Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra (2nd ed.)

Description

In the introduction, the author writes, "Continuous group theory, Lie algebras and differential geometry play an important role in the understanding of the structure of nonlinear partial differential equations, in particular for generating integrable equations, finding Lax pairs, recursion operators, Bäcklund transformations and finding exact analytical solutions.'' This book provides a self-contained and very accessible introduction to these topics.
The book begins with basic material on Lie groups, Lie algebras, group actions and their infinitesimal generators, differential forms and tensors. Examples arising from symmetry groups of some elementary differential equations and from the study of invariant and conformally invariant tensors are given. The author then does a nice job of describing the basic techniques for finding the Lie algebra of infinitesimal symmetries for a system of differential equations, and conversely, for finding those differential equations with a prescribed group of symmetries. A list of some standard differential equations and their symmetries is given. The book concludes with an introduction to a variety of topics of current interest in the theory of integrable systems: Lie-Bäcklund or generalized symmetries; recursion operators; Lax representations; conservation laws; and Painlevé tests. None of these topics are dealt with in great detail or with a great amount of rigor, but nevertheless this text, by consistently presenting well-worked-out examples of all the main definitions, provides a good starting point for anyone seeking a quick introduction to these subjects. A novel feature of the text is the inclusion of computer algebra routines, written in REDUCE, at the end of each chapter.

Reviewed by Ian M. Anderson