Characteristics and the Geometry of Hyperbolic Equations in the Plane

This paper focuses on the implementation and application of Élie Cartan′s ideas on characteristics for second order non-linear hyperbolic equations in the plane. The topics include a discussion of both intrinsic and extrinsic contact equivalence, a description of the theory of Bäcklund and Laplace transformations based on characteristics, a geometric approach to the notion of Riemann invariants, and a study of the various possibilities for their existence, resulting in a new simple proof of Lie′s description of the contact orbit of the wave equation. The initial value problem is analyzed from the viewpoint of exterior differential systems and applied to establish a smooth existence theorem with an application to a shock wave problem.