Wilhelm FUSHCHYCH & Renat ZHDANOV

Symmetries and Exact Solutions of Nonlinear Dirac Equations

Mathematical Ukraina Publisher, Kyiv, 1997, ISSN 966-02-0144-3

Abstract

Today there exists a growing interest to the nonlinear equations of mathematical and theoretical physics. But till now there are no books, where nonlinear Dirac-type equations are treated in a unified and consistent way. The present book is aimed to fill this gap and to give a comprehensive group-theoretical analysis of systems of nonlinear partial differential equations (PDEs) for spinor field invariant under the Poincare and Galilei groups, with a particular emphasis on developing efficient methods for constructing their exact solutions.

The authors give a detailed information about symmetry (Lie, non-Lie, conditional) of nonlinear PDEs for spinor, vector and scalar fields; using advanced methods of group-theoretical, symmetry analysis construct wide families of classical solutions of the nonlinear Dirac, Yang-Mills, Maxwell-Dirac, Dirac-d'Alembert, d'Alembert-Hamilton equations; expound a new symmetry approach to variable separation in linear and nonlinear PDEs, which allows, in particular, to classify separable Schroedinger equations.

The book offers a uniform and relatively simple presentation of a considerable amount of material that is otherwise not easily available. The basic part of the book contains original results obtained by the authors. It is sure to be of interest to mathematical and theoretical physicists, particularly those working on classical and quantum field theories and on nonlinear dynamical systems.

Table of Contents

• Preface
• Introduction
• Symmetry of Nonlinear Spinor Equations
• Local and nonlocal symmetry of the Dirac equation
• Nonlinear spinor equations
• Systems of nonlinear second-order equations for the spinor field
• Symmetry of systems of nonlinear equations for spinor, vector and scalar fields
• Conditional symmetry and reduction of partial differential equations
• Conservation laws
• Exact Solutions
• On compatibility and general solution of the d'Alembert-Hamilton system
• Ansatzes for the spinor field
• Reduction of Poincare-invariant spinor equations
• Exact solutions of nonlinear spinor equations
• Nonlinear spinor equations and special functions
• Construction of fields with spins s = 0, 1, 3/2 via the Dirac field
• Exact solutions of the Dirac-d'Alembert equation
• Exact solutions of the nonlinear electrodynamics equations
• Two-Dimensional Spinor Models
• Two-dimensional spinor equations invariant under the infinite-parameter groups
• Nonlinear two-dimensional Dirac-Heisenberg equations
• Two-dimensional classical electrodynamics equations
• General solutions of Galilei-invariant spinor equations
• Nonlinear Galilei-Invariant equations
• Nonlinear equations for the spinor field invariant under the group G(1,3) and its extensions
• Exact solutions of Galilei-invariant spinor equations
• Galilei-invariant second-order spinor equations
• Separation of Variables
• Separation of variables and symmetry of systems of partial differential equations
• Separation of variables in the Galilei-invariant spinor equation
• Separation of variables in the Schroedinger equation
• Conditional Symmetry and Reduction of Spinor Equations
• Non-Lie reduction of Poincare-invariant spinor equations
• Non-Lie reduction of Galilei-invariant spinor equations
• Reduction and Exact Solutions of SU(2) Yang-Mills Equations
• Symmetry reduction and exact solutions of the Yang-Mills equations
• Non-Lie reduction of the Yang-Mills equations
• Appendix 1: The Poincare Group and Its Representations
• Appendix 2: The Galilei Group and Its Representations
• Appendix 3: Representations of the Poincare and Galilei Algebras by Lie Vector Fields
• Index
• Bibliography