Jet Bundles and the Formal Theory of Partial Differential Equations

Abstract

Systems of partial differential equations lie at the heart of physics. Despite this, the general theory of these systems has remained rather obscure in comparison to numerical approaches such as finite element models and various other discretisation schemes. There are, however, several theoretical approaches to systems of PDEs, including schemes based on differential algebra and geometric approaches including the theory of exterior differential systems [5] and the so-called "formal theory" [4] built on the jet bundle formalism. This paper is a brief introduction to jet bundles focusing on the completion of systems to equivalent involutive systems for which power series solutions may be constructed order by order. We will not consider the mathematical underpinnings of involution (which lie in the theory of combinatorial decompositions of polynomial modules [2,3]) nor other applications of the theory of jet bundles such as the theory of symmetries of systems of PDEs [6] or discretisation schemes based on discrete approximations to jet bundles [1].

Canonical involution on double jet bundles

“…(iii) If M is a real vector space of dimension m, then J 1 p M inherits a vector space structure: for any j 1 f, j 1 g ∈ J 1 p M and λ ∈ R, operations are given by…”

“…In this definition, a k -jet is an equivalence class determined by an equivalence relation ∽ k . Two sections of a fibered manifold are called k -related by the relation ∽ k if they have the same Taylor polynomial expansion at the point x truncated at order k. This definition usually leads to a geometric approach, which is applied to the study of systems of differential equations (we refer the reader to [1,2,4,[9][10][11] for more details).…”