The Geometry of Darboux Integrable Elliptic Systems

TL;DR

This paper characterizes Darboux integrable elliptic systems via their relation to holomorphic Pfaffian systems and symmetry groups, providing a geometric framework for understanding solutions of elliptic PDEs.

Contribution

It introduces a geometric characterization of Darboux integrable elliptic systems using complex Lie group symmetries and holomorphic structures, extending classical integrability concepts.

Findings

Elliptic systems are characterized by Darboux integrability under certain conditions.

Solutions can be expressed through holomorphic solutions of associated Pfaffian systems.

Examples include first- and second-order elliptic PDEs in the plane.

Abstract

We characterize real elliptic differential systems whose solutions can be expressed in terms of holomorphic solutions to an associated holomorphic Pfaffian system $\mathcal H$ on a complex manifold. In particular, these elliptic systems arise as quotients by a group $G$ of the real differential system generated by the real and imaginary parts of $\mathcal H$, such that $G$ is the real form of a complex Lie group $K$ which is a symmetry group of $\mathcal H$. Subject to some mild genericity assumptions, we show that such elliptic systems are characterized by a property known as Darboux integrability. Examples discussed include first- and second-order elliptic PDE and PDE systems in the plane.