Nonlinear Evolution Equations: Bäcklund Transformations and Bäcklund Charts

Backlund Charts have been introduced to depict links, via Backlund Transformations, which relate different Nonlinear Evolution Equations. Notably, the links established among different Nonlinear Evolution Equations can be extended to the whole Hierarchies of Nonlinear Evolution Equations. This approach proved to be very fruitful, as well known, when applied to scalar Nonlinear Evolution Equations since it induces very many interesting results. Some of them are here reviewed and further new research perspectives are shown. Specifically, when the non commutative analogue Nonlinear Evolution Equations are introduced new problems arise. Here, hierarchies of non-commutative Nonlinear Evolution Equations together with their links via Backlund Transformations are considered. Specifically, a non-commutative analogues of Cole-Hopf and of Miura transformation are shown, respectively, to connect the operator versions of Burgers equation to heat equation and the operator version of KdV equation to the corresponding operator version of modified KdV equation. Again, the links are connecting not only the base member equations but all the corresponding equations in the hierarchy generated by the recursion operator it admits. Finally, it is pointed out how the method here presented allows to construct new non-commutative operator equation.