×

zbMATH — the first resource for mathematics

Discretizations of the Landau-Lifshits equation. (English. Russian original) Zbl 0984.39010
Theor. Math. Phys. 124, No. 1, 897-908 (2000); translation from Teor. Mat. Fiz. 124, No. 1, 48-61 (2000).
Summary: 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.

MSC:
39A12 Discrete version of topics in analysis
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] E. K. Skylyanin,Funct. Anal. Appl.,16, 263–270 (1982). · Zbl 0513.58028
[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). · Zbl 0632.58003
[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). · Zbl 0646.35010
[4] A. B. Shabat and R. I. Yamilov,Leningrad. Math. J.,2, 377–400 (1990).
[5] O. Ragnisco and P. M. Santini,Inverse Problems,6, 441–452 (1990). · Zbl 0725.45005
[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. · Zbl 1253.37073
[7] A. I. Bobenko and Yu. B. Suris,Commun. Math. Phys.,204, 147–188 (1999). · Zbl 0945.70010
[8] A. V. Mikhailov and A. B. Shabat,Phys. Lett. A,116, No. 4, 191–194 (1986).
[9] Ya. I. Granovskii and A. S. Zhedanov,Theor. Math. Phys.,71, 438–445 (1987).
[10] A. P. Veselov,Theor. Math. Phys.,71, 446–450 (1987).
[11] A. P. Veselov,Russ. Math. Surv.,46, 1–51 (1991). · Zbl 0785.58027
[12] A. P. Veselov,Dokl. Akad. Nauk SSSR,270, 1094–1096 (1983).
[13] V. E. Adler and A. B. Shabat,Theor. Math. Phys.,112, 935–948 (1997). · Zbl 0978.37504
[14] V. G. Marikhin and A. B. Shabat,Theor. Math. Phys.,118, 173–182 (1999). · Zbl 0984.37093
[15] I. Z. Golubchik and V. V. Sokolov,Theor. Math. Phys.,124, 909–917 (2000). · Zbl 1112.37324
[16] S. I. Svinolupov,Commun. Math. Phys.,143, 559–575 (1992). · Zbl 0753.35100
[17] V. E. Adler, S. I. Svinolupov and R. I. Yamilov,Phys. Lett. A,254, 24–36 (1999). · Zbl 0983.37082
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.