This textbook comprehensively introduces students and researchers to the application of continuous symmetries and their Lie algebras to ordinary and partial differential equations. Covering all the modern techniques in detail, it relates applications to cutting-edge research fields such as Yang–Mills theory and string theory.

Aimed at readers in applied mathematics and physics rather than pure mathematics, the material is ideally suited to students and researchers whose main interest lies in finding solutions to differential equations and invariants of maps.

A large number of worked examples and challenging exercises help readers to work independently of teachers, and by including SymbolicC++ implementations of the techniques in each chapter, the book takes full advantage of the advancements in algebraic computation.

Twelve new sections have been added in this edition, including: Haar measure, Sato's theory and sigma functions, universal algebra, anti-self dual Yang–Mills equation, and discrete Painlevé equations.

Contents:

• Groups
• Lie Groups
• Lie Transformation Groups
• Infinitesimal Transformations
• Lie Algebras
• Introductory Examples
• Differential Forms and Tensor Fields
• Lie Derivative and Invariance
• Invariance of Differential Equations
• Lie–Bäcklund Vector Fields
• Differential Equation for a Given Lie Algebra
• A List of Lie Symmetry Vector Fields
• Recursion Operators
• Bäcklund Transformations
• Lax Representations
• Conservation Laws
• Symmetries and Painlevé Test
• Ziglin’s Theorem and Integrability
• Lie Algebra Valued Differential Forms
• Bose Operators and Lie Algebras
• Maps and Invariants
• Computer Algebra
• Differential Manifolds