Applications of moving frames to group foliation of differential equations

Abstract

The classical group foliation algorithm uses the continuous symmetries of a differential equation to aid in its integration. This is accomplished by transforming the differential equation into two alternative systems, called the resolving and automorphic systems. Incorporating the theory of equivariant moving frames for Lie pseudogroups, a completely symbolic and systematic version of the group foliation algorithm is introduced. In this version of the algorithm, the resolving system is derived using only knowledge of the structure of the differential invariant algebra, requiring no explicit formulae for differential invariants. Additionally, the automorphic system is replaced by an equivalent reconstruction system, again requiring only symbolic computation. The efficacy of this approach is illustrated through several examples. Further applications of aspects of group foliation are given, including the construction of Backlund transformations using resolving systems and a reconstruction process for an invariant submanifold flow corresponding to a given invariant signature evolution.