Acta Applicandae Mathematicae

, Volume 122, Issue 1, pp 93–106

Nonlinear Evolution Equations: Bäcklund Transformations and Bäcklund Charts

Article

DOI: 10.1007/s10440-012-9729-8

Cite this article as:
Carillo, S. Acta Appl Math (2012) 122: 93. doi:10.1007/s10440-012-9729-8

Abstract

Bäcklund Charts have been introduced to depict links, via Bäcklund 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 Bäcklund 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.

Keywords

Bäcklund transformationsBäcklund chartNon-commutative evolution equationsHierarchies of non-commutative evolution equations

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Dipartimento di Scienze di Base e Applicate per l’Ingegneria, sez. MatematicaSapienza Università di RomaRomeItaly