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## An overview of hidden symmetries.

Approaches to nding solutions to di erential equations are usually ad hoc.

One of the more successful methods is that of group theory, due to Sophus

Lie. In the case of ordinary di erential equations, the subsequent symmetries

obtained allow one to reduce the order of the equation. In the case

of partial di erential equations, the symmetries are used to nd (particular)

group invariant solutions by reducing the number of variables in the original

equation. In the latter case, these solutions are particularly popular in applications

as they are often the only physically signi cant ones obtainable.

As a result, it is now becoming traditional to apply this symmetry method

to nd solutions to di erential equations in a systematic manner.

Based upon the Lie algebra of symmetries of the equation, we expect a certain

number of symmetries after the reductions. However, it has become increasingly

observed that, after reduction, more symmetries than expected are

often obtained. These are called Hidden Symmetries and they provide new

routes for further reduction. The idea of our research is to give an overview

of this phenomenon. In particular, we investigate the possible origins of these

symmetries. We show that they manifest themselves as nonlocal symmetries

(or potential symmetries), contact symmetries or nonlocal contact symmetries

of the original equation as well as point symmetries of another equation

of same order. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2012.

Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:ukzn/oai:http://researchspace.ukzn.ac.za:10413/10605 |

Date | January 2012 |

Creators | Bujela, Ntobeko Isaac. |

Contributors | Govinder, Kesh S. |

Source Sets | South African National ETD Portal |

Language | en_ZA |

Detected Language | English |

Type | Thesis |

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