Skip to main content
SpringerLink
Log in

Initial Value Problems of the Sine-Gordon Equation and Geometric Solutions

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

Recent results using inverse scattering techniques interpret every solution φ(x, y) of the sine-Gordon equation as a nonlinear superposition of solutions along the axes x=0 and y=0. This has a well-known geometric interpretation, namely that every weakly regular surface of Gauss curvature K=−1, in arc length asymptotic line parametrization, is uniquely determined by the values φ(x, 0) and φ(0, y) of its coordinate angle along the axes. We introduce a generalized Weierstrass representation of pseudospherical surfaces that depends only on these values, and we explicitely construct the associated family of pseudospherical immersions corresponding to it.

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. Kricever, I. M.: An analogue of D’Alembert’s Formula for the equations of the principal chiral field and for the sine-Gordon Equation, Soviet Math. Dokl. 22(1) (1980), 79–84.

    Google Scholar 

  2. Dorfmeister, J., Pedit, F. and Wu, H.: Weierstrass type representations of harmonic maps into symmetric spaces, Comm. Anal. Geom. 6 (1998), 633–668.

    Google Scholar 

  3. Bobenko, A. I. and Kitaev, A. V.: On asymptotic cones of surfaces with constant curvature and the third Painleve equation, Manuscripta Math. 97 (1998), 489–516; SFB288, Preprint {315} (1998).

    Google Scholar 

  4. Bobenko, A. I., Matthes, D. and Suris, Y. B.: Nonlinear hyperbolic equations in surface theory: Integrable discretizations and approximations results, arXiv:Math.NA/0208042 v1 Aug 2002.

  5. Bobenko, A. I.: Surfaces in terms of 2 by 2 matrices, In: Harmonic Maps and Integrable Systems, Vieweg, 1994, pp. 83–128.

  6. Eisenhart, L. P.: A Treatise in the Differential Geometry of Curves and Surfaces, Dover, New York, 1909.

    Google Scholar 

  7. Wu, H.: Non-linear partial differential equations via vector fields on homogeneous Banach manifolds, Ann. Global Anal. Geom. 10 (1992), 151–170.

    Article  Google Scholar 

  8. Terng, C. L. and Uhlenbeck, K.: Bäcklund transformation and loop group actions, Comm. Pure Appl. Math. 53 (2000), 1–75, math.dg/9805074.

    Article  Google Scholar 

  9. Dorfmeister, J. and Sterling, I.: Finite type Lorentz harmonic maps and the method of Symes, J. Differential Geom. Appl. 17 (2002), 43–53.

    Article  Google Scholar 

  10. Sym, A.: Soliton surfaces and their applications (Soliton geometry from spectral problems), In: {Lecture Notes in Physics} 239, Springer, New York, 1985, pp. 154–231.

    Google Scholar 

  11. Amsler, M. H.: Des surfaces a courbure negative constante dans l’espace a trois dimensions et de leurs singularites, Math. Ann. 130 (1955), 234–256.

    Article  Google Scholar 

  12. Melko, M. and Sterling, I.: Integrable systems, harmonic maps and the classical theory of surfaces, In: Harmonic Maps and Integrable Systems, Vieweg, 1994, pp. 129–144.

  13. Melko, M. and Sterling, I.: Applications of Soliton theory to the construction of pseudospherical surfaces in R3, Ann. Global Anal. Geom. 11 (1993), 65–107.

    Article  Google Scholar 

  14. Toda, M.: Weierstrass representation of weakly regular pseudospherical surfaces in euclidean space, Balkan J. Geom. Appl. 7(2) (2002), 87–136.

    Google Scholar 

  15. Pressley, A. and Segal, G.: Loop Groups, Oxford Math. Monogr., Oxford University Press, 1986.

  16. Bobenko, A. I.: Constant mean curvature surfaces and integrable equations, Russ. Math. Surveys 46(4) (1991), 1–45.

    Google Scholar 

  17. Chern, S. S. and Terng, C. L.: An analogue of Bäcklund’s Theorem in affine geometry, Rocky Mountain J. Math. 10(1) (1980), 439–458.

    Google Scholar 

  18. Dorfmeister, J. and Haak, G.: Meromorphic potentials and smooth surfaces of constant mean curvature, Math. Z. 224 (1997), 603–640.

    Google Scholar 

  19. Lund, F.: Soliton and geometry, In: A. O. Barut (ed.), Proceedings of the NATO ASI on Nonlinear Equations in Physics and Mathematics, Reidel, Dordrecht, 1978.

  20. Toda, M.: Pseudospherical surfaces via moving frames and loop groups, PhD Dissertation, hardbound, University of Kansas, 2000, 114 pp.

  21. Wu, H.: A simple way for determining the normalized potentials for harmonic maps, Ann. Global Anal. Geom. 17 (1999), 189–199.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, 79409, U.S.A.

    Magdalena Toda

Authors
  1. Magdalena Toda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Magdalena Toda.

Additional information

Mathematics Subject Classifications (2000): 53A10, 58E20.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Toda, M. Initial Value Problems of the Sine-Gordon Equation and Geometric Solutions. Ann Glob Anal Geom 27, 257–271 (2005). https://doi.org/10.1007/s10455-005-1582-9

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-005-1582-9

Keywords

Search

Navigation