This article is dedicated to solving the equivalence problem for a pair of third-order differential operators on the line under general fiber-preserving transformation using the Cartan method of equivalence. We will treat 2 versions of equivalence problems: First, the direct equivalence problem, and second, an equivalence problem to determine conditions on 2 differential operators such that there exists a fiber-preserving transformation mapping one to the other according to gauge equivalence. | Turkish Journal of Mathematics Research Article Turk J Math (2013) 37: 949 – 958 ¨ ITAK ˙ c TUB ⃝ doi: Cartan equivalence problem for third-order differential operators Mehdi NADJAFIKHAH,1 Rohollah BAKHSHANDEH CHAMAZKOTI2,∗ School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran 2 Department of Mathematics, Faculty of Basic Science, Babol University of Technology, Babol, Iran 1 Received: • Accepted: • Published Online: • Printed: Abstract: This article is dedicated to solving the equivalence problem for a pair of third-order differential operators on the line under general fiber-preserving transformation using the Cartan method of equivalence. We will treat 2 versions of equivalence problems: first, the direct equivalence problem, and second, an equivalence problem to determine conditions on 2 differential operators such that there exists a fiber-preserving transformation mapping one to the other according to gauge equivalence. Key words: Differential operator, Cartan equivalence, gauge equivalence, invariant, pseudogroup, Lie algebra 1. Introduction The classification of linear differential equations is a special case of the general problem of classifying differential operators, which has a variety of important applications, including quantum mechanics and the projective geometry of curves [9]. The general equivalence problem is to recognize when 2 geometrical objects are mapped on each other by a certain class of diffeomorphisms. E. Cartan developed the general equivalence problem and provided a systematic procedure for determining the necessary and sufficient conditions [1, 2]. In Cartan’s approach, the conditions of equivalence of 2 objects must be reformulated in terms of differential forms. We associate a collection of one-forms to an object under investigation in the original coordinates; the corresponding object in .
