Ideals and solutions of nonlinear field equations
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.
scientific article; zbMATH DE number 4171874