Parametric Factorizations of Second-, Third- and Fourth-Order Linear Partial Differential Operators with a Completely Factorable Symbol on the Plane

TL;DR

This paper investigates parametric factorizations of second, third, and fourth-order linear partial differential operators with completely factorable symbols, revealing limited cases where irreducible parametric factorizations exist and characterizing their parameterization.

Contribution

It characterizes the conditions under which irreducible parametric factorizations occur for certain linear PDE operators with factorable symbols and provides explicit examples and parameterizations.

Findings

Irreducible parametric factorizations are rare and occur only for specific operator types.

For second and third-order operators, factorization families depend on a single univariate function.

Explicit examples of parametric families are provided for each factorization type.

Abstract

Parametric factorizations of linear partial operators on the plane are considered for operators of orders two, three and four. The operators are assumed to have a completely factorable symbol. It is proved that ``irreducible'' parametric factorizations may exist only for a few certain types of factorizations. Examples are given of the parametric families for each of the possible types. For the operators of orders two and three, it is shown that any factorization family is parameterized by a single univariate function (which can be a constant function).