Vessiot Structure for Manifolds of (p,q)-Hyperbolic type: Darboux Integrability & Symmetry

Peter J. Vassiliou

In this paper we employ a geometric object intrinsically associated with any such partial differential equation: its hyperbolic structure, to study the Darboux integrability of the class E of semilinear second order hyperbolic partial differential equations in one dependent and two independent variables. The main tool used is E. Vessiot's formulation of the Cartan-Kähler theorem. It is shown that the problem of classifying the Darboux integrable equations in E contains, as a subproblem, that of classifying the manifolds of (p,q)- hyperbolic type of rank 4 and dimension 2k+3, k\geq 2; p=2,q\geq 2.

In turn, it is shown that the problem of classifying these manifolds in the two (lowest) cases (p,q)=(2,2), (2,3) contains, as a subproblem, the classification problem for Lie groups. This generalises classical results of E. Vessiot. It follows that part of the moduli space for the Darboux integrable equations in E is filled out by isomorphism classes of Lie groups.

Each (2,2)- or (2,3)-Darboux integrable equation in the class E determines a finite Lie group. This Lie group is the group of automorphisms of the characteristic systems of the given equation which leaves invariant the foliation induced by the characteristic (or, Riemann) invariants of the equation, the tangential characteristic symmetries. The isomorphism class of the tangential chartacteristic symmetries is a contact invariant of the corresponding Darboux integrable partial differential equation.

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