Symmetries and Exact Solutions of Nonlinear Dirac Equations

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


Different nonlinear generalizations of the classical Dirac equation were suggested many years ago as the foundation of the Unified Quantum Field Theory.

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.

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