Theoretical and Mathematical Physics

, Volume 118, Issue 2, pp 173–182 | Cite as

Integrable lattices

  • V. G. Marikhin
  • A. B. Shabat


We propose a method for constructing integrable lattices starting from dynamic systems with two different parameterizations of the canonical variables and hence two independent Bäcklund flows. We construct integrable lattices corresponding to generalizations of the nonlinear Schrödinger equation. We discuss the Toda, Volterra, and Heisenberg models in detail. For these systems, as well as for the Landau-Lifshitz model, we obtain totally discrete Lagrangians. We also discuss the relation of these systems to the Hirota equations.


Discrete Equation Heisenberg Model Toda Lattice Integrable Lattice Toda Chain 
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  1. 1.
    R. Hirota,J. Phys. Soc. Japan,43, 2074 (1977);50, 3785 (1981).MathSciNetCrossRefADSGoogle Scholar
  2. 2.
    V. E. Adler and A. B. Shabat,Theor. Math. Phys.,112, 935 (1997).MathSciNetzbMATHGoogle Scholar
  3. 3.
    V. G. Marikhin,JETP Lett.,66, 705 (1997).CrossRefADSGoogle Scholar
  4. 4.
    R. M. Miura,J. Math. Phys.,9, 1202 (1968).CrossRefMathSciNetzbMATHADSGoogle Scholar
  5. 5.
    A. V. Mikhailov, A. B. Shabat, and R. I. Yamilov,Russ. Math. Surv.,42, 1 (1987).CrossRefMathSciNetGoogle Scholar
  6. 6.
    Y. B. Suris,J. Phys. A.,30, 2235 (1997).CrossRefADSMathSciNetzbMATHGoogle Scholar
  7. 7.
    F. W. Nijhoff, O. Ragnisco, and V. B. Kuznetsov,Commun. Math. Phys.,176, 681 (1996).CrossRefADSMathSciNetzbMATHGoogle Scholar
  8. 8.
    R. I. Yamilov, “Classification of Toda type scalar lattices,” in:Nonlinear Evolution Equations and Dynamical Systems, World Scientific, Singapore (1993), p. 423.Google Scholar
  9. 9.
    V. É. Adler and R. I. Yamilov, Private communication.Google Scholar
  10. 10.
    F. W. Nijhoff and V. Papageorgiou,Phys. Lett. A,141, 269 (1989).CrossRefADSMathSciNetGoogle Scholar
  11. 11.
    V. É. Adler and A. B. Shabat,Theor. Math. Phys.,115, 639 (1998).MathSciNetzbMATHGoogle Scholar
  12. 12.
    I. Krichever, O. Lipan, P. Wiegmann, and A. Zabrodin, “Quantum integrable systems and elliptic solutions of classical discrete nonlinear equations,” Preprint hep-th/9604080 (1996).Google Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 1999

Authors and Affiliations

  • V. G. Marikhin
    • 1
  • A. B. Shabat
    • 1
  1. 1.Landau Institute for Theoretical PhysicsRussian Academy of SciencesMoscowRussia

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