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.

Second-order generalized Monge--Ampère equations on a plane and its geometric singular solutions

“…where z = z(x, y) is an unknown function and A, B, C and D are given functions of x, y, z, z x and z y . By Example 4.2 of [12], the GMAE (3.1) corresponds to the GMAS generated by the following 1-forms:…”

“…For example, the symplectic Monge-Ampère equation and the third-order Monge-Ampère equation are well-known generalizations of MAEs ( [1,2,6]). In our previous study, the author and Shibuya define the generalized Monge-Ampère equation (or GMAE for short), which includes the above -mentioned generalizations of MAEs and other important differential equations ( [12]). Moreover, there is a natural correspondence between (local) solutions of kth order GMAEs and (local) integral manifolds of special exterior differential systems on (k − 1)-jet spaces (called generalized Monge-Ampère systems, or GMAS for short).…”