Algebraic and differential geometric aspects of the integrability of nonlinear dynamical systems on infinite-dimensional functional manifolds

Isospectral integrability analysis of dynamical systems on discrete manifolds

“…Therefore, it is possible to apply Lemma 1.4 to any one of the dynamical systems (2.32) if the related vector fields commuting with (1.1) are assumed known. To solve equation (1.15) for an element ϕ ∈ T * (M (N ) ) one can, in the case of a polynomial dynamical system (1.1), make use of the well-known asymptotic small parameter method [6,15]. When applying this approach, it is necessary to take into account the following expansions at zero -elementũ = 0 ∈ M (N ) with respect to the small parameter µ → 0 :…”

“…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.…”

“…Accordingly we need to construct invariants with respect to its functions, called conservation laws, which are mutually commuting with respect to the Poisson bracket (1.4). The following Lax's criterion [3,6,13] proves to be very useful.…”

“…One says that two Poissonian structures ϑ, η : [6,8,10,18], if for any λ, µ ∈ R the linear combination λϑ + µη :…”

“…Concerning the integrability of the infinite-dimensional dynamical system (1.1) on the discrete manifold M it is, in general, necessary, but not sufficient [4,6,13], to prove the existence of an infinite hierarchy of mutually commuting conservation laws with respect to the Poissonian structure (1.4).…”