Reduction of exterior differential systems with infinite dimensional symmetry groups
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.
scientific article; zbMATH DE number 5343409
scientific article