Finite-dimensional discrete systems integrated in quadratures. (English) Zbl 1178.37078

Kazakova, T. G.

Theor. Math. Phys. 138, No. 3, 356-369 (2004); translation from Teor. Mat. Fiz. 138, No. 3, 356-369 (2004).

Summary: We consider finite-dimensional reductions (truncations) of discrete systems of the type of the Toda chain with discrete time that retain the integrability. We show that for finite-dimensional chains, in addition to integrals of motion, we can construct a rich family of higher symmetries described by the master symmetry. We reduce the problem of integrating a finite-dimensional system to the implicit function theorem.

MSC:

37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K15	Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems

Keywords:

integrability; truncation condition; zero-curvature equation; classical symmetry; master symmetry; integrals of motion