O.1, Vakhnenko V.O.2

1 Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine
(14-B, Metrologichna Str., Ky¨ıv 03680, Ukraine; e-mail: vakhnenko@bitp.kiev.ua)
2 Department of Dynamics of Deformable Solids, Subbotin Institute of Geophysics, Nat. Acad. of Sci. of Ukraine
(63-B, Bohdan Khmel’nyts’kyy Str., Kyïv 01054, Ukraine; e-mail: vakhnenko@ukr.net)

Linear Analysis of Extended Integrable Nonlinear Ladder Network System

Section: Nonlinear processes
Original Author's Text: English

Abstract: The nontrivial integrable extension of a nonlinear ladder electric network system characterized by two coupling parameters is presented. Relying upon the lowest local conservation laws, the concise form of the general semidiscrete integrable system is given, and two versions of its self-consistent reduction in terms of four true field variables are found. The comprehensive analysis of the dispersion equation for low-amplitude excitations of the system is made. The criteria distinguishing the two-branch and four-branch realizations of the dispersion law are formulated. The critical values of adjustable coupling parameter are found, and a collection of qualitatively distinct realizations of the dispersion law is graphically presented. The looplike structure of the low-amplitude dispersion law of a reduced system emerging within certain windows of the adjustable coupling parameter turns out to reproduce the loop-like structure of the dispersion law typical of beam-plasma oscillations in hydrodynamic plasma. The richness of the low-amplitude spectrum of the proposed ladder network system as a function of the adjustable coupling parameter is expected to stimulate even the more rich dynamical behavior in an essentially nonlinear regime.

Key words: nonlinear ladder electric network system, dispersion law, hydrodynamic plasma.

References:

  1. S.V. Manakov, Sov. Phys. – JETP 40, 269 (1975).
  2. H. Flaschka, Progr. Theor. Phys. 51, 703 (1974).
    https://doi.org/10.1143/PTP.51.703
  3. M.J. Ablowitz and J.F. Ladik, J. Math. Phys. 16, 598 (1975).
    https://doi.org/10.1063/1.522558
  4. M.J. Ablowitz and J.F. Ladik, J. Math. Phys. 17, 1011 (1976).
    https://doi.org/10.1063/1.523009
  5. D.N. Christodoulides and R.I. Joseph, Opt. Lett. 13, 794 (1988).
    https://doi.org/10.1364/OL.13.000794
  6. C. Waschke, H.G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. K¨ohler, Phys. Rev. Lett. 70, 3319 (1993).
    https://doi.org/10.1103/PhysRevLett.70.3319
  7. V.G. Lyssenko, G. Valuˇsis, F. L¨oser, T. Hasche, K. Leo, M.M. Dignam, and K. K¨ohler, Phys. Rev. Lett. 79, 301 (1997).
    https://doi.org/10.1103/PhysRevLett.79.301
  8. P. Marquie, J.M. Bilbault, and M. Remoissenet, Phys. Rev. E 49, 828 (1994).
    https://doi.org/10.1103/PhysRevE.49.828
  9. A.S. Davydov, Biology and Quantum Mechanics (Pergamon Press, Oxford–New York, 1981).
  10. J.W. Mintmire, B.I. Dunlap, and C.T. White, Phys. Rev. Lett. 68, 631 (1992).
    https://doi.org/10.1103/PhysRevLett.68.631
  11. O.O. Vakhnenko, J. Nonlin. Math. Phys. 18, 401 (2011).
  12. O.O. Vakhnenko, J. Nonlin. Math. Phys. 18, 415 (2011).
  13. P.J. Caudrey, Physica D 6, 51 (1982).
    https://doi.org/10.1016/0167-2789(82)90004-5
  14. P.J. Caudrey, in Wave Phenomena: Modern Theory and Applications, edited by C. Rogers and T.B. Moodie (Elsevier, Amsterdam, 1984), p. 221.
    https://doi.org/10.1016/S0304-0208(08)71267-2
  15. O.O. Vakhnenko, J. Phys. A 36, 5405 (2003).
    https://doi.org/10.1088/0305-4470/36/20/305
  16. L.D. Faddeev and L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer, Berlin, 1987).
    https://doi.org/10.1007/978-3-540-69969-9
  17. T. Tsuchida, J. Phys. A 35, 7827 (2002).
    https://doi.org/10.1088/0305-4470/35/36/310
  18. O.O. Vakhnenko, J. Phys. A 39, 11013 (2006).
    https://doi.org/10.1088/0305-4470/39/35/005
  19. O.O. Vakhnenko, J. Nonlin. Math. Phys. 20, 606 (2013).
  20. M. Wadati, H. Sanuki, and K. Konno, Progr. Theor. Phys. 53, 419 (1975).
    https://doi.org/10.1143/PTP.53.419
  21. R. Hirota and K. Suzuki, Proc. of IEEE 61, 1483 (1973).
    https://doi.org/10.1109/PROC.1973.9297
  22. R. Hirota, J. Phys. Soc. Japan 35, 289 (1973).
    https://doi.org/10.1143/JPSJ.35.289
  23. K. Daikoku, Y. Mizushima, and T. Tamama, Jap. J. Appl. Phys. 14, 367 (1975).
    https://doi.org/10.1143/JJAP.14.367
  24. R. Hirota and J. Satsuma, J. Phys. Soc. Japan 40, 891 (1976).
    https://doi.org/10.1143/JPSJ.40.891
  25. L.E. Dickson, Elementary Theory of Equations (Wiley, New York, 1914).
  26. E.L. Rees, Amer. Math. Monthly 29, 51 (1922).
    https://doi.org/10.2307/2972804
  27. J.E. Cremona, LMS J. Comput. Math. 2, 62 (1999).
    https://doi.org/10.1112/S1461157000000073
  28. R.W.D. Nickalls, Math. Gazette 93, 66 (2009).
    https://doi.org/10.1017/S0025557200184190
  29. P.A. Sturrock, Phys. Rev. 112, 1488 (1958).
    https://doi.org/10.1103/PhysRev.112.1488
  30. A.I. Akhiezer, I.A. Akhiezer, R.V. Polovin, A.G. Sitenko, and K.N. Stepanov, Plasma Electrodynamics. Vol. 1. Linear Theory (Pergamon Press, Oxford–New York, 1975).
  31. L.V. Postnikov, V.I. Korolev, T.M. Tarantovich, V.A. Mel'nikova, and S.Ya. Vyshkind, Collection of Problems of the Theory of Oscillations (Nauka, Moscow, 1978) (in Russian).