Symmetry Methods for Differential Equations

There are many ingenious techniques for obtaining exact solutions of differential equations, but most work only for a very limited class of problems. How can one solve differential equations of an unfamiliar type?

Surprisingly, most well-known techniques have a common feature: they exploit symmetries of differential equations. It is often quite easy to find symmetries of a given differential equation (even an unfamiliar one) and to use them systematically to obtain exact solutions. Symmetries can also be used to simplify problems and to understand bifurcations of nonlinear systems.

More than a century ago, the Norwegian mathematician Sophus Lie put forward many of the fundamental ideas behind symmetry methods. Most of these ideas are essentially simple, but are so far reaching that they are still the basis of much research. As an applied mathematician, I have found symmetry methods to be invaluable. They are fairly easy to master and provide the user with a powerful range of tools for studying new equations. I believe that no one who works with differential equations can afford to be ignorant of these methods.

This book introduces applied mathematicians, engineers, and physicists to the most useful symmetry methods. It is aimed primarily at postgraduates and those involved in research, but there is sufficient elementary material for a one-semester undergraduate course. (Over the past five years, I have taught these methods to both undergraduates and postgraduates.) Bearing in mind the interests and needs of the intended readership, the book focuses on techniques.