Approximate Lie symmetries and singular perturbation theory

Perturbation theory plays a central role in the approximate solution of nonlinear differential equations. However, its naïve application often yields divergent series solutions. While these can be made convergent using singular perturbation methods of various types, the procedures used can be subtle owing to the lack of globally applicable algorithms. Inspired by the fact that all exact solutions of differential equations are consequences of their (Lie) symmetries, we reformulate perturbation theory for differential equations as a series expansion of their solutions’ symmetries. This is a change in perspective from the usual method of obtaining series expansions of the solutions themselves. We show that these approximate symmetries are straightforward to calculate and are never singular; their integration is therefore an easier way of constructing uniformly convergent solutions. This geometric viewpoint naturally subsumes the renormalization group-inspired approach of Chen, Goldenfeld and Oono, the method of multiple scales and the Poincare–Lindstedt method, by exploiting a fundamental class of symmetries that we term ‘hidden scale symmetries’. It also clarifies when and why these singular perturbation methods succeed and just as importantly, when they fail. More broadly, direct, algorithmic identification and integration of these hidden scale symmetries permits solution of problems where other methods are impractical.