On weakly commutative triples of partial differential operators

TL;DR

This paper explores the algebraic structure of weakly commutative triples of partial differential operators, linking them to commuting elements in skew Ore fields, and proves a version of the Burchnall-Chaundy theorem using algebraic methods.

Contribution

It introduces an algebraic approach to weakly commutative triples, relating them to skew Ore fields and establishing a new algebraic proof of the Burchnall-Chaundy theorem.

Findings

Weakly commutative triples are related to commuting elements in skew Ore fields.

A version of the Burchnall-Chaundy theorem is proved algebraically.

The approach avoids analytical complexities typical in integrable systems.

Abstract

We investigate algebraic properties of weakly commutative triples, appearing in the theory of integrable nonlinear partial differential equations. Algebraic technique of skew fields of formal pseudodifferential operators as well as skew Ore fields of fractions are applied to this problem, relating weakly commutative triples to commuting elements of skew Ore field of formal fractions of ordinary differential operators. A version of Burchnall-Chaundy theorem for weakly commutative triples is proved by algebraic means avoiding analytical complications typical for its proofs known in the theory of integrable equations.