Associate symmetries: A novel procedure for finding contact symmetries

Abstract

A new method for finding contact symmetries is proposed for both ordinary and partial differential equations. Symmetries more general than Lie point are often difficult to find owing to an increased dependency of the infinitesimal functions on differential quantities. As a consequence, the invariant surface condition is often unable to be “split” into a reasonably sized set of determining equations, if at all. The problem of solving such a system of determining equations is here reduced to the problem of finding its own point symmetries and thus subsequent similarity solutions to these equations. These solutions will (in general) correspond to some subset of symmetries of the original differential equations. For this reason, we have termed such symmetries associate symmetries. We use this novel method of associate symmetries to determine new contact symmetries for a non-linear PDE and a second order ODE which could not previously be found using computer algebra packages; such symmetries for the latter are particularly difficult to find. We also consider a differential equation with known contact symmetries in order to illustrate that the associate symmetry procedure may, in some cases, be able to retrieve all such symmetries.