Fundamental Models on the Lattice

  • Ludwig D. Faddeev
  • Leon A. Takhtajan
Part of the Springer Series in Soviet Mathematics book series (CLASSICS)


Here we shall give a complete list of results pertaining to the Toda model, a fundamental model on the lattice. We will show that the r-matrix approach applies to this case and may be used to prove the complete integrability of the model in the quasi-periodic case. For the rapidly decreasing boundary conditions we will analyze the mapping F from the initial data of the auxiliary linear problem to the transition coefficients and outline a method for solving the inverse problem, i.e. for F−1 constructing. On the basis of the r-matrix approach it will be shown that F is a canonical trans formation to action-angle type variables establishing the complete integrability of the Toda model in the rapidly decreasing case. We shall also define a lattice version of the LL model, the most general integrable lattice system with two-dimensional auxiliary space.


Integral Equation Inverse Problem Global Analysis Linear Problem Type Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A 1979]
    Adler, M.: On a trace functional for formal pseudodifferential operatorsand symplectic structure of the Korteweg-de Vries type equations. Invent. Math. 50, 219–248 (1979)ADSCrossRefGoogle Scholar
  2. [BFT 1986]
    Buslaev, V. S., Faddeev, L. D., Takhtajan, L. A.: Scattering theory for the Korteweg-de Vries equation and its Hamiltonian interpretation. Physica D18, 255–266 (1986)MathSciNetMATHGoogle Scholar
  3. [CK 1973]
    Case, K. M., Kac, M.: A discrete version of the inverse scattering problem. J. Math. Phys. 14, 594–603 (1973)ADSMathSciNetCrossRefGoogle Scholar
  4. [E 1981]
    Eilenberger, G.: Solitons. Mathematical method for Physicists. Berlin, Springer 1981MATHGoogle Scholar
  5. [F 1974 a]
    Flaschka, H.: The Toda lattice. I. Existence of integrals. Phys. Rev. B9, 1924–1925 (1974)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. [F 1974b]
    Flaschka, H.: On the Toda lattice. II. Inverse scattering solution. Prog. Theor. Phys. 51, 703–716 (1974)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. [FM 1976]
    Flaschka, H., McLaughlin, D.: Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions. Prog. Theor. Phys. 55, 438–456 (1976)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. IK 1981 a] Izergin, A. G., Korepin, V. E.: The lattice sine-Gordon model. Vestn.Leningr. Univ. Ser. Fiz. Khim. 22,84–87 (1981) [Russian]Google Scholar
  9. [IK 1981 b]
    Izergin, A. G., Korepin, V. E.: The lattice quantum Sine-Gordon model. Lett. Math. Phys. 5, 199–205 (1981)ADSMathSciNetCrossRefGoogle Scholar
  10. IK 1982 a] Isergin, A. G., Korepin, V. E.: Lattice regularizations of two-dimensional quantum field models. In: Problems in quantum field theory and statistical physics. 3. Zap. Nauchn. Semin. LOMI 120,75–91 (1982) [Russian]Google Scholar
  11. [IK 1982b]
    Isergin, A. G., Korepin, V. E.: Lattice versions of quantum field theory models in two dimensions. Nucl. Phys. B 205, 401–413 (1982)MathSciNetGoogle Scholar
  12. K 1978] Krichever, I. M.: Algebraic curves and nonlinear difference equations. Usp. Mat. Nauk 33 (4), 215–216 (1978) [Russian]Google Scholar
  13. [M 1974a] Manakov, S. V.: Complete integrability and stochastization of discrete dynamical systems. Zh. Exp. Teor. Fiz. 67, 543-555 (1974)
    [Russian]; English transi. in Soy. Phys. JETP 40, 269–274 (1975)Google Scholar
  14. [M 1974 b] Manakov, S. V.: Inverse scattering method as applied to problems of the physics of waves in nonlinear media. Candidate thesis, Chernogolovka 1974 [Russian]Google Scholar
  15. [S 1982] Sklyanin, E. K.: Algebraic structures connected with the Yang-Baxter equation. Funk. Anal. Priloz. 16 (4), 27-34 (1982)
    Russian]; English transi. in Funct. Anal. Appt 16, 263–270 (1982)Google Scholar
  16. [T 1981]
    Toda, M.: Theory of non-linear lattices. Springer series in solid state physics 20. Berlin-Heidelberg-New York, Springer 1981Google Scholar
  17. [T 1982] Tarasov, V. O.: Classical version of the lattice sine-Gordon model. In: Problems in quantum field theory and statistical physics. 3. Zap. Nauchn. Semin. LOMI 120,173–187 (1982) [Russian]Google Scholar
  18. [TF 1979]
    Takhtajan, L. A., Faddeev, L. D.: The quantum inverse problem method and the XYZ Heisenberg model. Usp. Mat. Nauk 34 (5), 13–63 (1979)[Russian]; English transi. in Russian Math. Surveys 34 (5), 11–68 (1979)Google Scholar
  19. [V 1985] Volkov, A. Yu.: A discrete version of the Landau-Lifshitz equation. In: Problems in quantum field theory and statistical physics. 5. Zap. Nauchn. Semin. LOMI 145, 62–71 (1985) [Russian]Google Scholar
  20. [Z 1973] Zakharov, V. E.: On stochastization of one-dimensional chains of nonlinear oscillators. Zh. Exp. Teor. Fiz. 65, 219-225 (1973)
    Russian]; English transi. in Sov. Phys. JETP 38, 108–110 (1974)Google Scholar
  21. [ZMNP 1980]
    Zakharov, V. E., Manakov, S. V., Novikov, S. P., Pitaievski, L. P.: Theory of Solitons. The Inverse Problem Method. Moscow, Nauka 1980 [Russian]; English transi.: New York, Plenum 1984Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Ludwig D. Faddeev
    • 1
  • Leon A. Takhtajan
    • 1
  1. 1.Steklov Mathematical InstituteLeningradUSSR

Personalised recommendations