Algorithms for Symmetric Differential Systems

Abstract.

Over-determined systems of partial differential equations may be studied using differential—elimination algorithms, as a great deal of information about the solution set of the system may be obtained from the output. Unfortunately, many systems are effectively intractable by these methods due to the expression swell incurred in the intermediate stages of the calculations. This can happen when, for example, the input system depends on many variables and is invariant under a large rotation group, so that there is no natural choice of term ordering in the elimination and reduction processes. This paper describes how systems written in terms of the differential invariants of a Lie group action may be processed in a manner analogous to differential—elimination algorithms. The algorithm described terminates and yields, in a sense which we make precise, a complete set of representative invariant integrability conditions which may be calculated in a ``critical pair'' completion procedure. Further, we discuss some of the profound differences between algebras of differential invariants and standard differential algebras. We use the new, regularized moving frame method of Fels and Olver [11], [12] to write a differential system in terms of the invariants of a symmetry group. The methods described have been implemented as a package in \MAPLE. The main example discussed is the analysis of the (2+1 )-d'Alembert—Hamilton system

u_{xx}+u_{yy}- u_{zz}&=& f(u), u_x^2+u_y^2- u_z^2&=&1. (1)

We demonstrate the classification of solutions due to Collins [7] for f\ne 0 using the new methods.