SIGMA 9 (2013), 017, 22 pages arXiv:1210.2370
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
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.
Cauchy problem; Darboux integrability; exterior differential systems; d'Alembert's formula.
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