Geodesic
The Cauchy Problem for Darboux Integrable Systems and Non-Linear d'Alembert Formulas
Symmetry, integrability and geometry: methods and applications, Tome 9 (2013) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: Cauchy problem; Darboux integrability; exterior differential systems; d'Alembert's formula. 
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Ian. M. Anderson; Mark E. Fels. The Cauchy Problem for Darboux Integrable Systems and Non-Linear d'Alembert Formulas. Symmetry, integrability and geometry: methods and applications, Tome 9 (2013). http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a16/

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