Journal of Applied Mechanics and Technical Physics

, Volume 15, Issue 5, pp 628–631 | Cite as

Damping of steady-state waves in systems described by a nonlinear Klein-Gordon equation

  • E. N. Pelinovskii
  • S. Kh. Shavratskii


The damping of a nonsinusoidal wave in systems described by a Klein-Gordon equation is investigated by the method of averaging. An explicit solution is given for an initial-value problem. It is shown that in certain cases the prolonged existence of a steady-state wave is impossible. Dissipation can lead to the damping out of the wave. The characteristic features of the boundary-value problem are discussed. Formulas are obtained describing the damping of single pulses (solitons).


Mathematical Modeling Mechanical Engineer Soliton Characteristic Feature Industrial Mathematic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature cited

  1. 1.
    M. Shirobokov, “On the theory of the mechanism of magnetization of ferromagnets,” Zh. Éksp. Teor. Fiz.,15, 2 (1945).Google Scholar
  2. 2.
    Ya. I. Frenkel', “On the theory of plastic deformation and twinning,” in: Collection of Selected Works [in Russian], Vol. 2, Akad. Nauk SSSR, Moscow-Leningrad (1958).Google Scholar
  3. 3.
    G. L. Lamb, “Analytical descriptions of ultrashort optical pulse propagation in a resonant medium,” Rev. Mod. Phys.,43, 99 (1971).Google Scholar
  4. 4.
    I. O. Kulik and I. K. Yanson, The Josephson Effect in Superconducting Tunnel Structures [in Russian], Nauka, Moscow (1970).Google Scholar
  5. 5.
    J. K. Perring and T. H. R. Skyrme, “A model unified field equation,” Nucl. Phys.,31, 550 (1962).Google Scholar
  6. 6.
    A. C. Scott, “A nonlinear Klein-Gordon equation,/rd Amer. J. Phys.,37, 52 (1969).Google Scholar
  7. 7.
    L. Bianchi, Lezioni di Geometria Differenziale, Vol. 1, Part 2, N. Zanicheltura, Bologna (1927).Google Scholar
  8. 8.
    J. D. Gibbon and J. C. Eilbeck, “A possible N-soliton solution for a nonlinear optics equation,” J. Phys. A5, 2122 (1972).Google Scholar
  9. 9.
    R. Hirota, “Exact solution of the sine-Gordon equation for multiple collisions of solitons,” J. Phys. Soc. Japan,33, 1459 (1972).Google Scholar
  10. 10.
    A. J. Callegari and E. L. Reiss, “Nonlinear stability problems for the sine-Gordon equation,” J. Math. Phys.,14, 267 (1973).Google Scholar
  11. 11.
    A. Z. Averbukh and E. V. Venitsianov, “Steady-state solutions of a dispersive system for nonlinear waves,” Prikl. Matem, i Mekhan.,36, No. 5 (1972).Google Scholar
  12. 12.
    I. O. Kulik, “Wave propagation in a Josephson tunnel junction in the presence of vortices, and the electrodynamics of weak superconductivity,” Zh. Éksp. Teor. Fiz.,51, 1952 (1966).Google Scholar
  13. 13.
    P. Lebwohl and M. J. Stephen, “Properties of vortex lines in superconducting barriers,” Phys. Rev.163, 376 (1967).Google Scholar
  14. 14.
    A. C. Scott, “Waveform stability on a nonlinear Klein-Gordon equation,” Proc. IEEE,57, 1338 (1969).Google Scholar
  15. 15.
    E. N. Pelinovskii, “On the absorption of nonlinear waves in dispersive media,” Zh. Prikl. Mekhan. Tekh. Fiz., No. 2 (1971).Google Scholar
  16. 16.
    L. A. Ostrovskii and E. N. Pelinovskii, “Method of averaging and the generalized variational principle for nonsinusoidal waves,” Prikl. Matem, i Mekhan.,36, 71 (1972).Google Scholar
  17. 17.
    P. G. Kryukov and V. S. Letokhov, “Propagation of a light pulse in a resonantly amplifying (absorbing) medium,” Usp. Fiz. Nauk,99, 169 (1969).Google Scholar

Copyright information

© Plenum Publishing Corporation 1976

Authors and Affiliations

  • E. N. Pelinovskii
    • 1
  • S. Kh. Shavratskii
    • 1
  1. 1.Gor'kii

Personalised recommendations