Vessiot structure for manifolds of $(p,q)$-hyperbolic type: Darboux integrability and symmetry

It is well known that if a scalar second order hyperbolic partial differential equation in two independent variables is Darboux integrable, then its local Cauchy problem may be solved by ordinary differential equations. In addition, such an equation has infinitely many non-trivial conservation laws. Moreover, Darboux integrable equations have properties in common with infinite dimensional completely integrable systems.

In this paper we employ a geometric object intrinsically associated with any hyperbolic 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. 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 $2k+3$, $k\ge2$; $p=2,q\ge 2$.

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 generalizes classical results of E. Vessiot.

The main result is that if an equation in $E$ is (2,2)- or (2,3)-Darboux integrable on the $k$-jets, $k\ge 2$, then its intrinsic hyperbolic structure admits a Lie group of symmetries of dimension $2k-1$ or $2k-2$, respectively. It follows that part of the moduli space for the Darboux integrable equations in $E$ is determined by isomorphism classes of Lie groups.

The Lie group in question 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 characteristic symmetries is a contact invariant of the corresponding Darboux integrable partial differential equation.