Bäcklund transformations and non-Abelian nonlinear evolution equations

Abstract: Bäcklund transformations are well known to represent a powerful tool in investigating nonlinear differential equations. In particular, we are concerned about so-called soliton equations since they admit soliton type solutions. The aim of the present study is twofold since, on one side, we consider the connections which can be established and the induced structural properties; on the other side, we consider Bäcklund transformations as a tool to construct solutions, admitted by nonlinear evolution equations. Hence, first of all, we consider the links which can be established among different nonlinear evolution equations via Bäcklund transformations. Accordingly, a net of connections among different nonlinear evolution equations is depicted in a Bäcklund Chart, as we term such a net of links. The attention is focussed on third order, nonlinear evolution questions in particular, the comparison between the commutative (Abelian) and the non-commutative cases is analyzed. Notably, a richer structure can be observed when the commutativity condition is removed. Then, via Bäcklund transformations, solutions of matrix modified KdV equation can be constructed. Finally, some new results as well as some problems, currently under investigation, concerning fifth order nonlinear evolution equations are mentioned. Most of the presented results are part of a joint research project with Cornelia Schiebold, Sundsvall University, Sweden which involves also, in alphabetical order, M. Lo Schiavo, Rome, E. Porten, Sundsvall, and F. Zullo, Brescia.

mathematical physics

Audience: researchers in the topic

Seminar-Type Workshop on Noncommutative Integrable Systems