Existence and Construction of Vessiot Connections

TL;DR

This paper rigorously formulates Vessiot's vector field approach for analyzing PDE systems, establishing its equivalence to the formal theory and linking its applicability to involutivity, with a new characterization of transversal integral elements.

Contribution

It provides a rigorous formulation of Vessiot's approach, proves its equivalence to the formal theory, and introduces a novel characterization of transversal integral elements.

Findings

Vessiot's approach is equivalent to the formal theory of differential equations.

The approach is applicable if and only if the system is involutive.

A new characterization of transversal integral elements via the contact map is provided.

Abstract

A rigorous formulation of Vessiot's vector field approach to the analysis of general systems of partial differential equations is provided. It is shown that this approach is equivalent to the formal theory of differential equations and that it can be carried through if, and only if, the given system is involutive. As a by-product, we provide a novel characterisation of transversal integral elements via the contact map.