Reduction and completion algorithms for partial differential equations

Abstract

An algorithm InvolutionSystem is described that transforms a system of partial differential equations (pale's) into invohdive form, i, e. into a differential Grobner base. This algorithm rests on the theory of Riquier and Janet of pale's that has been created at the beginning of this century. It is shown how it leads to a constructive method for determining the size of the group of Lie symmetries of any system of pale's, If all consistent orderings for the terms in a system of pale's are known, the algorithm InvolutionSystem may be applied repeatedly to determine a universal differential Grobner base. Several examples for them are given and it is shown how in this way various apparently disconnected concepts like the Cole-Hopf transformation, the Miura transformation and certain Backlund transformations are related and may be obtained in a purely algorithmic way.