Ricerc@Sapienza

Third order nonlinear evolution equations, that is the Korteweg–de Vries (KdV), modified Korteweg–de Vries (mKdV) equation and other ones are considered: they all are connected via Bäcklund transformations. These links can be depicted in a wide Bäcklund Chart which further extends the previous one constructed in [22]. In particular, the Bäcklund transformation which links the mKdV equation to the KdV singularity manifold equation is reconsidered and the nonlinear equation for the KdV eigenfunction is shown to be linked to all the equations in the previously constructed Bäcklund Chart.

Connections via B ̈acklund transformations among different non- linear evolution equations are investigated aiming to compare corresponding Abelian and non Abelian results. Specifically, links, via B ̈acklund transfor- mations, connecting Burgers and KdV-type hierarchies of nonlinear evolution equations are studied. Crucial differences as well as notable similarities be- tween Ba ̈cklund charts in the case of the Burgers - heat equation, on one side and KdV -type equations are considered.