All Physics Faculty Publications
Document Type
Article
Journal/Book Title/Conference
Communications in Mathematical Physics
Volume
212
Issue
3
Publisher
Springer Verlag
Publication Date
2000
First Page
653
Last Page
686
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.
Recommended Citation
I.M. Anderson, M.E. Fels, and C.G. Torre, Group Invariant Solutions Without Transversality, Communications in Mathematical Physics, vol. 212(3), 2000, pp. 653-686.
http://arxiv.org/abs/math-ph/9910015
Comments
Originally published by Springer Verlag. Publisher's PDF can be accessed through Communications in Mathematical Physics. Article also available online through arXiv.org.