An overview of hidden symmetries.

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.