Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 017, 22 pages      arXiv:1210.2370      https://doi.org/10.3842/SIGMA.2013.017
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

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

Ian M. Anderson and Mark E. Fels
Utah State University, Logan Utah, USA

Received October 08, 2012, in final form February 20, 2013; Published online February 27, 2013

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.

Key words: Cauchy problem; Darboux integrability; exterior differential systems; d'Alembert's formula.

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