Nonlinear Dynamical Systems of Mathematical Physics

The following sections are included:

• Preface

• Contents

We begin in this chapter with a brief treatment of some elements of dynamical systems theory, mainly in finite dimensions, that are indispensable for modern mathematical physics.

The following sections are included:

• The Poisson structures and Lie group actions on manifolds: Introduction

• Lie group actions on Poisson manifolds and the orbit structure

• The canonical reduction method on symplectic spaces and related geometric structures on principal fiber bundles

• The form of reduced symplectic structures on cotangent spaces to Lie group manifolds and associated canonical connections

• The geometric structure of abelian Yang–Mills type gauge field equations via the reduction method

• The geometric structure of non-abelian Yang–Mills gauge field equations via the reduction method

• Classical and quantum integrability

The following sections are included:

• Introduction

• Preliminaries

• Integral submanifold embedding problem for an abelian Lie algebra of invariants

• Integral submanifold embedding problem for a nonabelian Lie algebra of invariants

• Examples

• Existence problem for a global set of invariants

• Additional examples

The following sections are included:

• Preliminary remarks

• Implectic operators and dynamical systems

• Symmetry properties and recursion operators

• Bäcklund transformations

• Properties of solutions of some infinite sequences of dynamical systems

• Integro-differential systems

The following sections are included:

• The Lax representation

• Recursive operators and conserved quantities

• Existence criteria for a Lax representation

• The current Lie algebra on a cycle: A symmetry subalgebra of compatible bi-Hamiltonian nonlinear dynamical systems

The following sections are included:

• A non-isospectrally Lax integrable KdV dynamical system

• Algebraic structure of the gradient-holonomic algorithm for Lax integrable systems

• Analysis of a Whitham type nonlocal dynamical system for a relaxing medium with spatial memory

• A regularization scheme for a generalized Riemann hydrodynamic equation and integrability analysis

• The generalized Riemann hydrodynamic regularization

• Differential-algebraic integrability analysis of the generalized Riemann and KdV hydrodynamic equations

• Symplectic analysis of the Maxwell equations

• Symplectic analysis of vortex helicity in magneto-hydrodynamics

• Algebraic-analytic structure of integrability by quadratures of Abel–Riccati equations

The following sections are included:

• Introduction: -actions

• Lie-algebraic structure of the -action

• Casimir functionals and reduction problem

• Associated momentum map and versal deformations of the action

The following sections are included:

• Short introduction

• Lie-algebra of Lax integrable (2+1)-dimensional dynamical systems

The following sections are included:

• Problem setting

• Factorization properties

• Hamiltonian analysis

• Tensor products of Poisson structures and source like factorized operator dynamical systems

• Remarks

The following sections are included:

• Spectral operators and generalized eigenfunctions expansions

• Semilinear forms, generalized kernels and congruence of operators

• Congruent kernel operators and related Delsarte transmutation maps

• Differential-geometric structure of the Lagrangian identity and related Delsarte transmutation operators

• The general differential-geometric and topological structure of Delsarte transmutation operators: A generalized de Rham–Hodge theory

• A special case: Relations with Lax systems

• Geometric and spectral theory aspects of Delsarte–Darboux binary transformations

• The spectral structure of Delsarte–Darboux transmutation operators in multi-dimensions

• Delsarte–Darboux transmutation operators for special multi-dimensional expressions and their applications

The following sections are included:

• Differential-geometric problem setting

• The differential invariants

The following sections are included:

• Introduction

• Mathematical preliminaries: Fock space and realizations

• The Fock space embedding method, nonlinear dynamical systems and their complete linearization

• Summary remarks and preview

• A geometric approach to quantum holonomic computing algorithms

The following sections are included:

• Introductory setting

• Classical relativistic electrodynamics revisited

• Vacuum field theory electrodynamics: Lagrangian analysis

• Vacuum field electrodynamics: Hamiltonian analysis

• Quantization of electrodynamics models in vacuum field theory: No-geometry approach

• Some relevant observations

• Introduction to further analysis

• The charged point particle least action principle revisited: Vacuum field theory approach

• A new hadronic string model: Least action principle and relativistic electrodynamics analysis in the vacuum field theory approach

Our intention here is to provide a fairly complete treatment of the basics of differential geometry that are employed throughout the book, so as to render the overall exposition essentially complete.

The following sections are included:

• Bibliography

• Index