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.