Abstract
We incorporate the new theory of equivariant moving frames for Lie pseudogroups into Vessiot’s method of group foliation of differential equations. The automorphic system is replaced by a set of reconstruction equations on the pseudogroup jets. The result is a completely algorithmic and symbolic procedure for finding both invariant and noninvariant solutions of differential equations admitting a symmetry group.
Highlights
The method of group foliation is a procedure for obtaining solutions of differential equations invariant under a symmetry group
Group foliation In the first part of this section, we review the classical method of group foliation, mostly following Ovsiannikov’s treatment [47]
As Example 39 shows, this final step of the group foliation method may result in a problem no easier to solve than the original differential equation
Summary
The method of group foliation ( called group splitting, or group stratification) is a procedure for obtaining solutions of differential equations invariant under a symmetry group. Continuing Example 3, the right-invariant Maurer–Cartan forms of the Lie pseudogroup (3) satisfy the linear relations μYx = μUx = 0, μy = 0, μu = −U μxX ,. The left-invariant Maurer–Cartan forms are obtained by successively Lie differentiating with respect to m μa = d za −. For the implementation of the group foliation method it will be useful to know the relation between left-invariant and right-invariant Maurer–Cartan forms. The linear relations among the higher-order Maurer–Cartan forms are obtained by Lie differentiating (14) with respect to (13): m μaA = −.
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