Cartan equivalence method on fifth-order differential operator

TL;DR

This paper applies Cartan's equivalence method to solve the classification problem for fifth-order differential operators under fiber-preserving transformations, establishing criteria for their equivalence.

Contribution

It introduces a systematic approach to determine when two fifth-order differential operators are equivalent under fiber-preserving transformations using Cartan's method.

Findings

Solved direct equivalence problem for fifth-order operators.

Established necessary and sufficient conditions for gauge equivalence.

Provided a framework for classifying fifth-order differential operators.

Abstract

In this paper, we carry out the equivalence problem for fifth-order differential operators on the line under general fiber-preserving transformation using the Cartan method of equivalence. Two versions of equivalence problems have been solved. We consider the direct equivalence problem and an equivalence problem to determine the sufficient and necessary conditions on fifth-order differential operators such that there exists a fiber--preserving transformation mapping one to the other according to gauge equivalence.