On a generalization of Monge-Ampère equations and Monge-Ampère systems

Abstract: We discuss Monge-Amp`ere equations from the view point of differential geometry. It is known that a Monge-Amp`ere equation corresponds to a special exterior differential system on a 1-jet space. In this paper, we generalize Monge-Amp`ere equations and prove that a $(k+1)$st order generalized Monge-Amp`ere 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.