On a Generalization of Monge–Ampère Equations and Monge–Ampère Systems

Abstract

We discuss Monge-Ampère equations from the view point of differential geometry. It is known that a Monge–Ampère equation corresponds to a special exterior differential system on a 1-jet space. In this paper, we generalize Monge–Ampère equations and prove that a (k+ 1)st order generalized Monge–Ampère equation corresponds to a special exterior differential system on a k-jet space. Then its solution naturally corresponds to an integral manifold of the corresponding exterior differential system. Moreover, we verify that the Korteweg-de Vries (KdV) equation and the Cauchy–Riemann equations are examples of our equation. 2010 Mathematics Subject Classification. Primary 58A15; Secondary 58A17.