Symmetry Classification Using Noncommutative Invariant Differential Operators

Given a class F of differential equations, the symmetry classification problem is to determine for each member f ∈ F the structure of its Lie symmetry group G f , or equivalently of its Lie symmetry algebra. The components of the symmetry vector fields of the Lie algebra are solutions of an associated over-determined ‘defining system’ of differential equations. The usual computer classification method which applies a sequence of total derivative operators and eliminations to this associated system often fails on problems of interest due to the excessive size of expressions generated in intermediate computations. We provide an alternative classification method which exploits the knowledge of an equivalence group G preserving the class. A non-commutative differential elimination procedure due to Lemaire, Reid and Zhang, where each step of the procedure is invariant under G , can be applied and an existence and uniqueness theorem for the output used to classify the structure of symmetry groups for each f ∈ F . The method is applied to a class of nonlinear diffusion convection equations vx = u, vt = B(u)ux−K(u) which is invariant under a large but easily determined equivalence group G . In this example the complexity of the calculations is much reduced by the use of G -invariant differential operators. AMS Classification: 35N10, 58J70, 53A55, 13P10, 12H05. ∗School of Information Sciences and Engineering, University of Canberra, ACT Australia. 2600. Email: [email protected] †Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada. Email: [email protected], Web: www.apmaths.uwo.ca/ ̃reid ‡GJR gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada. He also acknowledges many helpful discussions with Elizabeth Mansfield on the topic of this paper, and support of an ESPRC grant from the U.K. government for a visit to the University of Kent in Canterbury. Support from University of Canberra is also acknowledged.