Bäcklund Transformations and Non-Abelian Nonlinear Evolution Equations: a Novel Bäcklund Chart

Abstract

Abstract. Classes of third order non-Abelian evolution equations linked to that of Korteweg-de Vries-type are investigated and their connections represented in a non-commutative Bäcklund chart, generalizing results in [Fuchssteiner B., Carillo S., Phys. A 154 (1989), 467-510]. The recursion operators are shown to be hereditary, thereby allowing the results to be extended to hierarchies. The present study is devoted to operator nonlinear evolution equations: general results are presented. The implied applications referring to finite-dimensional cases will be considered separately.

Matrix Solitons Solutions of the Modified Korteweg–de Vries Equation

“…A very short outline on how this method can be adopted to construct solutions of the matrix mKdV equation is provided. Indeed, the study in [10], [7], [8], is devoted to study nonlinear evolution equations in which the unknown is an operator on a Banach space. The idea of the method can be sketched as follows:…”

“…So called soliton equations are widely investigated since the late 1960's when the first exact solution of the Korteweg-de Vries equation was obtained, as reported, for instance, in the well known book by Calogero and Degasperis [2]. The name soliton equations is generally used to indicate nonlinear evolution equations which admit exact solutions in the Schwartz space of smooth rapidly 6 termed also non-Abelian arXiv:1902.05823v3 [math-ph] 12 Jun 2019 decreasing functions 7 . The relevance of Bäcklund transformations in studying nonlinear evolution equations is well known both under the viewpoint of finding solutions to given initial boundary value problems, see [25] - [27] as well as in giving insight in the study of their structural properties, such as symmetry properties, admitted conserved quantities and Hamiltonian structure, see e.g.…”

“…The important role played by Bäcklund transformations to investigate properties enjoyed by nonlinear evolution equations is based on the fact that most of the properties of interest are preserved under Bäcklund transformations [16], [17]. Thus, the construction of a net of Bäcklund transformation to connect nonlinear evolution equations allowed to prove new results both in the case of scalar equations [3]- [5], [18]- [19], as well as in the generalised case of operator equations [7]- [12] 8 A comparison between the scalar (Abelian) and the operator (non-Abelian) cases referring to third order KdV-type equations, Bäcklund transformations connecting them and related properties is comprised in [9]. Notably, Bäcklund transformations indicate a way to construct solutions to nonlinear evolution equations [20] and also to nonlinear ordinary differential equations, [13] and [14].…”

Abelian versus non-Abelian Bäcklund charts: Some remarks

“…Specifically, to start with, the Bäcklund chart constructed in [10,9] is given. Then, the operator Bäcklund chart obtained in [12] and its extensions in [15,17], are considered. Finally, a comparison between the two different situations closes this section.…”

“…Thus, the Bäcklund chart in [10,9] is recalled in the subsection devoted to the classical commutative setting. The following subsection is concerned about the Bäcklund chart, constructed in [12] and later extended [15,17]. The last subsection is concerned about pointing out differences and analogies between the two cases.…”