Title: Group Invariant Solutions of Differential Equations

Abstract: Due to the nature of differential equations, no single method exists to find solutions of all equations (when those solutions exist). However, the symmetry method of Sophus Lie has proven to be enormously successful in finding many solutions and/or classifying equations in terms of their invariance properties. Lie observed that many equations that were solvable via ad hoc techniques were invariant under a one--parameter family (group) of transformations. 

Having found the symmetries of an equation one can utilise them in a number of ways. The most common use is to reduce the order of an ODE by one. Another use is in the classification of equations. In particular, one can determine equivalence classes of equations via Lie algebraic representations. One can also calculate first integrals/constants of the motion (though this can be rather technical).

In the case of PDEs, a standard approach is to  determine group invariant solutions by calculating reduction variables via symmetries. The idea of group invariant solutions can also be applied to ODEs though this approach is not often exploited. In this case, they can lead to singular solutions. We will show how group invariant solutions arise in a number of equations in general relativity. We will also illustrate a link between group invariant solutions and the singularity analysis of Paul Painleve.