Contact Geometry of Hyperbolic Equations of Generic Type

TL;DR

This paper explores the contact geometry of generic hyperbolic equations, classifies their invariants, and identifies the maximum symmetry algebra, revealing that maximally symmetric cases are Darboux integrable.

Contribution

It provides a classification of generic hyperbolic equations using contact invariants and establishes the maximum symmetry dimension, along with explicit normal forms and integrability results.

Findings

Maximum contact symmetry algebra dimension is nine.

Normal forms for maximally symmetric equations are derived.

Maximally symmetric equations are Darboux integrable.

Abstract

We study the contact geometry of scalar second order hyperbolic equations in the plane of generic type. Following a derivation of parametrized contact-invariants to distinguish Monge-Ampere (class 6-6), Goursat (class 6-7) and generic (class 7-7) hyperbolic equations, we use Cartan's equivalence method to study the generic case. An intriguing feature of this class of equations is that every generic hyperbolic equation admits at most a nine-dimensional contact symmetry algebra. The nine-dimensional bound is sharp: normal forms for the contact-equivalence classes of these maximally symmetric generic hyperbolic equations are derived and explicit symmetry algebras are presented. Moreover, these maximally symmetric equations are Darboux integrable. An enumeration of several submaximally symmetric (eight and seven-dimensional) generic hyperbolic structures is also given.