Vessiot structure for manifolds of \((p,q)\)-hyperbolic type: Darboux integrability and symmetry. (English) Zbl 0984.58003
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.
The 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.
The paper contains a number of rather explicit examples.
The 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.
The paper contains a number of rather explicit examples.
Reviewer: Otto Liess (Bologna)
MSC:
| 58D27 | Moduli problems for differential geometric structures |
| 58J45 | Hyperbolic equations on manifolds |
| 58J70 | Invariance and symmetry properties for PDEs on manifolds |
| 35L70 | Second-order nonlinear hyperbolic equations |
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