| This thesis presents the geometric investigation of hyperbolic partial differential equations in the plane as carried out by Niky Kamran, Ian Anderson, and Martin Juras. In particular, the relation between the Darboux integrability of an arbitrary hyperbolic equation and the Laplace invariants of the linearization of this equation is established. This extends to non-linear hyperbolic equations in the plane a classical result of Goursat for linear hyperbolic equations. The formal setting for this geometric investigation is afforded by the constrained variational bicomplex, which allows the solution to a partial differential equation to be viewed as a manifold on which standard differential geometric operations such as exterior differentiation and Lie differentiation can be performed. The key element in this investigation is the judicious construction and use of appropriate moving coframes which will reflect the properties of the equations under investigation. |
