Reduction of exterior differential systems with infinite dimensional symmetry groups
type: Item
instance of: scholarly article
title: Reduction of exterior differential systems with infinite dimensional symmetry groups
zbMATH Open document ID: 1147.58003
author: Juha Pohjanpelto
published in: BIT
publication date: 2008-09-12T00:00:00Z
review text: Let \(\Delta=0\) be a system of differential equations, and let \({\mathcal S}\) be its solution space. A symmetry group of this system is a group \({\mathcal G}\) which acts on the space \({\mathcal S}\). Usually a symmetry group of a system is known, but the solution space is not. There is an equivalent exterior differential system \({\mathcal I}\) invariant under \({\mathcal G}\) so that \({\mathcal S}\) corresponds to integral manifolds \({\mathcal I}\). The \({\mathcal G}\)-invariant system \({\mathcal I}\) gives rise to a reduced system \(\widetilde{\mathcal I}\) specified on a cross section to the pseudogroup orbits, and the solution space \({\mathcal S}\) to \(\Delta=0\) can be reconstructed from integral manifolds \(\widetilde{\mathcal I}\) by solving an equation of generalized Lie type for the jets of pseudogroup transformations. In this paper, the author uses the moving frames algorithm to model the orbit space of the pseudogroup action by a cross section to the pseudogroup orbits, and employs the bundle \({\mathcal G}^ k\) of jets of the pseudogroup transformations as a substitute for the Lie group used in the finite dimensional situation. As a consequence, the reconstruction problem becomes equivalent to solving a system of differential equations of a generalized Lie type on \({\mathcal G}^ k\). The author presents this process to construct explicit solutions to a nonlinear wave equation.
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 58A15
Mathematics Subject Classification ID: 58A20
Mathematics Subject Classification ID: 58H05
Mathematics Subject Classification ID: 58J70
zbMATH DE Number: 5343409
zbMATH Keywords: pseudogroup
zbMATH Keywords: moving frame
zbMATH Keywords: differential invariant
zbMATH Keywords: exterior differential system
MaRDI profile type: Publication
full work available at URL: s10543-008-0177-9
OpenAlex ID: W2148713308
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DOI: 10.1007/S10543-008-0177-9
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description: scientific article; zbMATH DE number 5343409