An overview of hidden symmetries.
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Date
2012
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Abstract
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.
Description
Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2012.
Keywords
Symmetry., Symmetry groups., Theses--Mathematics.