The Geometry of Fourth Order Differential Operator

TL;DR

This paper applies Cartan's method to analyze the equivalence of fourth order differential operators under fiber-preserving transformations, establishing conditions for their classification and gauge equivalence.

Contribution

It introduces a comprehensive Cartan-based framework for solving the equivalence problem of fourth order differential operators, including gauge equivalence conditions.

Findings

Derived necessary and sufficient conditions for operator equivalence

Established criteria for gauge equivalence of differential operators

Provided a systematic approach for classifying fourth order operators

Abstract

In present paper, the equivalence problem for fourth order differential operators with one variable under general fiber-preserving transformation using the Cartan method of equivalence is applied. Two versions of equivalence problems are considered. First, the direct equivalence problem and second equivalence problem is to determine the sufficient and necessary conditions on two fourth order differential operators such that there exists a fiber-preserving transformation mapping one to the other according to gauge equivalence.