B ̈acklund Charts: commutative versus non-commutative Equation Hierarchies

Here different Baecklund Charts are considered both in the case of Commutative Equation Hierarchies as well as in the case of their Non-Commutative analogues. The aim is to point out differences and analogies[1]. Specifically, the case of the Cole-Hopf link between Burgers and Heat Equations [2, 3] and its extension to the corresponding Hierarchies are considered [4]. Furthermore, links connecting third order nonlinear evolution equations, such as KdV, mKdV are analysed, again, in both the commutative [5] and non- commutative case [6]. Notably, the latter give rise to a wider variety of equations. Correspondingly, various different hierarchies of non- commutative equations are generated while, in the commutative case, there was only one hierarchy. Furthermore, the related recursion operators are considered pointing out their peculiar properties in the non-commutative case. These properties, already directly proved in previous works, can be, now, verified on use of a computer algebra program ad hoc devised [7] to verify the algebraic requirements which characterise recursion operators.