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 ∆ = 0 with a symmetry group G an exterior differential system I invariant under G so that solutions of ∆ = 0 correspond to integral manifolds I . The G -invariant exterior differential system gives rise to a reduced system I specified on a cross section to the pseudogroup orbits, and it is shown that solutions to ∆ = 0 can be reconstructed from integral manifolds 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. (2000): 58A15, 58A20, 58H05, 58J70. AMS subject classification

Group Foliation of Differential Equations Using Moving Frames

“…The group foliation/inductive moving frames approach outlined in Section 6 does not have this limitation. It would be worthwhile to pursue the possibility of constructing new Bäcklund transformations by realizing systems of interest as resolving systems for infinite-dimensional Lie pseudogroups; these ideas are also most likely closely related to the reduction methods for infinite-dimensional Lie pseudogroups introduced by Pohjanpelto [48]. The investigation of nonmaximal rank resolving systems could also produce interesting examples.…”

“…This pseudogroup is also used in [48] to illustrate the method of symmetry reduction of exterior differential systems admitting an infinite-dimensional symmetry group; we reproduce these results in Examples 28 and 47. In Section 5, the group foliation method is applied to several equations of interest, including a nonlinear wave equation studied by Calogero [5], the equation of a transonic gas flow, and a nonlinear second-order ordinary differential equation.…”

“…The reconstruction result in Example 47 was derived in [48] using the machinery of symmetry reduction of exterior differential systems.…”

“…Central to this adaptation is the introduction of the pseudogroup jet differential expressions which, after pull-back by a moving frame, generalize Cartan's structure equations of a moving frame for Lie group actions [14], and play the role of Mansfield's 'curvature matrix' equation in the reconstruction process. The reconstruction step is also related to the reconstruction procedure appearing in symmetry reduction of exterior differential systems [2,3,48].…”

Invariant discretization of partial differential equations admitting infinite-dimensional symmetry groups

“…The pseudo-group (1.1) was introduced by Lie, [15, p.373], in his study of second order partial differential equations integrable by the method of Darboux. It also appears in Vessiot's work on group splitting and automorphic systems, [29], in Kumpera's investigation of Lie's theory of differential invariants based on Spencer's cohomology, [13], and recently in [21,22,25] to illustrate a new theoretical foundation of moving frames. The differential invariants of the pseudo-group action (1.1) can be found in [22].…”