Kavli Affiliate: L. Mahadevan
| First 5 Authors: Alexander J. Dear, L. Mahadevan, , ,
| Summary:
Singular perturbation theory plays a central role in the approximate solution
of nonlinear differential equations. However, applying these methods is a
subtle art owing to the lack of globally applicable algorithms. Inspired by the
fact that all exact solutions of differential equations are consequences of
(Lie) symmetries, we reformulate perturbation theory for differential equations
in terms of expansions of the Lie symmetries of the solutions. 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 RG-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.
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