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UWO ORCCA TR-03-03 Summary

Non-commutative Riquier Theory in Moving Frames of Differential Operators, November, 2003, Francois Lemaire, Greg Reid and Yang Zhang, 29 pages

Abstract: Moving frames chosen to be invariant under a known Lie group G} provide a powerful generalization of the idea of choosing G-invariant coordinates to cases where G-invariant coordinates do not exist. Such G-invariant formulations are of great current interest in areas such as Geometric Integration where G-invariant integrators (e.g. symplectic integrators), can often substantially outperform non-invariant integrators. They are also of substantial interest in applications where one would like to factor out a known group.

One form of classical existence and uniqueness theory for analytic PDE referred to (standard) commuting partial derivatives is that of Riquier, which was formulated and generalized by Rust using a Gr\"obner style development.

We extend the Rust-Riquier existence and uniqueness theory to analytic PDE written in terms of moving frames of non-commuting Partial Differential Operators (PDO). The main idea for the theoretical development is to use the commutation relations between the PDO to place them in a standard order. This normalization is exploited to generalize the corresponding steps of the commuting Rust-Riquier Theory to the non-commuting case.

Given an equivalence group G Lisle has given a G-invariant method for determining the structure of Lie symmetry groups of classes of PDE. Lisle's method for such group classification problems was illustrated on a number of challenging examples, which lead to unmanageable expression explosion for computer algebra programs using the standard (commuting) frame. He obtained new results, which for want of an existence and uniqueness theorem for PDE in non-commuting frames, had to be individually checked. We provide an existence and uniqueness theorem making rigorous the output from Lisle's method. For the finite parameter group case, the output is reformulated in terms of the integration of a system of Frobenius type, which can be numerically integrated by integrating an ODE system along a curve.

AMS Classification: 35N10, 58J70, 53A55, 13P10, 12H05.

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