Normal forms for parabolic Monge-Ampere equations

TL;DR

This paper establishes normal forms for parabolic Monge-Ampere equations, classifies their symmetry properties, and provides explicit conditions for integrals, advancing the understanding of their geometric structure and solution methods.

Contribution

It introduces a comprehensive classification of parabolic Monge-Ampere equations based on their symmetry and integral properties, with explicit equations for integrals and a geometric approach.

Findings

Normal forms for all parabolic Monge-Ampere equations with integrals

Explicit determining equations for complete integrals

Classification of vector fields in contact structures

Abstract

We find normal forms for parabolic Monge-Ampere equations. Of these, the most general one holds for any equation admitting a complete integral. Moreover, we explicitly give the determining equation for such integrals; restricted to the analytic case, this equation is shown to have solutions. The other normal forms exhaust the different classes of parabolic Monge-Ampere equations with symmetry properties, namely, the existence of classical or nonholonomic intermediate integrals. Our approach is based on the equivalence between parabolic Monge-Ampere equations and particular distributions on a contact manifold, and involves a classification of vector fields lying in the contact structure. These are divided into three types and described in terms of the simplest ones (characteristic fields of first order PDE's).