Recursion Operators admitted by non-Abelian Burgers equations: Some Remarks

TL;DR

This paper investigates the structural properties of recursion operators for non-Abelian Burgers equations, demonstrating their role as strong symmetries and hereditary operators, and explores the hierarchy they generate, extending classical Burgers equations.

Contribution

It provides a detailed analysis of recursion operators for non-Abelian Burgers equations, including their properties and the hierarchy they generate, using both direct and computer-assisted methods.

Findings

Recursion operator is a strong symmetry for the non-Abelian Burgers equation.

The recursion operator is hereditary.

The hierarchy reduces to the classical Burgers hierarchy when the operator is scalar.

Abstract

The recursion operators admitted by different operator Burgers equations, in the framework of the study of nonlinear evolution equations, are here con- sidered. Specifically, evolution equations wherein the unknown is an operator acting on a Banach space are investigated. Here, the mirror non-Abelian Burgers equation is considered: it can be written as $r_t = r_{xx} + 2r_x r$. The structural properties of the admitted recursion operator are studied; thus, it is proved to be a strong symmetry for the mirror non-Abelian Burgers equation as well as to be the hereditary. These results are proved via direct computations as well as via computer assisted manipulations; ad hoc routines are needed to treat non-Abelian quantities and relations among them. The obtained recursion operator generates the $mirror$ non-Abelian Burgers hierarchy. The latter, when the unknown operator $r$ is replaced by a real valued function reduces to the usual (commutative) Burgers hierarchy. Accordingly, also the recursion operator reduces to the usual Burgers one.