11 - Discrete Symmetries

Summary

There is a place apart

Beyond the solar ray,

Where parallel straight lines can meet

in an unofficial way.

(G. K. Chesterton: The Higher Mathematics)

Some Uses of Discrete Symmetries

In Chapter 1, the discrete symmetries of a triangle are used to introduce the reader to Lie symmetries. It seems appropriate that this closing chapter introduces a method that uses Lie symmetries to reveal the discrete symmetries of a given differential equation. Here are some reasons why it is important to determine the discrete symmetries.

1. (i) Discrete point symmetries are used to increase the efficiency of computational methods. If a boundary-value problem (BVP) is symmetric and the solution is known to be unique, computation can be carried out on a reduced domain. Alternatively, a spectral method can be used, with basis functions that are invariant under the symmetry. The presence of a discrete symmetry also improves the accuracy of some numerical methods.

2. (ii) Many nonlinear B VPs have multiple solutions, and it is necessary to identify when and how the system changes its behaviour as any parameters vary. Discrete symmetries must be taken into account, because the behaviour of “generic” nonsymmetric systems is usually quite different from that of systems with symmetries. It is important to identify all of the symmetries in a problem in order to understand its behaviour correctly.

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