Factorization of linear partial differential operators and Darboux integrability of nonlinear PDEs

TL;DR

This paper introduces a new approach to factorizing linear partial differential operators using generalized divisors, with implications for solving nonlinear PDEs and computing Groebner bases.

Contribution

It develops a modular lattice structure for generalized divisors and extends classical factorization theorems to PDEs, linking algebraic and differential operator theory.

Findings

Lattice of generalized divisors is modular.

Analogues of Loewy-Ore factorization theorems are established.

Application to Darboux integrability criteria for nonlinear PDEs.

Abstract

Using a new definition of generalized divisors we prove that the lattice of such divisors for a given linear partial differential operator is modular and obtain analogues of the well-known theorems of the Loewy-Ore theory of factorization of linear ordinary differential operators. Possible applications to factorized Groebner bases computations in the commutative and non-commutative cases are discussed, an application to finding criterions of Darboux integrability of nonlinear PDEs is given.