Abelian versus non-Abelian Bäcklund charts : Some remarks

Connections via Backlund transformations among different nonlinear evolution equations are investigated aiming to compare corresponding Abelian and non Abelian results. Specifically, links, via Backlund transformations, connecting Burgers and KdV-type hierarchies of nonlinear evolution equations are studied. Crucial differences as well as notable similarities between Backlund charts in the case of the Burgers -heat equation, on one side, and KdV-type equations, on the other, are considered. The Backlund 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.