On the classification of polynomial differential operators

TL;DR

This paper classifies first order polynomial differential operators based on an order concept, revealing only five possible orders and linking finite orders to Liouvillian integrals, with explicit solutions and examples.

Contribution

It introduces a new classification scheme for polynomial differential operators using an order concept and characterizes their integrability properties.

Findings

Only four finite possible orders for the operators: 0, 1, 2, 3, or infinity.

Finite order operators have expansions generated by the operator and a differential polynomial A.

Operators with order 0, 1, or 2 possess Liouvillian first integrals.

Abstract

This paper gives a classification of first order polynomial differential operators of form $\mathscr{X} = X_1(x_1,x_2)\delta_1 + X_2(x_1,x_2)\delta_2$, $(\delta_i = \partial/\partial x_i)$. The classification is given through the order of an operator that is defined in this paper. Let $X=\mathscr{X}y$ to be the differential polynomial associated with $\mathscr{X}$, the order of $\mathscr{X}$, $\mathrm{ord}(\mathscr{X})$, is defined as the order of a differential ideal $\Lambda$ of differential polynomials that is a nontrivial expansion of the ideal $\{X\}$ and with the lowest order. In this paper, we prove that there are only four possible values for the order of a differential operator, 0, 1, 2, 3, or $\infty$. Furthermore, when the order is finite, the expansion $\Lambda$ is generated by $X$ and a differential polynomial $A$, which can be obtained through a rational solution of a partial differential equation that is given explicitly in this paper. When the order is infinite, the expansion $\Lambda$ is just the unit ideal. In additional, if, and only if, the order of $\mathscr{X}$ is 0, 1, or 2, the polynomial differential equation associating with $\mathscr{X}$ has Liouvillian first integrals. Examples for each class of differential operators are given at the end of this paper.