Vessiot structure for manifolds of \((p,q)\)-hyperbolic type: Darboux integrability and symmetry (Q2701677)

The paper deals with Darboux-integrability of hyperbolic semilinear second order partial differential equations in two variables, which, after a change of variables, can be written in the form \(u _{xy}= f(x,y, u , u _{x}, u _{y})\). It is explained that every such equation is intrinsically associated with a geometric object referred to as a manifold of \((p,q)\)-hyperbolic type of rank \(4\). Roughly speaking, this is a smooth manifold of dimension at least \(6\) with a hyperbolic structure on it. NEWLINENEWLINENEWLINEThe problem of classifying these manifolds contains as a subproblem the classification problem for Lie groups and is thus extremely complex. It is solved therefore only under some additional assumptions. NEWLINENEWLINENEWLINEThe paper contains a number of rather explicit examples.