Ideals and solutions of nonlinear field equations
type: Item
instance of: scholarly article
title: Ideals and solutions of nonlinear field equations
zbMATH Open document ID: 0712.53018
DOI: 10.1007/BF00673909
published in: International Journal of Theoretical Physics
publication date: 1990-01-01T00:00:00Z
review text: Families of horizontal ideals of contact manifolds of finite order are studied. Each horizontal ideal is shown to admit an n-dimensional module of Cauchy characteristic vector fields that is also a module of annihilators (in the sense of Cartan) of the closed contact ideal. Since horizontal ideals are generated by 1-forms, any closed horizontal ideal leads to a foliation of the contact manifold by submanifolds of dimension n on which the horizontal ideal and the contact ideal vanish. Explicit conditions are obtained under which an open subset of a leaf of this foliation is the graph of a solution map of the fundamental ideal that characterizes a given system of PDE of finite order with n independent variables. These conditions are tantamount to determining the intersections of the leaves of the foliation generated by a closed horizontal ideal with the intersection of a collection of hypersurfaces that is determined by the system of PDE under study. The problem of solving systems of PDE is thereby reduced to manifold- intersection problems. Every smooth solution map of the fundamental ideal is shown to be obtained in this manner. Conditions are also obtained under which every leaf of the foliation is a graph of a solution map. Solution maps are also shown to be determined by complete systems of first integrals that are obtained by integrating the orbital equations of a canonical basis for the module of Cauchy characteristic vector fields of a horizontal ideal. The Lie algebra of isovectors of horizontal ideals is analyzed and the flows generated by such a Lie algebra is shown to generate automorphisms of the foliation generated by the horizontal ideal. Extended canonical transformations are studied. These transformations map a closed horizontal ideal onto another closed horizontal ideal and thus provide the means for obtaining the explicit representations of closed horizontal ideals demanded by the method.
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 35Q80
zbMATH DE Number: 4171874
zbMATH Keywords: Cauchy characteristics
zbMATH Keywords: geometric methods
zbMATH Keywords: horizontal ideals
zbMATH Keywords: contact manifolds
zbMATH Keywords: foliation
zbMATH Keywords: solution map
zbMATH Keywords: first integrals
zbMATH Keywords: Lie algebra of isovectors
author: Dominic G. B. Edelen
MaRDI profile type: Publication
cites work: Q3411975
cites work: Alternatives to the Cartan-Kähler theorem
cites work: Q3715336
cites work: Q3209062
description: scientific article; zbMATH DE number 4171874