A novel noncommutative KdV-type equation, its recursion operator, and solitons

Abstract

A noncommutative KdV-type equation is introduced extending the Bäcklund chart in [4]. This equation, called meta-mKdV here, is linked by Cole-Hopf transformations to the two noncommutative versions of the mKdV equations listed in [22, Theorem 3.6]. For this meta-mKdV, and its mirror counterpart, recursion operators, hierarchies and an explicit solution class are derived. 1991 Mathematics Subject Classification. 35Q53; 46L55; 37K35.

Matrix Solitons Solutions of the Modified Korteweg–de Vries Equation

“…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 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. [6] and references therein. 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].…”

Abelian versus non-Abelian Bäcklund charts: Some remarks

“…Then, aiming to construct a an operator counterpart of the Bäcklund chart in [25], the previous Bäcklund chart is further extended in [15,17] together with the investigation of properties enjoyed by KdV-type non-Abelian equation and the corresponding hierarchies generated on application of the constructed recursion operators. Some of these hierarchies were already known in the literature, while some other ones are new.…”

“…Some of these hierarchies were already known in the literature, while some other ones are new. In particular, in [17], new non-commutative hierarchies are constructed on the basis of results in previous works [15] and by Athorne and Fordy [3]. These hierarchies, mirror to each other, when commutativity is assumed, reduce to the nonlinear equation for the KdV eigenfunction, (KdV eigenfunction equation) [38] which takes its origin in the early days of soliton theory when the inverse spectral transform (IST) method was introduced [50].…”

“…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.…”