Secondary differential operators

Abstract

Let Ч be an arbitrary sustem of partial (non-linear differential equations. Higher infinitesimal sysmmetries of Ч may be interpreted as vector fields on the «manifoldå Sol Ч of all local solution of this system. The paper deals with construction of differential operators of arbitrary orders on Sol Ч. These approaches to construction of the theory of these operators, geometric and functional are presented, and their equivalence is proved when Ч is the trivial equation. Coincidence of «extrinsicå and «intrinsicå geometric secondary operator is proved for an arbitrary system Ч. It is shown that each geometric secondary operator may be approximated by a sum of compositions of evolution differentiations with any possible accuracy, a description of geometric secondary operators in local coordinates is algo given. These results are obtained by studying the geometry of finite jets and infinitely prolonged equations.