Second-order generalized Monge--Amp\`ere equations on a plane and its geometric singular solutions

TL;DR

This paper investigates generalized Monge--Ampère equations on a plane using exterior differential systems, constructs their geometric singular solutions via Cauchy characteristics, and classifies these solutions based on their equivalence to known singularities.

Contribution

It introduces a method to construct and classify geometric singular solutions of generalized Monge--Ampère equations using differential geometry techniques.

Findings

Constructed geometric singular solutions using Cauchy characteristics.

Provided criteria for solutions to be equivalent to cuspidal edge, swallowtail, and butterfly.

Enhanced understanding of singular solutions in the context of generalized Monge--Ampère equations.

Abstract

In the present paper, we study some generalized Monge--Amp\`ere equations in terms of special exterior differential systems on a jet space. Moreover, we construct geometric singular solutions of the generalized Monge--Amp\`ere equations by using the method of Cauchy characteristics. Furthermore, we give criteria for geometric singular solutions to be right-left equivalent to the cuspidal edge, swallowtail and butterfly.