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Toda systems and exponents of simple Lie groups. (English) Zbl 1017.37028

The authors study some aspects of the Bogoyavlensky Toda systems of \(A_n\), \(B_n\) and \(C_n\) types. The results include master symmetries, recursion operators, higher Poisson brackets and invariants presented both in Flaschka’s and in natural coordinates. A conjecture that relates the degrees of higher Poisson brackets and the exponents of the corresponding Lie group is verified for these systems.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
34A05 Explicit solutions, first integrals of ordinary differential equations
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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[1] Adler, M., On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-de-Vries type equations, Invent. Math., Vol. 50, 219-248 (1979) · Zbl 0393.35058
[2] Bogoyavlensky, O. I., On perturbations of the periodic Toda lattices, Comm. Math. Phys., Vol. 51, 201-209 (1976)
[3] Damianou, P. A., Master symmetries and R-matrices for the Toda Lattice, Letters in Mathematical Physics, Vol. 20, 101-112 (1990) · Zbl 0714.70014
[4] Damianou, P. A., Multiple Hamiltonian structures for Toda-type systems, J. Math. Phys., Vol. 35, 5511-5541 (1994) · Zbl 0822.58017
[5] Damianou, P. A., Multiple Hamiltonian structures for Toda systems of type A-B-C, Regular and Chaotic Dynamics, Vol. 5, 1, 17-32 (2000) · Zbl 0947.37050
[6] Das, A.; Okubo, S., A systematic study of the Toda lattice, Ann. Phys., Vol. 190, 215-232 (1989)
[7] Fernandes, R. L., On the mastersymmetries and bi-Hamiltonian structure of the Toda lattice, J. of Phys. A, Vol. 26, 3797-3803 (1993) · Zbl 0811.58035
[8] Flaschka, H., The Toda lattice I. Existence of integrals, Phys. Rev., Vol. 9, 1924-1925 (1974) · Zbl 0942.37504
[9] Flaschka, H., On the Toda lattice II. Inverse-scattering solution, Progr. Theor. Phys., Vol. 51, 703-716 (1974) · Zbl 0942.37505
[10] Fokas, A. S.; Fuchssteiner, B., The hierarchy of the Benjamin-Ono equation, Phys. Lett., Vol. 86A, 341-345 (1981)
[11] Fuchssteiner, B., Mastersymmetries, higher order time-dependent symmetries and conserved densities of non-linear evolution equations, Progr. Theor. Phys., Vol. 70, 1508-1522 (1983) · Zbl 1098.37536
[12] Henon, M., Integrals of the Toda lattice, Phys. Rev. B, Vol. 9, 1921-1923 (1974) · Zbl 0942.37503
[13] Kac, M.; van Moerbeke, P., On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices, Adv. Math., Vol. 16, 160-169 (1975) · Zbl 0306.34001
[14] Kostant, B., The solution to a generalized Toda lattice and representation theory, Adv. Math., Vol. 34, 195-338 (1979) · Zbl 0433.22008
[15] Kupershmidt, B., Discrete Lax equations and differential-difference calculus, Asterisque, Vol. 123, 1-222 (1985) · Zbl 0565.58024
[16] Magri, F., A simple model of the integrable Hamiltonian equations, J. Math. Phys., Vol. 19, 1156-1162 (1978) · Zbl 0383.35065
[17] Manakov, S., Complete integrability and stochastization of discrete dynamical systems, Zh. Exp. Teor. Fiz., Vol. 67, 543-555 (1974)
[18] Morosi, C.; Tondo, G., Some remarks on the bi-Hamiltonian structure of integral and discrete evolution equations, Inv. Probl., Vol. 6, 557-566 (1990) · Zbl 0731.58022
[19] Moser, J., Finitely many mass points on the line under the influence of an exponential potential - an integrable system, Lecture Notes in Physics, Vol. 38, 97-101 (1976)
[20] Moser, J., Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math., Vol. 16, 197-220 (1975) · Zbl 0303.34019
[21] Nunes da Costa, J. M.; Marle, C.-M., Master symmetries and bi-Hamiltonian structures for the relativistic Toda lattice, J. Phys. A, Vol. 30, 7551-7556 (1997) · Zbl 1004.37510
[22] Oevel, W.; Fuchssteiner, B., Explicit formulas for symmetries and conservation laws of the Kadomtsev-Petviashvili equation, Phys. Lett., Vol. 88A, 323-327 (1982)
[23] Oevel, W., Topics in Soliton Theory and Exactly Solvable Non-Linear Equations (1987), World Scientific Publ.
[24] Olver, P. J., Evolution equations possessing infinitely many symmetries, J. Math. Phys., Vol. 18, 1212-1215 (1977) · Zbl 0348.35024
[25] Olshanetsky, M. A.; Perelomov, A. M., Explicit solutions of classical generalized Toda models, Invent. Math., Vol. 54, 261-269 (1979) · Zbl 0419.58008
[26] Toda, M., One-dimensional dual transformation, J. Phys. Soc. Japan, Vol. 22, 431-436 (1967)
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