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Infinite Lie pseudogroups

We have already discussed our effective methods for calculating the structure constants of finite dimensional Lie symmetry algebras of PDEs. Although he was optimistic Lie was unable to extend his infinitesimal structure theory to an infinitesimal structure theory of infinite parameter Lie pseudogroups.

Cartan developed a structure theory of such infinite pseudogroups. However this structure theory is difficult to apply to physical PDEs because it requires the analysis of associated overdetermined (pseudogroup) determining systems which are often highly nonlinear and can be very large (e.g. the pseudogroup defining system for the KP equation occupies 30 megabytes when generated automatically by computer). In comparison Lie's method yields infinitesimal determining systems which are linear and generally much smaller.

My first effort [61] towards a practical algorithm for calculating Cartan structure of infinite Lie pseudogroups of PDEs was a hybrid Lie-Cartan algorithm. To apply the method the infinitesimal determining system is reduced to standard form. This standard form is used to predict a model of the pseudogroup defining system. By requiring the model system to admit the infinitesimal symmetries we obtain a system of PDEs for the unspecified functions in the model. Cartan's algorithm is then run on the model system subject to this differential characterization. Our success in calculating Cartan structure in examples using this method inspired Lisle to conjecture an infinitesimal form of the method. This infinitesimal method was rigorously justified [45,44] for Lie pseudogroups of transitive type using the theory of Singer and Sternberg [84].

In summary we obtained an effective algorithm which from the infinitesimal determining system can determine whether a pseudogroup of symmetries is isomorphic to a transitive pseudogroup. If so, then the algorithm can determine the Cartan structure of the pseudogroup. We applied a preliminary implementation of this algorithm to physical PDEs with infinite pseudogroups, including the KP and Liouville equations.


Next: Solving PDEs Up: Determination of the Size Previous: Finite parameter Lie groups
Greg Reid 2003-11-24