On the Complete Integrability of Nonlinear Dynamical Systems on Functional Manifolds Within the Gradient-Holonomic Approach

A gradient-holonomic approach for the Lax-type integrability analysis of differential-discrete dynamical systems is described. The asymptotic solutions to the related Lax equation are studied, the related gradient identity subject to its relationship to a suitable Lax-type spectral problem is analyzed in detail. The integrability of the discrete nonlinear Schrödinger, Ragnisco–Tu and Burgers–Riemann type dynamical systems is treated, in particular, their conservation laws, compatible Poissonian structures and discrete Lax-type spectral problems are obtained within the gradient-holonomic approach.