×

Integrability conditions for an analogue of the relativistic Toda chain. (English. Russian original) Zbl 1121.37056

Theor. Math. Phys. 151, No. 1, 492-504 (2007); translation from Teor. Mat. Fiz. 151, No. 1, 66-80 (2007).
Summary: We consider a class of discrete-differential equations that contains the relativistic Toda chain and is characterized by one arbitrary function of six variables. We derive three conditions that allow testing the integrability of any given equation in this class. In deriving these conditions, we use higher symmetries distinguishing the equations that are integrable via the inverse scattering method.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems

Keywords:

higher symmetry
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] S. N. M. Ruijsenaars, Comm. Math. Phys., 133, 217–247 (1990). · Zbl 0719.58019
[2] Yu. B. Suris, J. Phys. A, 30, 1745–1761 (1997). · Zbl 1001.37508
[3] V. É. Adler and A. B. Shabat, Theor. Math. Phys., 111, 647–657 (1997). · Zbl 0978.37502
[4] V. V. Sokolov and A. B. Shabat, Sov. Sci. Rev. Sect. C, 4, 221–280 (1984); A. V. Mikhailov, A. B. Shabat, and V. V. Sokolov, ”The symmetry approach to classification of integrable equations,” in: What is Integrability? (V. E. Zakharov, ed.), Springer, Berlin (1991), p. 115–184. · Zbl 0554.35107
[5] A. V. Mikhailov, A. B. Shabat, and R. I. Yamilov, Russ. Math. Surveys, 42, No. 4, 1–63 (1987). · Zbl 0646.35010
[6] V. É. Adler, A. B. Shabat, and R. I. Yamilov, Theor. Math. Phys., 125, 1603–1661 (2000). · Zbl 1029.37041
[7] R. I. Yamilov, Uspekhi Mat. Nauk, 38, No. 6, 155–156 (1983); R. I. Yamilov, ”Classification of Toda type scalar lattices,” in: Nonlinear Evolution Equations and Dynamical Systems NEEDS’92 (Proc. 8th Intl. Workshop, Dubna, Russia, 1992, V. Makhankov, I. Puzynin, and O. Pashaev, eds.), World Scientific, River Edge, N. J. (1993), p. 423–431.
[8] D. Levi and R. Yamilov, J. Math. Phys., 38, 6648–6674 (1997); R. Yamilov and D. Levi, J. Nonlinear Math. Phys., 11, 75–101 (2004). · Zbl 0896.34057
[9] R. Yamilov, J. Phys. A, 39, R541–R623 (2006). · Zbl 1105.35136
[10] R. I. Yamilov, Theor. Math. Phys., 139, 623–635 (2004). · Zbl 1178.37094
[11] A. V. Mikhailov, A. B. Shabat, and R. I. Yamilov, Comm. Math. Phys., 115, 1–19 (1988). · Zbl 0659.35091
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.