General solutions of the Monge-Amp\`{e}re equation in $n$-dimensional space

TL;DR

This paper derives the general solution to the homogeneous Monge-Ampère equation in n-dimensional space by linking it to an implicitly integrable system and providing a method to construct solutions explicitly.

Contribution

It establishes a connection between the Monge-Ampère equation and an integrable system, enabling explicit construction of general solutions.

Findings

General solution expressed via an implicitly integrable system

Method to construct solutions explicitly

Enhanced understanding of Monge-Ampère equations in multiple dimensions

Abstract

It is shown that the general solution of a homogeneous Monge-Amp\`{e}re equation in $n$-dimensional space is closely connected with the exactly (but only implicitly) integrable system \frac {\partial \xi_{j}}{\partial x_0}+\sum_{k=1}^{n-1} \xi_{k} \frac {\partial \xi_{j}}{\partial x_{k}}=0 \label{1} Using the explicit form of solution of this system it is possible to construct the general solution of the Monge-Amp\`{e}re equation.