On the Lie and Cartan Theory of Invariant Differential Systems, II

TL;DR

This paper explores the application of Lie groupoids and Cartan's methods to analyze and solve systems of partial differential equations, focusing on integration, equivalence, and the structure of solutions.

Contribution

It advances the understanding of integrating PDE systems using Lie and Cartan techniques, emphasizing the role of contact structures and local equivalence in solution existence.

Findings

Criteria for solution existence in PDE systems

Analysis of local equivalence problems for PDEs

Insights into non-integrable Pfaffian systems

Abstract

We start discussing basic properties of Lie groupoids and Lie pseudo-groups in view of applying these techniques to the analysis of Jordan-H\"older resolutions and the subsequent integration of partial differential equations which is the summit of Lie and Cartan's work. Next, we discuss the integration problem for systems of partial differential equations in one unknown function and special attention is given to the first order systems. The Grassmannian contact structures are the basic setting for our discussion and the major part of our considerations inquires on the nature of the Cauchy characteristics in view of obtaining the necessary criteria that assure the existence of solutions. In all the practical applications of partial differential equations, what is mostly needed and what is in fact hardest to obtains are the solutions of the system or, occasionally, some specific solutions. We continue our discussion by examining the local equivalence problem for partial differential equations, illustrating it with some examples, since almost any integration process or method is actually a local equivalence problem involving a suitable model. We terminate the discussion by inquiring on non-integrable Pfaffian systems and their integral manifolds of maximal dimension. This work is based on four most enlightening M\'emoires written by \'Elie Cartan in the beginning of the last century.