Cartan equivalence problem for third order differential operators

TL;DR

This paper addresses the Cartan equivalence problem for third order differential operators on the line, using the Cartan method to determine when two such operators are equivalent under fiber-preserving transformations.

Contribution

It develops three approaches to the equivalence problem, including direct equivalence and gauge equivalence conditions, for third order differential operators.

Findings

Derived criteria for equivalence under fiber-preserving transformations

Established conditions for gauge equivalence of differential operators

Applied Cartan's method to classify third order operators

Abstract

This article is dedicated to solve the equivalence problem for two third order differential operators on the line under general fiber--preserving transformation using the Cartan method of equivalence. We will do three versions of the equivalence problems: first via the direct equivalence problem, second equivalence problem is to determine conditions on two differential operators such that there exists a fiber-preserving transformations mapping one to the other according to gauge equivalence.