Abstract
The quintessential example of a Darboux integrable differential equation is the Liouville equation \(u_{xy} = -2 {\rm e}^u\)u_{xy} = -2 {\rm e}^u,
whose general solution is given by \(u= \log \frac{ f'(x) g'(y)}{(f(x)-g(y))^2\).
Daboux integrability is classically related to the existence of intermediate integrals (or Riemann invariants) which in turn allow an explicit closed form formula to be derived for these equations.
Motivated by work of E. Vessiot, I will describe a differential geometric construction which provides a fundamental description of Darboux integrable systems in terms of superposition of differential systems and the quotient theory of differential systems by Lie groups.
The general theory will be discussed (I won't assume familiarity with differential systems), and demonstrated with examples. If time permits some interesting properties of these systems will be shown based on the existence of the quotient representation of Darboux integrable systems.