Moving Frames and Calculation of Lie symmetry of PDE

Ian Lisle
School of Mathematics and Statistics
University of Canberra, Australia
email:  lisle@ise.canberra.edu.au

Algorithmic calculation of Lie symmetries of PDE relies on differential
elimination/completion methods, which bring the defining system to a canonical
form where an existence-uniqueness theorem can be applied, and the dimension
and structure of the symmetry algebra found.  These methods can perform poorly
when applied to classification of symmetries of classes of PDE, where algebraic
expressions can grow to the point where the calculation may fail, or lead to
results so complex as to defy interpretation.  To overcome these limitations,
we make use of the notion of the equivalence group of the class of PDE, that is
a group of transformations mapping equations in the class to other equations in
the class.  A moving frame which is invariant under the action of the
equivalence group is sought, and we show how to execute a change of frame in
the symmetry defining system.

It is shown how to perform differential elimination/completion with respect to
the noncommuting basis of differential operators provided by the frame.
Existence and uniqueness of formal solutions of the canonical form is
demonstrated.  These results are used to find the dimension and structure of
the Lie symmetry algebra, in a transparent form.  Application to symmetries of
a class of PDE leads to traversing a binary tree of subcases, each with its own
symmetry algebra.  Computational examples are given, including complete
symmetry classifications of nonlinear diffusion-convection equations, and
linear hyperbolic equations.