An exploration of hyperbolic exterior differential systems and their integrability by the method of Darboux

I consider the partial classification and Darboux integrability for a general first-order quasi-linear hyperbolic system of partial differential equations for three functions of two independent variables. The general system of PDE is formulated as an exterior differential system (EDS), consisting of a 5-manifold M, an ideal in the exterior algebra I , and an independence condition phi ≠ 0. Integral manifolds of the EDS are in one-to-one correspondence to local solutions of the system of PDE. The EDS thus defined is generalized to include other geometric systems that share properties with the systems of PDE. I use tools from exterior differential systems and Cartan's method of equivalence to study these EDS.;There are relative invariants associated to the EDS, determined by properties of the EDS. I find normal forms for important classes of PDE, such as first order semi-linear hyperbolic systems, and conditions on relative invariants and, in some cases, their derivatives, that determine when an EDS is locally equivalent under gauge transformations to these systems. Under certain conditions on the relative invariants, an invariant K is introduced -- under these conditions and when K = 0, the three-fold foliation of solution surfaces by characteristic curves can be straightened.;The main results center around the question of when such an EDS or its first prolongation is integrable by the method of Darboux, which allows the construction of solutions using only ordinary differential equations. EDS which are Darboux integrable without prolongation are systems of conservation laws and are classified by one function of five variables. I find a normal form for such EDS which are equivalent to quasi-linear hyperbolic systems of PDE. These are classified by three functions of three variables.;Under certain genericity conditions, hyperbolic systems of PDE whose first prolongation is Darboux integrable can be put into normal forms. These systems are also classified by three functions of three variables.