Reduction of exterior differential systems with infinite dimensional symmetry groups

Abstract

A symmetry-based method for constructing solutions to systems of differential equations founded on the reduction of exterior differential systems invariant under the action of an infinite dimensional pseudogroup is proposed. One can associate to any system of differential equations Δ=0 with a symmetry group \(\mathcal{G}\) an exterior differential system \(\mathcal{I}\) invariant under \(\mathcal{G}\) so that solutions of Δ=0 correspond to integral manifolds \(\mathcal{I}\). The \(\mathcal{G}\)-invariant exterior differential system gives rise to a reduced system \(\overline{\mathcal{I}}\) specified on a cross section to the pseudogroup orbits, and it is shown that solutions to Δ=0 can be reconstructed from integral manifolds \(\overline{\mathcal{I}}\) by solving an equation of generalized Lie type for the jets of pseudogroup transformations. In particular, as opposed to the classical method of symmetry reduction, every solution to the system of differential equations can, under some mild regularity assumptions, be constructed by the present algorithm.