The general differential-geometric structure of multidimensional Delsarte transmutation operators in parametric functional spaces and their applications in soliton theory. Part 2

The structure properties of multidimensional Delsarte transmutation operators in parametirc functional spaces are studied by means of differential-geometric tools. It is shown that kernels of the corresponding integral operator expressions depend on the topological structure of related homological cycles in the coordinate space. As a natural realization of the construction presented we build pairs of Lax type commutive differential operator expressions related via a Darboux-Backlund transformation having a lot of applications in solition theory. Some results are also sketched concerning theory of Delsarte transmutation operators for affine polynomial pencils of multidimensional differential operators.Comment: 10 page

The Electromagnetic Lorentz Problem and the Hamiltonian Structure Analysis of the Maxwell-Yang-Mills Type Dynamical Systems within the Reduction Method

Based on analysis of reduced geometric structures on fi bered manifolds, invariant under action of an abelian functional symmetry group, we construct the symplectic structures associated with connection forms on the related principal fi ber bundles with abelian functional structure groups. The Marsden-Weinstein reduction procedure is applied to the Maxwell and Yang-Mills type dynamical systems. The geometric properties of Lorentz type constraints, describing the electromagnetic fi eld properties in vacuum and related with the well known Dirac-Fock-Podolsky problem, are discussed

On the Lagrangian and Hamiltonian aspects of infinite -dimensional dynamical systems and their finite-dimensional reductions

A description of Lagrangian and Hamiltonian formalisms naturally arisen from the invariance structure of given nonlinear dynamical systems on the infinite--dimensional functional manifold is presented. The basic ideas used to formulate the canonical symplectic structure are borrowed from the Cartan's theory of differential systems on associated jet--manifolds. The symmetry structure reduced on the invariant submanifolds of critical points of some nonlocal Euler--Lagrange functional is described thoroughly for both differential and differential discrete dynamical systems. The Hamiltonian representation for a hierarchy of Lax type equations on a dual space to the Lie algebra of integral-differential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Backlund transformation. The connection of this hierarchy with integrable by Lax spatially two-dimensional systems is studied.Comment: 30 page