Applications to linear overdetermined systems of PDE

In the past two decades there has been rapid development in methods for gaining insight into a given physical PDE by formulating auxiliary overdetermined systems whose solutions determine features of the given PDE. The auxiliary overdetermined system of PDEs for the infinitesimal Lie symmetries of a given PDE can be automatically produced by many symbolic programs [34], and may contain thousands of PDEs. Other symbolic programs have been found to be very useful for simplifying and often solving these systems. Applications of Lie symmetries include the determination of invariant or similarity solutions, Bäcklund transformations and soliton solutions. We contributed to this area with our Standard Form package [67].

There has been considerable activity analysing properties of physical PDEs through generalizations of Lie's methods [3,21]. Often the associated overdetermined systems are nonlinear. Thus systematic methods for simplifying large nonlinear systems are needed. Few programs and methods have been developed in this area compared to the well-developed theory, methods and implemented algorithms for linear systems. This motivated us to develop the rif algorithm for simplifying and determining properties of such systems.