Applied Mathematical Sciences, 81. New York, NY etc.: Springer-Verlag. xiii, 412 p. DM 114.00 (1989).
This book is aimed, first of all, at applied mathematicians, physicists and engineers. The authors treat Lie groups of transformations with an emphasis on solving ordinary and partial differential equations. The idea is that if a symmetry can be discovered in a differential equation then, usually, it makes the construction of a solution easier.
Mainly, results achieved in the past fifteen years are summed up, i.e. results that have been published since the publication of G. W. Bluman and J. D. Cole, Similarity methods for differential equations, Springer (1974; Zbl 0292.35001)].
The applied character of the book is emphasised by that that the first chapter treats dimensional analysis and its applications. The “Buckingham Pi Theorem” is treated thoroughly and illustrated by several examples. The authors show that dimensional analysis is a special case of reduction from invariance under groups of scaling transformations. The second chapter gives a rigorous treatment of Lie groups of transformations and infinitesimal transformations.
Multiparameter Lie groups and Lie algebras are also discussed here. The third and the fourth chapters deal with ordinary and with partial differential equations, respectively. Methods are presented for constructing solutions, for reducing the order of differential equations, for finding invariant solutions etc. Noethers theorem (about the existence of a conservation law under some conditions) and Lie-Bäcklund transformations are treated in detail in chapter five. Chapter six deals with mappings that carry solutions of a differential equation into solutions of another one. The construction of such a mapping is also related to Lie-algebra technique. Of special interest are algorithms which may decide whether a given nonlinear differential equation can be mapped into a linear one. In the last chapter “potential symmetries” are dealt with which are, in a sense, non-local, and related to the existence of generalized potential functions.
The text is richly illustrated with examples and exercises. Many of these are related to the wave equation, to the nonlinear heat conduction equation, and to boundary value problems in general. The book is an important contribution to the manipulative theory of differential equations both ordinary and partial, aimed at the construction of solutions.