Skip to main content
SpringerLink
Log in

An initial-boundary value problem for the sine-Gordon equation in laboratory coordinates

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider the sine-Gordon equation in laboratory coordinates with both x and t in [0, ∞). We assume that u(x, 0), ut(x, 0), u(0, t) are given, and that they satisfy u(x, 0)→2πq, ut(x, 0)→0, for large x, u(0, t)→2πp for large t, where q, p are integers. We also assume that ux(x, 0), ut(x, 0), ut(0, t), u(0, t)-2πp, u(x, 0)-2πq ε L2. We show that the solution of this initial-boundary value problem can be reduced to solving a linear integral equation which is always solvable. The asymptotic analysis of this integral equation for large t shows how the boundary conditions can generate solitons.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Fokas A.S., Proceedings of the III Potsdam — V Kiev International Workshop 1991, Berlin: Springer-Verlag, 1992; Fokas A.S., Its A.R, Phys. Rev. Lett. 1992. V. 68. p. 3117.

    Google Scholar 

  2. Rajaraman R.,Solitons and Instantons, North Holland, 1982.

  3. Jain A.K., Likharev K.K., Lukens J.E., Sauvageau J.E., Phys. Rep.109 (1984), 309.

    Google Scholar 

  4. Yukon S.P., Liu N.C., IEEE Transactions on Magnets.27 (1991), 2736.

    Google Scholar 

  5. Zakharov V.E., Shabat A.B., Funkts. Anal. Prilozh.13 (1979), no. 3, 13–22. (Russian)

    Google Scholar 

  6. Manakov S.V., Sov. Phys.-JETP38 (1974), no. 4, 693–696.

    Google Scholar 

  7. Its A.R., Sov. Math. Dokl.24 (1981), no. 3, 452–456.

    Google Scholar 

  8. Deift P., Zhou X., Announcement in Bull. Amer. Math. Soc. (New Series).26 (1992), 119–123.

    Google Scholar 

  9. Ablowitz M.J., Kaup D.J., Newell A.C., Segur H., Phys. Rev. Lett.30 (1973), 1262–1964.

    Google Scholar 

  10. Zakharov V.E., Faddeev L.D., Takhtajan L.A., Dokl. Akad. Nauk SSSR.219 (1974), no. 6, 1334–1337 (Russian)

    Google Scholar 

  11. Zhou X., SIAM. J. Math. Anal.20 (1989), 966–986.

    Google Scholar 

Download references

Authors
  1. A. S. Fokas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. R. Its
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The authors dedicate this paper to the memory of M. C. Polivanov

Department of Mathematics and Computer Science; Institute for Nonlinear Studies, Clarkson University, Postdam, New York. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 3, pp. 387–403, September, 1992.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fokas, A.S., Its, A.R. An initial-boundary value problem for the sine-Gordon equation in laboratory coordinates. Theor Math Phys 92, 964–978 (1992). https://doi.org/10.1007/BF01017074

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01017074

Keywords

Search

Navigation