Yarema A. Prykarpatsky

The electromagnetic Lorentz condition problem and symplectic properties of Maxwell-and Yang-Mills-type dynamical systems

Symplectic structures associated to connection forms on certain types of principal fiber bundles are constructed via analysis of reduced geometric structures on fibered manifolds invariant under naturally related symmetry groups. This approach is then applied to nonstandard Hamiltonian analysis of of dynamical systems of Maxwell and Yang-Mills type. A symplectic reduction theory of the classical Maxwell equations is formulated so as to naturally include the Lorentz condition (ensuring the existence of electromagnetic waves), thereby solving the well known Dirac -Fock - Podolsky problem. Symplectically reduced Poissonian structures and the related classical minimal interaction principle for the Yang-Mills equations are also considered. 1.

The Integrability of Lie-invariant Geometric Objects Generated by Ideals in the Grassmann Algebra

We investigate closed ideals in the Grassmann algebra serving as bases of Lie-invariant geometric objects studied before by E.Cartan. Especially, the E.Cartan theory is enlarged for Lax integrable nonlinear dynamical systems to be treated in the frame workof the Wahlquist Estabrookprolongation structures on jet-manifolds and CartanEhresmann connection theory on fibered spaces. General structure of integrable oneforms augmenting the two-forms associated with a closed ideal in the Grassmann algebra is studied in great detail. An effective Maurer-Cartan one-forms construction is suggested that is very useful for applications. As an example of application the developed Lie-invariant geometric object theory for the Burgers nonlinear dynamical system is considered having given rise to finding an explicit form of the associated Lax type representation.

Finite dimensional local and nonlocal reductions of one type of hydrodynamic systems

Abstract In this article we describe finite dimensional local and nonlocal reductions and integrability of one type of hydrodynamic systems.

The de Rham-Hodge-Skrypnik theory of Delsarte transmutation operators in multidimension and its applications*

We study differential-geometric and topological structures related with Delsarte transmutations of multi-dimensional differential operators in Hilbert spaces. Based on the naturally defined de Rham-Hodge-Skrypnik differential complex the relationships with spectral theory and special Berezansky type congruence properties of Delsarte transmuted operators are stated. Some applications to multi-dimensional differential operators are done including three-dimensional Laplace operator, two-dimensional classical Dirac operator and its multidimensional affine extension, related with self-dual Yang-Mills equations. The soliton-like solutions to the related set of nonlinear dynamical systems are discussed.

Imbeddings of integral submanifolds and associated adiabatic invariants of slowly perturbed integrable Hamiltonian systems

Abstract A new method is developed for characterizing the evolution of invariant tori of slowly varying perturbations of completely integrable (in the sense of Liouville-Arnold [1–3]) Hamiltonian systems on cotangent phase spaces. The method is based on Cartans theory of integral submanifolds, and it provides an algebro-analytic approach to the investigation of the embedding [4–10] of the invariant tori in phase space that can be used to describe the structure of quasi-periodic solutions on the tori. In addition, it leads to an adiabatic perturbation theory [3,12,13] of the corresponding Lagrangian asymptotic submanifolds via the Poincare-Cartan approach [4], a new Poincare-Melnikov type [5,11,14] procedure for determining stability, and fresh insights into the existence problem for adiabatic invariants [2,3] of the Hamiltonian systems under consideration.

Differential-algebraic and bi-Hamiltonian integrability analysis of the Riemann hierarchy revisited

A differential-algebraic approach to studying the Lax integrability of the generalized Riemann type hydrodynamic hierarchy is revisited and its new Lax representation is constructed in exact form. The bi-Hamiltonian integrability of the generalized Riemann type hierarchy is discussed by means of the gradient-holonomic and symplectic methods and the related compatible Poissonian structures for N = 3 and N = 4 are constructed.

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 Schrodinger, 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.

A geometrical approach to quantum holonomic computing algorithms

The article continues a presentation of modern quantum mathematics backgrounds started in [Quantum Mathematics and its Applications. Part 1. Automatyka, vol. 6, AGH Publisher, Krakow, 2002, No. 1, pp. 234-2412; Quantum Mathematics: Holonomic Computing Algorithms and Their Applications. Part 2. Automatyka, vol. 7, No. 1, 2004]. A general approach to quantum holonomic computing based on geometric Lie-algebraic structures on Grassmann manifolds and related with them Lax type flows is proposed. Making use of the differential geometric techniques like momentum mapping reduction, central extension and connection theory on Stiefel bundles it is shown that the associated holonomy groups properly realizing quantum computations can be effectively found concerning diverse practical problems. Two examples demonstrating two-form curvature calculations important for describing the corresponding holonomy Lie algebra are presented in detail.

THE MARSDEN–WEINSTEIN REDUCTION STRUCTURE OF INTEGRABLE DYNAMICAL SYSTEMS AND A GENERALIZED EXACTLY SOLVABLE QUANTUM SUPERRADIANCE MODEL

An approach to describing nonlinear Lax type integrable dynamical systems of modern mathematical and theoretical physics, based on the Marsden–Weinstein reduction method on canonically symplectic manifolds with group symmetry, is proposed. Its natural relationship with the well-known Adler–Kostant–Souriau–Berezin–Kirillov method and the associated R-matrix approach is analyzed. A new generalized exactly solvable spatially one-dimensional quantum superradiance model, describing a charged fermionic medium interacting with external electromagnetic field, is suggested. The Lax type operator spectral problem is presented, the related R-structure is calculated. The Hamilton operator renormalization procedure subject to a physically stable vacuum is described, the quantum excitations and quantum solitons, related with the thermodynamical equilibrity of the model, are discussed.

The Bogolubov representation of the polaron model and its completely integrable RPA-approximation

The polaron model in ionic crystal is studied in the Bogolubov representation using a special RPA-approximation. A new exactly solvable approximated polaron model is derived and described in detail. Its free energy at nite temperature is calculated analytically. The polaron free energy in the constant magnetic eld at nite temperature is also discussed. Based on the structure of the Bogolubov unitary transformed polaron Hamiltonian there is stated a very important new result: the full polaron model is exactly solvable.