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

Masahiro Kawamata, Kazuhiro Shibuya

Published 2020-08-24Version 1

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.