On a generalization of Monge-Amp\`ere equations and Monge-Amp\`ere systems

TL;DR

This paper generalizes Monge-Ampère equations within differential geometry, linking higher-order equations to exterior differential systems and providing examples like the KdV and Cauchy-Riemann equations.

Contribution

It introduces a higher-order generalization of Monge-Ampère equations and establishes their correspondence with exterior differential systems on jet spaces.

Findings

Generalized Monge-Ampère equations correspond to exterior differential systems.

Solutions are integral manifolds of these systems.

Examples include KdV and Cauchy-Riemann equations.

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.