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The Geometry of Darboux Integrable Elliptic Systems

Mark E. Fels, Thomas A. Ivey

TL;DR

This work extends the Darboux integrability framework from hyperbolic to elliptic exterior differential systems by characterizing real elliptic decomposable DI systems as quotients of holomorphic Pfaffian systems under a transverse symmetry by a real form $G$ of a complex Lie group $K$. The construction provides a holomorphic-to-real reconstruction: maximally DI elliptic systems on a real manifold $M$ arise as the realification of holomorphic systems on a complex manifold $N$, with a quotient by $G$ yielding $M=N/G$ and the singular system $V$ controlling holomorphic Darboux invariants. A central contribution is the Quotient Theorem (and its converse) establishing a precise correspondence between holomorphic data on $N$ and DI elliptic systems on $M$, together with a local normal form (Vessiot coframe) and an integrable extension framework that yields closed-form solutions via holomorphic data. The Vessiot algebra, the Lie algebra of $G$, emerges as a key invariant capturing the symmetry structure and distinguishing inequivalent elliptic DI systems. Collectively, the results generalize AFV’s group-quotient approach to elliptic PDEs, enabling explicit solution formulas and deeper structural understanding through holomorphic and differential-geometric techniques.

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.

The Geometry of Darboux Integrable Elliptic Systems

TL;DR

This work extends the Darboux integrability framework from hyperbolic to elliptic exterior differential systems by characterizing real elliptic decomposable DI systems as quotients of holomorphic Pfaffian systems under a transverse symmetry by a real form of a complex Lie group . The construction provides a holomorphic-to-real reconstruction: maximally DI elliptic systems on a real manifold arise as the realification of holomorphic systems on a complex manifold , with a quotient by yielding and the singular system controlling holomorphic Darboux invariants. A central contribution is the Quotient Theorem (and its converse) establishing a precise correspondence between holomorphic data on and DI elliptic systems on , together with a local normal form (Vessiot coframe) and an integrable extension framework that yields closed-form solutions via holomorphic data. The Vessiot algebra, the Lie algebra of , emerges as a key invariant capturing the symmetry structure and distinguishing inequivalent elliptic DI systems. Collectively, the results generalize AFV’s group-quotient approach to elliptic PDEs, enabling explicit solution formulas and deeper structural understanding through holomorphic and differential-geometric techniques.

Abstract

We characterize real elliptic differential systems whose solutions can be expressed in terms of holomorphic solutions to an associated holomorphic Pfaffian system on a complex manifold. In particular, these elliptic systems arise as quotients by a group of the real differential system generated by the real and imaginary parts of , such that is the real form of a complex Lie group which is a symmetry group of . 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.
Paper Structure (17 sections, 38 theorems, 315 equations)

This paper contains 17 sections, 38 theorems, 315 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Elliptic Distributions and Elliptic Decomposable Systems
  4. Darboux Integrability
  5. Maximal Darboux Integrability
  6. Examples
  7. Symmetry Reduction of Maximally Darboux Integrable Elliptic Systems
  8. Group Action Conventions
  9. The Quotient Theorem
  10. The Local Normal Form
  11. Examples
  12. The Converse of the Quotient Theorem and the Vessiot Algebra
  13. The Vessiot Coframe
  14. Constructing an Integrable Extension
  15. Proof of the Converse to the Quotient Theorem
  16. ...and 2 more sections

Key Result

Theorem 1.1

A differential system $(\mathcal{I},M)$ is a Darboux integrable decomposable (hyperbolic or elliptic) differential system where the singular systems are Pfaffian, if and only if there exists and EDS $\mathcal{E}$ on a manifold $N$ and a symmetric pair $(K,G)$ of Darboux type such that

Theorems & Definitions (97)

  • Theorem 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • proof
  • ...and 87 more