The Cauchy Problem for Darboux Integrable Systems and Non-Linear d'Alembert Formulas

TL;DR

This paper demonstrates that the Cauchy problem for Darboux integrable systems can be reduced to Lie group equations, enabling explicit solutions similar to d'Alembert's formula when the associated Vessiot group is solvable.

Contribution

It establishes a method to solve the Cauchy problem for Darboux integrable PDEs via Lie group techniques, providing explicit integral formulas.

Findings

Reduction of Cauchy problem to Lie type equations

Explicit integral formulas for solvable Vessiot groups

Generalization of d'Alembert's formula for Darboux systems

Abstract

To every Darboux integrable system there is an associated Lie group $G$ which is a fundamental invariant of the system and which we call the Vessiot group. This article shows that solving the Cauchy problem for a Darboux integrable partial differential equation can be reduced to solving an equation of Lie type for the Vessiot group $G$. If the Vessiot group $G$ is solvable then the Cauchy problem can be solved by quadratures. This allows us to give explicit integral formulas, similar to the well known d'Alembert's formula for the wave equation, to the initial value problem with generic non-characteristic initial data.