Journal of Engineering Mathematics

, Volume 29, Issue 4, pp 347–369

Numerical solutions of a damped Sine-Gordon equation in two space variables

Article

Abstract

Numerical solutions of the perturbed Sine-Gordon equation in two space variables, arising from a Josephson junction are presented. The method proposed arises from a two-step, one parameter method for the numerical solution of second-order ordinary differential equations. Though implicit in nature, the method is applied explicitly. Global extrapolation in both space and time is used to improve the accuracy. The method is analysed with respect to stability criteria and numerical dispersion. Numerical results are obtained for various cases involving line and ring solitons.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R.K. Dodd, J.C. Eilbeck, J.D. Gibbons and H.C. Morris, Solitons and Nonlinear Wave Equations. London Academic Press (1982).Google Scholar
  2. 2.
    P.G. Drazin and R.S. Johnson, Solitons: An Introduction. University Press, Cambridge (1989).Google Scholar
  3. 3.
    J.D. Josephson, Supercurrents through barriers, Advances in Physics 14 (1965) 419–451.Google Scholar
  4. 4.
    P.L. Christiansen and P.S. Lomdahl, Numerical solution of 2+1 dimensional Sine-Gordon solitons. Physica 2D (1981) 482–494.Google Scholar
  5. 5.
    R. Hirota, Exact three-soliton solution of the two-dimensional Sine-Gordon equation. J. Phys. Soc. Japan 35 (1973) 1566.Google Scholar
  6. 6.
    J. Zagrodzinsky, Particular solutions of the Sine-Gordon equation in 2+1 dimensions. Phys. Lett. 72A (1979) 284–286.Google Scholar
  7. 7.
    P.L. Christiansen, and O.H. Olsen, Ring-shaped quasi-soliton solutions to the two and three-dimensional Sine-Gordon equations. Physica Scripta 20 (1979) 531–538.Google Scholar
  8. 8.
    G. Leibbrandt, New exact solutions of the classical Sine-Gordon equation in 2+1 and 3+1 dimensions. Phys. Rev. Lett. 41 (1978) 435–438.Google Scholar
  9. 9.
    P. Kaliappan and M. Lakhshmanan, Kadomtsev-Petviashvili, two dimensional Sine-Gordon equations: reduction to Painleve transcendents. J. Phys. A.: Math. Gen. 12 (1979) L249-L252.Google Scholar
  10. 10.
    J. Argyris, M. Haase and J.C. Heinrich, Finite element approximation to two-dimensional Sine-Gordon solitons. Computer Methods in Applied Mechanics and Engineering 86 (1991) 1–26.Google Scholar
  11. 11.
    K. Nakajima, Y. Onodera, T. Nakamura and R. Sato, Numerical analysis of vortex motion on Josephson structures. Journal of Applied Physics 45(9) (1974) 4095–4099.Google Scholar
  12. 12.
    B.A. Malomed, Dynamics of quasi-one-dimensional kinks in the two dimensional Sine-Gordon model. Physica D, 52 (1991) 157–170.Google Scholar
  13. 13.
    I.S. Greig and J.Ll. Morris, A Hopscotch method for the Korteweg-de Vries equation. J. Comput. Phys. 20 (1976) 64–80.Google Scholar
  14. 14.
    E.H. Twizell, Y. Wang and W.G. Price, Chaos-free numerical solutions of reaction-diffusion equations. Proc. R. Soc. Lond. A 430 (1991) 541–576.Google Scholar
  15. 15.
    J.D. Lambert, Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. John Wiley and Sons 1991.Google Scholar
  16. 16.
    Y.I. Shokin, Springer-Verlag, Berlin, Heidelberg, The Method of Differential Approximation. New York (1983).Google Scholar
  17. 17.
    E. Turkel, Phase error and stability of second order methods for hyperbolic problems. J. Comput. Phys. 15 (1974) 226–250.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  1. 1.Department of Ship ScienceUniversity of SouthamptonSouthamptonEngland
  2. 2.Department of Mathematics and StatisticsBrunel UniversityUxbridgeEngland

Personalised recommendations