18 research outputs found
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
A vertex operator representation of solutions to the Gurevich-Zybin hydrodynamical equation
An approach based on the spectral and Lie - algebraic techniques for
constructing vertex operator representation for solutions to a Riemann type
Gurevicz-Zybin hydrodynamical hierarchy is devised. A functional representation
generating an infinite hirerachy of dispersive Lax type integrable flows is
obtaned.Comment: 6 page
The differential-algebraic and bi-Hamiltonian integrability analysis of the Riemann type hierarchy revisited
A differential-algebraic approach to studying the Lax type integrability of
the generalized Riemann type hydrodynamic hierarchy is revisited, its new Lax
type representation and Poisson structures constructed in exact form. The
related bi-Hamiltonian integrability and compatible Poissonian structures of
the generalized Riemann type hierarchy are also discussed.Comment: 18 page
On the Complete Integrability of Nonlinear Dynamical Systems on Discrete Manifolds within the Gradient-Holonomic Approach
A gradient-holonomic approach for the Lax type integrability analysis of
differentialdiscrete dynamical systems is devised. The asymptotical solutions
to the related Lax equation are studied, the related gradient identity is
stated. The integrability of a discrete nonlinear Schredinger type dynamical
system is treated in detail.Comment: 20 page
Isospectral integrability analysis of dynamical systems on discrete manifolds
It is shown how functional-analytic gradient-holonomic structures can be used for an isospectral integrability analysis of nonlinear dynamical systems on discrete manifolds. The approach developed is applied to obtain detailed proofs of the integrability of the discrete nonlinear Schrödinger, Ragnisco-Tu and Riemann-Burgers dynamical systems