Symmetry and Integrability of Classical Field Equations

Abstract

A number of characteristics of integrable nonlinear partial differential equations (PDE's) for classical fields are reviewed, such as Backlund transformations, Lax pairs, and infinite sequences of conservation laws. An algebraic approach to the symmetry problem of PDE's is described, suitable for PDE's the solutions of which are non-scalar fields (e.g., are in the form of matrices). A method is proposed which, in certain cases, may "unify" the symmetry and integrability properties of a nonlinear PDE. Application is made to the self-dual Yang-Mills equation and the Ernst equation of General Relativity.

31 pages, minor corrections on pp.26,28

References in corpus (2)

• The Differential Form Method for Finding Symmetries
• Nonlocal Symmetries and Generation of Solutions for Partial Differential Equations

Cited by in corpus (5)

• Symmetry, Conserved Charges, and Lax Representations of Nonlinear Field Equations: A Unified Approach
• Symmetry and Integrability of a Reduced, 3-Dimensional Self-Dual Gauge Field Model
• A Method for Constructing a Lax Pair for the Ernst Equation
• Symmetry operators and generation of symmetry transformations of partial differential equations
• The hidden symmetry and Mr. Higgs!