Approximate Lie symmetries and singular perturbation theory

Kavli Affiliate: L. Mahadevan

| First 5 Authors: Alexander J. Dear, L. Mahadevan, , ,

| Summary:

Perturbation theory plays a central role in the approximate solution of
nonlinear differential equations. The resultant series expansions are usually
divergent and require treatment by singular perturbation methods to generate
uniformly valid solutions. 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 approximate symmetries, via expansions of the Lie symmetries of the
solutions. This is a change in perspective from the usual method for 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 a powerful way of constructing uniformly convergent
solutions. This geometric viewpoint naturally implies that several key singular
perturbation methods such as the general perturbative RG-inspired approach of
Chen, Goldenfeld and Oono (CGO RG), the method of multiple scales (MMS), and
the Poincare-Lindstedt method (PLM), exploit a fundamental class of approximate
symmetries that we term “hidden scale symmetries”. In turn, this clarifies
when and why these methods succeed and just as importantly, when they fail. Our
algorithmic method directly identifies and integrates these hidden scale
symmetries, making it often simpler to implement, and permitting solution of
problems where other methods are impractical. Finally, we show how other kinds
of approximate symmetry can be exploited to solve systems that do not possess
integrable hidden scale symmetries.

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