Group Invariant Solutions Without Transversality

TL;DR

This paper extends Lie's method for finding group invariant solutions of PDEs by relaxing the transversality condition, allowing for broader applications and providing a theoretical foundation for analyzing reduced equations.

Contribution

It introduces a generalized approach to Lie symmetry reduction that handles non-transversal cases, supported by an existence theorem and illustrative examples.

Findings

Generalized Lie reduction method without transversality

Intrinsic characterization of reduced equations

Applications demonstrated in fluid mechanics, harmonic maps, and relativity

Abstract

We present a generalization of Lie's method for finding the group invariant solutions to a system of partial differential equations. Our generalization relaxes the standard transversality assumption and encompasses the common situation where the reduced differential equations for the group invariant solutions involve both fewer dependent and independent variables. The theoretical basis for our method is provided by a general existence theorem for the invariant sections, both local and global, of a bundle on which a finite dimensional Lie group acts. A simple and natural extension of our characterization of invariant sections leads to an intrinsic characterization of the reduced equations for the group invariant solutions for a system of differential equations. The characterization of both the invariant sections and the reduced equations are summarized schematically by the kinematic and dynamic reduction diagrams and are illustrated by a number of examples from fluid mechanics, harmonic maps, and general relativity. This work also provides the theoretical foundations for a further detailed study of the reduced equations for group invariant solutions.