| The purpose of this dissertation is twofold. We first address various geometric aspects of second-order scalar partial differential equations 1 uxy =fx,y,u,ux,uy . In particular, we examine some unresolved problems for equations of the form (1). Our second goal is to construct a framework for the geometric study of hyperbolic systems of m differential equations in m dependent variables and two independent variables 2 Fa x,y,ug,ugx,u gy,ugxx, ugxy,ugyy =0.; In the first part of the dissertation, we completely solve the local equivalence problem for partial differential equations (1) that are Darboux integrable at order two. We also explicitly calculate all generalized symmetries and conservation laws for equations (1) Darboux integrable at any order.; In the second part of the dissertation, we extend to systems (2) some techniques used in the geometric study of scalar second order partial differential equations. For a special subclass of (2) we identify obstructions to the existence of characteristic cohomology. We also examine the problem of determining when a system (2) is integrable by the method of Darboux and obtain some preliminary results in this direction. Finally, we demonstrate a complete solution to the inverse problem of the calculus of variations for the subclass of (2) known as the f-Gordon systems. |
