Anatoliy K. Prykarpatsky

Differential-algebraic integrability analysis of the generalized Riemann type and Korteweg–de Vries hydrodynamical equations

A differential-algebraic approach to studying the Lax-type integrability of the generalized Riemann-type hydrodynamic equations at N = 3, 4 is devised. The approach is also applied to studying the Lax-type integrability of the well-known Korteweg?de Vries dynamical system.

Nonlinear dynamical systems of mathematical physics : spectral and symplectic integrability analysis

This distinctive volume presents a clear, rigorous grounding in modern nonlinear integrable dynamics theory and applications in mathematical physics, and an introduction to timely leading-edge developments in the field - including some innovations by the authors themselves - that have not appeared in any other book. The exposition begins with an introduction modern integrable dynamical systems theory, treating such topics as Liouville-Arnold and Mischenko-Fomenko integrability. This sets the stage for such topics as new formulations of the gradient-holonomic algorithm for Lax integrability, novel treatments of classical integration by quadratures, Lie-algebraic characterizations of integrability, and recent results on tensor Poisson structures. Of particular note is the development via spectral reduction of a generalized de Rham-Hodge theory, related to Delsarte-Lions operators, leading to new Chern type classes useful for integrability analysis. Also included are elements of quantum mathematics along with applications to Whitham systems, gauge theories, hadronic string models models, and a supplement on fundamental differential-geometric concepts making this volume essentially self-contained. This book is ideal as a reference and guide to new directions in research for advanced students and researchers interested in the modern theory and applications of integrable (especially infinite-dimensional) dynamical systems.

On a Nonlocal Ostrovsky-Whitham Type Dynamical System, Its Riemann Type Inhomogeneous Regularizations and Their Integrability ?

Short-wave perturbations in a relaxing medium, governed by a special reduction of the Ostrovsky evolution equation, and later derived by Whitham, are studied using the gradient-holonomic integrability algorithm. The bi-Hamiltonicity and complete integrability of the corresponding dynamical system is stated and an infinite hierarchy of commuting to each other conservation laws of dispersive type are found. The well defined regularization of the model is constructed and its Lax type integrability is discussed. A generalized hyd- rodynamical Riemann type system is considered, infinite hierarchies of conservation laws, related compatible Poisson structures and a Lax type representation for the special case N = 3 are constructed.

The vacuum structure, special relativity theory, and quantum mechanics: A return to the field theory approach without geometry

We formulate the main fundamental principles characterizing the vacuum field structure and also analyze the model of the related vacuum medium and charged point particle dynamics using the developed field theory methods. We consider a new approach to Maxwell’s theory of electrodynamics, newly deriving the basic equations of that theory from the suggested vacuum field structure principles; we obtain the classical special relativity theory relation between the energy and the corresponding point particle mass. We reconsider and analyze the expression for the Lorentz force in arbitrary noninertial reference frames. We also present some new interpretations of the relations between special relativity theory and quantum mechanics. We obtain the famous quantum mechanical Schrödinger-type equations for a relativistic point particle in external potential and magnetic fields in the semiclassical approximation as the Planck constant ħ → 0 and the speed of light c→ ∞.

The Relativistic Electrodynamics Least Action Principles Revisited: New Charged Point Particle and Hadronic String Models Analysis

The classical relativistic least action principle is revisited from the vacuum field theory approach. New physically motivated versions of relativistic Lorentz type forces are derived, a new relativistic hadronic string model is proposed and analyzed in detail. The reasonings of R. Feynman, who argued that the relativistic dynamical expressions obtain true physical sense only with respect to the proper rest reference frames, are supported by analyzing the dynamical stability of a relativistic charged string model.

The gradient-holonomic integrability analysis of a Whitham-type nonlinear dynamical model for a relaxing medium with spatial memory

A new Whitham-type nonlinear evolution equation describing short-wave perturbations in a relaxing medium is studied. Making use of the gradient-holonomic analysis, the bi-Hamiltonicity and complete integrability of the corresponding dynamical system is stated. An infinite hierarchy of dispersive conservation laws which commute with each other is constructed. The two- and four-dimensional invariant reductions are studied in detail.

The non-polynomial conservation laws and integrability analysis of generalized Riemann type hydrodynamical equations

Based on the gradient-holonomic algorithm we analyse the integrability property of the generalized hydrodynamical Riemann type equation for arbitrary . The infinite hierarchies of polynomial and non-polynomial conservation laws, both dispersive and dispersionless are constructed. Special attention is paid to the cases N = 2, 3 and N = 4, for which the conservation laws, Lax type representations and Hamiltonian structures are analysed in detail. We also show that the case N = 2 is equivalent to a generalized Hunter–Saxton dynamical system, whose integrability follows from the results obtained. As a by-product of our analysis we demonstrate a new set of non-polynomial conservation laws for the related Hunter–Saxton equation.

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.

Replicator Dynamics and Mathematical Description of Multi-Agent Interaction in Complex Systems

Abstract We consider the general properties of the replicator dynamical system from the standpoint of its evolution and stability. Vector field analysis as well as spectral properties of such system has been studied. A Lyaponuv function for the investigation of the evolution of the system has been proposed. The generalization of replicator dynamics to the case of multi-agent systems is introduced. We propose a new mathematical model to describe the multi-agent interaction in complex system.

Quantum field theory with application to Quantum nonlinear optics

Methods of Studying Quantum Optical Phenomena: Quantum States and Statistics Creation and Annihilation Operators The One-Particle Greens Function The Dyson Equations Polarization Operators The Hartree-Fock Approximation Nonlinear Quantum Optics Models and Their Applications: Nonlinear Quantum Optics Bose-Systems The Dicke Model A Degenerate Parametric Quantum Optical Amplifier A Nonlinear Quantum-Optical System at Equilibrium Radiative State and other topics.