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Journal of Soviet Mathematics

, Volume 28, Issue 5, pp 800–806 | Cite as

Simple connection between the geometric and the Hamiltonian representations of integrable nonlinear equations

  • L. A. Takhtadzhyan
  • L. D. Faddeev
Article
  • 23 Downloads

Abstract

One gives a simple and general derivation of the well-known connection between the geometric and the Hamiltonian approaches in the classical method of the inverse problem. Namely, for the case of a two-dimensional auxiliary problem and periodic boundary conditions it is explicitly shown how the existence of the classical и-matrix for the integrable equations leads to their representation in the form of the condition of zero curvature.

Keywords

Boundary Condition Integrable Equation Inverse Problem Nonlinear Equation Periodic Boundary 
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Literature cited

  1. 1.
    V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: The Method of the Inverse Problem [in Russian], Nauka, Moscow (1980).Google Scholar
  2. 2.
    V. E. Zakharov and L. D. Faddeev, “The Korteweg-de Vries equation is a fully integrable Hamiltonian system,” Funkts. Anal. Prilozhen.,5, No. 4, 18–27 (1971).Google Scholar
  3. 3.
    E. K. Sklyanin, L. A. Takhtadzhyan, and L. D. Faddeev, “The quantum method of the inverse problem. I,” Teor. Mat. Fiz.,40, No. 2, 194–220 (1979).Google Scholar
  4. 4.
    L. D. Faddeev, “Quantum completely integrable models of field theory,” in: Problems of Quantum Field Theory (Proc. Int. Conf. on Nonlocal Field Theories, Alushta, 1979), Dubna (1979), pp. 249–299.Google Scholar
  5. 5.
    E. K. Sklyanin, “On complete integrability of the Landau-Lifschitz equation,” LOMI Preprint E-3-79, Leningrad (1979).Google Scholar
  6. 6.
    E. K. Sklyanin, “Quantum variant of the method of the inverse scattering problem,” J. Sov. Math.,19, No. 5 (1982).Google Scholar
  7. 7.
    A. G. Izergin and V. E. Korepin, “A lattice model connected with the nonlinear Schrödinger equation,” Dokl. Akad. Nauk SSSR,259, No. 1, 76–79 (1981).Google Scholar
  8. 8.
    P. P. Kulish and E. K. Sklyanin, “On the solutions of the Yang-Baxter equation,” J. Sov. Math.,19, No. 5 (1982).Google Scholar

Copyright information

© Plenum Publishing Corporation 1985

Authors and Affiliations

  • L. A. Takhtadzhyan
  • L. D. Faddeev

There are no affiliations available

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