# Symmetry and Integration Methods for Differential Equations

## By Peter Hydon

Applied Mathematical Sciences volume. 154,
Springer-Verlag, New York, 2002,
Hardback, x + 419 pages,
ISBN 0-387-98654-5.

Roughly speaking, a symmetry of an object is a mapping that leaves the object apparently unchanged. In other words, the object is mapped to itself. A symmetry of a differential equation maps the set of all solutions to itself. Any particular solution is mapped either to another solution, or to itself (in which case the solution is said to be invariant). Symmetries are ubiquitous in mathematical models; however, many applied mathematicians are unaware that they can be used systematically to simplify the model or to obtain exact solutions. Indeed, almost all of the well-known techniques for obtaining exact solutions of differential equations are based on symmetry methods.

Towards the end of the 19th century, the Norwegian mathematician Sophus Lie laid the foundations for symmetry methods for differential equations. In particular, he developed a powerful technique for determining the symmetries of a given differential equation by linearising the symmetry condition. Once the symmetries have been found, there are various ways of using them to obtain exact solutions.

The book by Bluman and Anco is an introduction to symmetry-based methods, and is a revision of the first four chapters of an earlier book, "Symmetries and Differential Equations" by Bluman and Kumei, which I will refer to as BK. (The authors plan to include the remaining material from BK in a second book.) It is intended for newcomers who have a modest background in mathematics and are familiar with undergraduate-level methods for solving differential equations. The authors use a `Theorem-Proof' format, although many theorems are left for the reader to prove. No hints are given, and I think that newcomers are likely to find it very difficult to prove some results.

The book is written in a clear and straightforward style, using many excellent worked examples. The authors have tried to present symmetry methods algorithmically, and practice is essential for the reader to master the techniques. There are many exercises (of varying difficulty), although no solutions are given. One weakness is that the book contains little reference to the computer algebra packages that are commonly used to carry out symmetry calculations. If the reader is to obtain symmetries of complicated systems of differential equations, then at least one such package should be used.

Chapter 1 describes dimensional analysis and the related topic of scaling-invariant (or similarity) solutions of partial differential equations (PDEs). The reader is introduced to scaling symmetries, and can see how they may be used to obtain families of exact solutions of a PDE.

Chapter 2 presents much of the theory that lies behind the rest of the book. Lie groups and Lie algebras are described from a fairly `applied' viewpoint. The emphasis is mainly on the action of symmetry groups on points and curves. Some of the formulae in this section look alarmingly complicated, but the authors have tried to show how they are a natural generalisation of the simpler formulae that are applicable to scalar ordinary differential equations (ODEs). I think that newcomers may struggle here.

Chapter 3 deals with ODEs. It runs to 195 pages, and is the heart of the book. It is also the only chapter from BK that has been extended significantly. All of the material from BK is included; most of this describes how to use symmetries to reduce the order of a given ODE and to obtain exact solutions. The (extensive) new material focuses on the dual problem of finding first integrals. One method of doing this is to obtain integrating factors, which can be constructed by solving the adjoint of the linearised symmetry condition. Thus symmetries and integrating factors can be found by the same techniques. The authors have presented a substantial body of recent results that extend the range of systematic methods for solving ODEs.

Chapter 4 introduces symmetry methods for PDEs. It includes the basic techniques for finding and using symmetries, but it does not deal with conservation laws. These are left to the forthcoming book.

There are a number of minor defects; most are editorial. The index has many inaccuracies, and should be revised in any subsequent printing. The line spacing is extremely uneven, as the authors include much mathematics within the body of the text. BK did not have such problems; has Springer-Verlag regressed?

As someone who teaches an undergraduate course on symmetry methods, I would recommend Bluman and Anco's book as a good source of examples and supplementary reading. However, I think that students who relied on it alone would struggle. There are several other good introductory texts available. However, for anyone wishing to master techniques for obtaining first integrals of ODEs, this book is outstanding. I look forward to the publication of the authors' second volume.

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