Structure of symmetry of PDE: exploiting partially integrated systems

Abstract

This work is part of a sequence in which we develop and refine algorithms for computer symmetry analysis of differential equations. We show how to exploit partially integrated forms of symmetry defining systems to assist the differential elimination algorithms that uncover structure of the Lie symmetry algebras. We thus incorporate a key advantage of heuristic integration methods, that of exploiting easy integrals of simple (e.g. one term) PDE that frequently occur in such analyses. A single unified method is given that computes structure constants whether the defining system is unsolved, or has been partially or completely integrated. We also give a symbolic-numeric algorithm which for the first time can determine the structure of Lie symmetry algebras specified by defining systems that contain floating point coefficients. This algorithm incorporates a numerical version of the Cartan-Kuranishi prolongation projection algorithm from the geometry of differential equations.