Theoretical and Mathematical Physics

, Volume 124, Issue 1, pp 897–908 | Cite as

Discretizations of the Landau-Lifshits equation

  • V. E. Adler


The relation between the Sklyanin chain and the Bäcklund transformations for the Landau-Lifshits equation is established. The stationary solutions of the chain determine an integrable mapping, which is a kind of classical Heisenberg spin chain. Some multifield generalizations are found.


Poisson Bracket Heisenberg Chain Nonlinear Schr6dinger Equation Jordan Triple System Discrete Integrable System 
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  1. 1.
    E. K. Skylyanin,Funct. Anal. Appl.,16, 263–270 (1982).CrossRefGoogle Scholar
  2. 2.
    L. A. Takhtadzhyan and L. D. Faddeev,The Hamiltonian Methods in the Theory of Solitons, [in Russian], Nauka, Moscow (1986); English transl L. D. Faddeev and L. A. Takhtajan, Berlin, Springer (1987).zbMATHGoogle Scholar
  3. 3.
    A. B. Shabat and R. I. Yamilov, “Factorization of nonlinear equations of the type of the Heisenberg model [in Russian],” Preprint, Bashkir Div. USSR Acad. Sci., Ufa (1987).Google Scholar
  4. 4.
    A. B. Shabat and R. I. Yamilov,Leningrad. Math. J.,2, 377–400 (1990).MathSciNetGoogle Scholar
  5. 5.
    O. Ragnisco and P. M. Santini,Inverse Problems,6, 441–452 (1990).CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    A. I. Bobenko, “Discrete integrable, systems and geometry,”, in:Proc. 12th Intl. Congress of Mathematical Physics (D. De Wit, A. J. Bracken, M. D. Gould and P. A. Pearc, eds.), International Press, Boston (1999), pp. 219–226.Google Scholar
  7. 7.
    A. I. Bobenko and Yu. B. Suris,Commun. Math. Phys.,204, 147–188 (1999).CrossRefADSMathSciNetGoogle Scholar
  8. 8.
    A. V. Mikhailov and A. B. Shabat,Phys. Lett. A,116, No. 4, 191–194 (1986).CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Ya. I. Granovskii and A. S. Zhedanov,Theor. Math. Phys.,71, 438–445 (1987).CrossRefMathSciNetGoogle Scholar
  10. 10.
    A. P. Veselov,Theor. Math. Phys.,71, 446–450 (1987).CrossRefMathSciNetGoogle Scholar
  11. 11.
    A. P. Veselov,Russ. Math. Surv.,46, 1–51 (1991).CrossRefMathSciNetGoogle Scholar
  12. 12.
    A. P. Veselov,Dokl. Akad. Nauk SSSR,270, 1094–1096 (1983).MathSciNetGoogle Scholar
  13. 13.
    V. E. Adler and A. B. Shabat,Theor. Math. Phys.,112, 935–948 (1997).MathSciNetGoogle Scholar
  14. 14.
    V. G. Marikhin and A. B. Shabat,Theor. Math. Phys.,118, 173–182 (1999).MathSciNetGoogle Scholar
  15. 15.
    I. Z. Golubchik and V. V. Sokolov,Theor. Math. Phys.,124, 909–917 (2000).MathSciNetGoogle Scholar
  16. 16.
    S. I. Svinolupov,Commun. Math. Phys.,143, 559–575 (1992).CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    V. E. Adler, S. I. Svinolupov and R. I. Yamilov,Phys. Lett. A,254, 24–36 (1999).CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 2000

Authors and Affiliations

  • V. E. Adler
    • 1
  1. 1.Institute of Mathematics, Ufa Scientific CenterRASUfaRussia

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