Constructive factorization of LPDO in two variables

TL;DR

This paper investigates conditions for factorizing linear partial differential operators in two variables, providing explicit criteria for second-order cases and outlining methods for higher orders, including recursive solutions and special cases.

Contribution

It offers explicit factorization conditions for second-order LPDOs and a recursive approach for higher orders, including degenerate cases like ODEs.

Findings

Factorization conditions are explicitly given for second-order operators.

Recursive systems of linear equations are used for general higher-order operators.

Degenerate cases reduce to solving Riccati equations.

Abstract

We study conditions under which a partial differential operator of arbitrary order $n$ in two variables or ordinary linear differential operator admits a factorization with a first-order factor on the left. The factorization process consists of solving, recursively, systems of linear equations, subject to certain differential compatibility conditions. In the generic case of partial differential operators one does not have to solve a differential equation. In special degenerate cases, such as ordinary differential, the problem is finally reduced to the solution of some Riccati equation(s). The conditions of factorization are given explicitly for second- and, and an outline is given for the higher-order case.