Advertisement

Mathematical Notes

, Volume 64, Issue 1, pp 98–109 | Cite as

Canonically conjugate variables for the Volterra lattice with periodic boundary conditions

  • A. V. Penskoi
Article

Abstract

The Volterra lattice is considered. This dynamical system is known to be Hamiltonian with respect to two compatible Poisson brackets (quadratic and cubic). For each of the two brackets, a set of canonically conjugate variables is found by using the spectral theory of the Jacobi operator.

Key words

integrable system Volterra model periodic boundary conditions Toda lattice Poisson bracket canonically conjugate variables Jacobi operator 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. A. Takhtadzhyan and L. D. Faddeev,Hamiltonian Approach in Soliton Theory [in Russian], Nauka, Moscow (1986).Google Scholar
  2. 2.
    P. A. Damianou, “The Volterra model and its relation to the Toda lattice,”Phys. Lett. A,155, No. 2–3, 126–132 (1991).MathSciNetGoogle Scholar
  3. 3.
    S. V. Manakov, “Complete integrability and stochastization in discrete dynamical systems,”Zhurn. Exp. Teor. Fiz.,67, No. 2, 543–555 (1974).MathSciNetGoogle Scholar
  4. 4.
    M. Kac and P. van Moerbeke, “On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices,”Adv. Math.,16, 160–169 (1975).CrossRefGoogle Scholar
  5. 5.
    H. Flaschka and D. W. McLaughlin, “Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions,”Progr. Theoret. Phys.,55, No. 2, 438–456 (1976).MathSciNetGoogle Scholar
  6. 6.
    V. L. Vereshchagin, “Spectral theory of single-phase solutions of the Volterra lattice,”Mat. Zametki [Math. Notes],48, No. 2, 145–148 (1990).zbMATHMathSciNetGoogle Scholar
  7. 7.
    I. M. Krichever, “Nonlinear equations and elliptic curves,” in:Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Fundamental'nye Napravleniya [in Russian], Vol. 23, VINITI, Moscow (1983), pp. 79–136.Google Scholar
  8. 8.
    B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, “Nonlinear equations of Korteweg-de Vries type, finite-gap linear operators, and Abel manifolds,”Uspekhi Mat. Nauk [Russian Math. Surveys],31, No. 1, 55–136 (1976).MathSciNetGoogle Scholar
  9. 9.
    A. P. Veselov, “Integrable systems with discrete time and difference operators,”Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.],22, No. 2, 1–13 (1988).zbMATHMathSciNetGoogle Scholar
  10. 10.
    F. Magri, “A simple model of integrable Hamiltonian equation,”J. Math. Phys.,19, 1156–1162 (1978).CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1999

Authors and Affiliations

  • A. V. Penskoi
    • 1
  1. 1.M. V. Lomonosov Moscow State UniversityUSSR

Personalised recommendations