1999/10/31 by I. Anderson, Ian M. Anderson, Mark E. Fels +3 · 1 citation
Mathematics · Physics and Astronomy · #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons #Numerical methods for differential equations #gr-qc #hep-th #math-ph #math.MP
paper · pdf · doi:10.1007/s002200000215
published as Commun.Math.Phys. 212 (2000) 653-686 · 41 pages; AMSTeX; typos have been fixed; title has been revised (formerly: "Symmetry reduction without transversality"); to appear in Communications in Mathematical Physics
arxiv created 2000/04/13 · openalex publication_date 2000/08/01 · arxiv updated 2015/06/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/01
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.