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

Abstract

In the present paper, we discuss Monge-Ampère equations from the viewpoint 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 and that its solution naturally corresponds to an integral manifold of the corresponding exterior differential system. Moreover, we show that the Korteweg-de Vries (KdV) equation and the Cauchy--Riemann equations are examples of our equations.