Tangential symmetries of Darboux integrable systems

TL;DR

This paper explores the tangential symmetries of Darboux integrable systems, generalizing classical notions, and provides a coordinate-free geometric construction of their symmetry Lie algebras applicable in any dimension.

Contribution

It introduces a new geometric method to construct Lie algebras of tangential symmetries for Darboux integrable systems without relying on adapted coordinates.

Findings

The general solution can be obtained by integration for these systems.

A coordinate-free geometric construction of symmetry Lie algebras is provided.

The method applies to systems in arbitrary dimensions and their prolongations.

Abstract

In this paper we analyze the tangential symmetries of Darboux integrable decomposable exterior differential systems. The decomposable systems generalize the notion of a hyperbolic exterior differential system and include the classic notion of Darboux integrability for first order systems and second order scalar equations. For Darboux integrable systems the general solution can be found by integration (solving ordinary differential equations). We show that this property holds for our generalized systems as well. We give a geometric construction of the Lie algebras of tangential symmetries associated to the Darboux integrable systems. This construction has the advantage over previous constructions that our construction does not require the use of adapted coordinates and works for arbitrary dimension of the underlying manifold. In particular it works for the prolongations of decomposable exterior differential systems.