Theoretical and Mathematical Physics

, Volume 136, Issue 1, pp 917–935 | Cite as

Spectral Curves and Parameterization of a Discrete Integrable Three-Dimensional Model

  • S. Z. Pakuliak
  • S. M. Sergeev


We consider a discrete classical integrable model on a three-dimensional cubic lattice. The solutions of this model can be used to parameterize the Boltzmann weights of various three-dimensional spin models. We find the general solution of this model constructed in terms of the theta functions defined on an arbitrary compact algebraic curve. Imposing periodic boundary conditions fixes the algebraic curve. We show that the curve then coincides with the spectral curve of the auxiliary linear problem. For a rational curve, we construct the soliton solution of the model.

three-dimensional integrable systems Bäcklund transformations spectral curves 


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  1. 1.
    S. M. Sergeev, Theor. Math. Phys., 118, 378-384 (1999); 124, 1187-1201 (2000); S. Sergeev, J. Phys. A, 32, 5693-5714 (1999); 34, 10493-10503 (2001); Phys. Lett. A, 265, 364-368 (2000).Google Scholar
  2. 2.
    I. M. Krichever, Russ. Math. Surveys, 33, 255-256 (1978).Google Scholar
  3. 3.
    R. Hirota, J. Phys. Soc. Japan, 50, 3785-3791 (1981).Google Scholar
  4. 4.
    D. Mumford, Tata Lectures on Theta, Vols. 1, 2 (Progr. Math., Vols. 28, 43), Birkhäuser, Boston (1983, 1984).Google Scholar
  5. 5.
    J. D. Fay, Theta Functions on Riemann Surfaces (Lect. Notes Math., Vol. 352), Springer, Berlin (1973).Google Scholar
  6. 6.
    G. Springer, Introduction to Riemann Surfaces, Chelsea, New York (1981).Google Scholar
  7. 7.
    S. Sergeev, J. Nonlinear Math. Phys., 27, 57-72 (2000).Google Scholar
  8. 8.
    S. Pakuliak and S. Sergeev, Int. J. Math. Math. Sci., 31, 513-554 (2002).Google Scholar
  9. 9.
    G. von Gehlen, S. Pakuliak, and S. Sergeev, “Explicit free parametrization of the modified tetrahedron equation,” Preprint MPI-2002-102, MPI, Bonn (2002).Google Scholar
  10. 10.
    S. Sergeev, “Functional equations and separation of variables for 3d spin models,” Preprint MPI-2002-46, MPI, Bonn (2002).Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • S. Z. Pakuliak
    • 1
  • S. M. Sergeev
    • 2
  1. 1.Bogoliubov Laboratory of Theoretical PhysicsJoint Institute for Nuclear Research, DubnaMoscow OblastRussia
  2. 2.Research School of Physical Sciences and EngineeringAustralian National UniversityCanberraAustralia

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