Elements of the integrability theory of discrete dynamical systems

Isospectral integrability analysis of dynamical systems on discrete manifolds

“…The algorithm, which was suggested in [11,12], makes it possible to readily construct an infinite hierarchy of conservation laws as well as to calculate their compatible co-symplectic structures.…”

“…First, we can easily show using the standard asymptotic small parameter approach [6,7,13] that the Nöther equation (1.6) on the manifold M (N ) possesses [11,12] the exact Poissonian operator solution…”

“…The Liouville integrability of this system is our next concern. To study the flow (3.1) on the manifold M (N ) , we shall make use of the Bogoyavlensky-Novikov [4,16] reduction scheme [4,6,8,12]. N ) ) be the standard finitely generated Grassmann algebra [2,6,13] of differential forms on the manifold M (N ) .…”

“…To analyze the integrability properties of the differential-difference dynamical system (1.1), we shall develop a gradient-holonomic scheme related to those devised in [6,7,13,15] for nonlinear dynamical systems defined on spatially one-dimensional functional manifolds and extended in [12] to include discrete manifolds.…”

The Bargmann type reduction for some Lax integrable two-dimensional generalization of the relativistic Toda lattice

“…The relationships (23) describe all periodic and quasiperiodic solutions of the system (8), (9) Proof. The exact symplectic structure on the invariant subspace M 4 N ⊂ M 4 can be found by means of the discrete analog [18], [19], [20] of the Gelfand-Dikii relationship on the functional manifold M 4 in the same manner as has been done in the paper [19] for the subspaces of critical points of local conservation laws.…”