Division of Differential operators, intertwine relations and Darboux Transformations

TL;DR

This paper explores the division of differential operators using generalized Bell polynomials, establishing intertwine relations, Darboux transformations, and a linearizable Burgers equation, with an alternative proof of Matveev's theorem.

Contribution

It introduces a novel approach to differential operator division using generalized Bell polynomials and derives new intertwine relations and Darboux transformations.

Findings

Explicit formulas for operator division in nonabelian rings

Intertwine relations leading to linearizable Burgers equations

Darboux-Matveev transformations expressed via generalized Bell polynomials

Abstract

The problem of a differential operator left- and right division is solved in terms of generalized Bell polinomials for nonabelian differential unitary ring. The definition of the polinomials is made by means of recurrent relations. The expresions of classic Bell polinomils via generalized one is given. The conditions of an exact factorization possibility leads to the intertwine relation and results in some linearizable generalized Burgers equation. An alternative proof of the Matveev theorem is given and Darboux - Matveev transformations formula for coefficients follows from the intertwine relations and also expressed in the generalized Bell polinomials.