Acta Applicandae Mathematica

, Volume 92, Issue 1, pp 63–76 | Cite as

The Generalized Weierstrass System for Nonconstant Mean Curvature Surfaces and the Nonlinear Sigma Model

  • Paul BrackenEmail author


A study of the generalized Weierstrass system which can be used to induce mean curvature surfaces in three-dimensional Euclidean space is presented. A specific transformation is obtained which reduces the initial system to a two-dimensional Euclidean nonlinear sigma model. Some aspects of integrability are discussed, in particular, a connection with a version of the sinh-Gordon equation is established. Finally, some specific solutions are given and a systematic way of calculating multisoliton solutions is presented.

Key words

Weirstrass system sinh-Gordon equation Euclidean space mean curvature 

Mathematics Subject Classifications (2000)

35Q51 53A10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Weierstrass, K.: Fortsetzung der Untersuchung über die Minimalflächen, Mathematische Werke. vol. 3, pp. 219–248. Verlagsbuchhandlung, Hillesheim (1866)Google Scholar
  2. 2.
    Enneper, A.: Nachrichten Königl. Gesell. Wissenschaft Georg-Augustus Universität Göttingen 12, 258 (1868)Google Scholar
  3. 3.
    Konopelchenko, B.: Induced surfaces and their integrable dynamics. Stud. Appl. Math. 96, 9–51 (1996)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Konopelchenko, B.G., Taimanov, I.A.: Constant mean curvature surfaces via an integrable dynamical system. J. Phys. A 29, 1261–1265 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bracken, P., Grundland, A.M., Martina, L.: The Weierstrass–Enneper system for constant mean curvature surfaces and the completely integrable sigma model. J. Math. Phys. 40, 3379–3402 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bracken, P., Grundland, A.M.: Symmetry properties and explicit solutions of the generalized Weierstrass system. J. Math. Phys. 42, 1250–1282 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bracken, P., Grundland, A.M.: On certain classes of solutions of the Weierstrass–Enneper system inducing constant mean curvature surfaces. J. Nonlinear Math. Phys. 6, 294–313 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bracken, P., Grundland, A.M.: On complete integrability of the generalized Weierstrass system. J. Nonlinear Math. Phys. 9, 229–247 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Zakrzewski, W.: Low Dimensional Sigma-models. Hilger, New York (1989)zbMATHGoogle Scholar
  10. 10.
    Nelson, D., Piran, T., Weinberg, S.: Statistical Mechanics of Membranes and Surfaces. World Scientific, Singapore (1992)Google Scholar
  11. 11.
    Gross, D.G., Pope, C.N., Weinberg, S.: Two-dimensional Quantum Gravity and Random Surfaces. World Scientific, Singapore (1992)Google Scholar
  12. 12.
    Grundland, A.M., Martina, L., Rideau, G.: Partial differential equations with differential constraints. In: CRM Proceedings and Lecture Notes, American Mathematical Society, Providence, Rhode Island, vol. 11, pp. 135–154 (1997)Google Scholar
  13. 13.
    Makhankov, V.G., Pashaev, O.K.: Integrable pseudospin models in condensed matter. Sov. Sci. Rev. Math. Phys. 9, 1–151 (1992)Google Scholar
  14. 14.
    Bracken, P.: Spin model equations, connections with integrable systems and applications to magnetic vortices. Int. J. Mod. Phys. B 17, 4325–4537 (2003)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Bracken, P.: Reductions of Chern–Simons theory to integrable systems which have geometric applications. Internat. J. Modern Phys. B 18, 1261–1275 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Polyakov, A.: Fine structure of strings. Nuclear Phys. B 286, 406–412 (1986)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Kleinert, H.: The membrane properties of condensing strings. Phys. Lett. B 174, 335–338 (1986)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TexasEdinburgUSA

Personalised recommendations