Algorithms for Symmetric Differential Systems |
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Authors: | Elizabeth L Mansfield |
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Institution: | (1) Institute of Mathematics and Statistics University of Kent Canterbury, CT2 7NF, UK E.L.Mansfield@ukc.ac.uk, GB |
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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.
October 13, 1999. Final version received: May 18, 2001. |
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Keywords: | AMS Classification 35N10 58J70 53A55 13P10 12H05 |
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