Abelian versus non-Abelian Baecklund Charts: some remarks

TL;DR

This paper compares Abelian and non-Abelian Baecklund transformations linking Burgers and KdV hierarchies, highlighting structural differences and similarities, and emphasizing the richer structures in non-commutative cases.

Contribution

It provides a comparative analysis of Baecklund charts for Abelian and non-Abelian hierarchies, revealing new structural insights especially in non-commutative scenarios.

Findings

Non-commutative Baecklund charts are richer than commutative ones.

Identified key differences and similarities between Burgers and KdV-type equations.

Established links between different nonlinear evolution equations via Baecklund transformations.

Abstract

Connections via Baecklund transformations among different non-linear evolution equations are investigated aiming to compare corresponding Abelian and non Abelian results. Specifically, links, via Baecklund transformations, connecting Burgers and KdV-type hierarchies of nonlinear evolution equations are studied. Crucial differences as well as notable similarities between Baecklund charts in the case of the Burgers - heat equation, on one side and KdV -type equations are considered. The Baecklund charts constructed in [16] and [17], respectively, to connect Burgers and KdV-type hierarchies of operator nonlinear evolution equations show that the structures, in the non-commutative cases, are richer than the corresponding commutative ones.