B\"acklund Transformations and Non-Abelian Nonlinear Evolution Equations: a Novel B\"acklund Chart

TL;DR

This paper investigates non-Abelian third order evolution equations related to Korteweg-de Vries equations, establishing a non-commutative Bäcklund chart and demonstrating hereditary recursion operators for hierarchies.

Contribution

It introduces a novel non-commutative Bäcklund chart for non-Abelian evolution equations and extends the analysis to hierarchies using hereditary recursion operators.

Findings

Connected classes of non-Abelian evolution equations are represented in a Bäcklund chart.

Recursion operators are shown to be hereditary, enabling hierarchy extensions.

Results are generalized from existing commutative cases to non-commutative operator equations.

Abstract

Classes of third order non-Abelian evolution equations linked to that of Korteweg-de Vries-type are investigated and their connections represented in a non-commutative B\"acklund chart, generalizing results in [Fuchssteiner B., Carillo S., Phys. A 154 (1989), 467-510]. The recursion operators are shown to be hereditary, thereby allowing the results to be extended to hierarchies. The present study is devoted to operator nonlinear evolution equations: general results are presented. The implied applications referring to finite-dimensional cases will be considered separately.