Symmetries and Exact Solutions of Nonlinear Dirac Equations

TL;DR

This paper explores symmetry methods for nonlinear PDEs related to Dirac and other field equations, providing new solutions, symmetry classifications, and a novel approach to variable separation, valuable for theoretical physics.

Contribution

It introduces a new symmetry approach to variable separation and offers a comprehensive analysis of symmetries and solutions for nonlinear Dirac and related equations.

Findings

Constructed families of classical solutions for nonlinear Dirac and related equations.

Classified separable Schrödinger equations using symmetry analysis.

Developed a new symmetry method for variable separation in PDEs.

Abstract

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.