Approximate Lie symmetries and singular perturbation theory

Abstract

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.

arXiv comment55 pages, 7 figures; references added, results sections re-ordered, SI created, material added on the Oseen equation, description of RG/E revised. Various formatting changes and minor phrasing changes. Findings & conclusions unchanged

References in corpus (10)

• The Renormalization Group and Singular Perturbations: Multiple-Scales, Boundary Layers and Reductive Perturbation Theory
• Renormalization Group Theory for Global Asymptotic Analysis
• A Geometrical Formulation of the Renormalization Group Method for Global Analysis
• Simple Viscous Flows: from Boundary Layers to the Renormalization Group
• The Renormalization-Group Method Applied to Asymptotic Analysis of Vector Fields
• Bogoliubov Renormalization Group and Symmetry of Solution in Mathematical Physics
• Group analysis and renormgroup symmetries
• Lie-Group Approach to Perturbative Renormalization-Group Method
• A Method to Construct Asymptotic Solutions Invariant under the Renormalization Group
• Surprises in a classic boundary-layer problem

Cited by in corpus (2)

• Kinetics of seeded protein aggregation: theory and application
• General integrated rate law for complex self-assembly reactions reveals the mechanism of amyloid-beta co-aggregation