Abelian versus non-Abelian Baecklund Charts: some remarks

Connections via Baecklund transformations among different non-linear evolution equations are investigated aiming to compare corresponding Abelian and non Abelian results. Specifically, links, via Baecklund transformations, connecting Burgers and KdV-type hierarchies of nonlinear evolution equations are studied. Crucial differences as well as notable similarities between Baecklund charts in the case of the Burgers - heat equation, on one side and KdV -type equations are considered. The Baecklund charts constructed in [16] and [17], respectively, to connect Burgers and KdV-type hierarchies of operator nonlinear evolution equations show that the structures, in the non-commutative cases, are richer than the corresponding commutative ones.


Introduction
Bäcklund transformations in soliton theory are well known to represent a key tool both for finding solutions to applicative problems as well as for investigating symmetry and structural properties admitted by nonlinear evolution equations, see [1,7,59,56,60,31] where Bäcklund and Darboux Transformations and applications to partial differential equations admitting soliton solutions are studied. The focus of the results in this study is on Bäcklund transformations as a tool to connect different nonlinear evolution equations both in the Abelian as well as in the non-Abelian setting. Noncommutative nonlinear equations are considered wherein the unknown is an operator on a Banach space. The investigation was initiated by Marchenko [45] and, later, further developped in [2,20]. The obtained results are part of a research project which on operator equations which takes its origin in [12] wherein, further to other results, connections, via Bäcklund transformation, among operator potential Korteweg-de Vries (pKdV), Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations are studied and extended to the corresponding hierarchies. Subsequent developments are comprised in [13] where the special case in which the operator unknowns can be represented as matrices, is considered; solutions are also constructed. Indeed, a great interest on non-Abelian nonlinear evolution equations is testified by a wide bibliography, see [54,42,40,39,34,35].
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. 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]. It was later studied in [38,46,67] together with many further nonlinear evolution equations. Notably, in the commutative case, the Dym equation is connected to all the other KdV-type nonlinear evolution equations, via reciprocal transformation [58,25]. Indeed, reciprocal transformations, according to the pioneering work [61], are closely connected to the IST method. Investigations aiming to classify integrable nonlinear evolution equations are comprised in [4,5,47,48,68].
The material is organised as follows. The opening Section 2 provides the definition of Bäcklund transformation and few properties needed throughout the subsequent parts. Section 3.1 is devoted to connections involving the Burgers equation, both in the Abelian as well as in the non-Abelian case.
The three subsections are concerned, in turn, to the well known commutative Burgers equation where some of its properties are recalled, to the two non-commutative counterparts of Burgers equation [11,14,16] and the last subsection points out differences and similarities between the two Abelian and non-Abelian cases. In the next Section 4 the attention is focussed on the wide Bäcklund charts involving KdV-type equations. Also Section 4 is organised in a way similar to the previous one. 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.

Background notions
Some background notions required in the presented study, are collected in this section. A wide literature, such as the books [1,7,59,56,60,31], to restrict the list to those ones more closely connected with the present investigation, are concerned about Bäcklund transformations and their applications to soliton equations. In this section, for sake of brevity, only definitions, not uniquely given in the literature, are shortly recalled. In particular, the definition of Bäcklund Transformation given by Fuchssteiner [23] and by Fokas and Fuchssteiner [22], is adopted. Consider two non linear evolution equations where the unknown functions u, v depend both on the independent variables x, t and, for fixed t, u(x, t) ∈ M , which denotes a smooth manifold, then its generic fiber T u M , at u ∈ M , can be identified with M itself. In addition, when soliton solutions are considered, M is assumed to coincide, for each fixed t, with the Schwartz space S 1 of rapidly decreasing functions on R I n . In detail, where M := M 1 ≡ M 2 , is usually assumed. Then, [22] a Bäcklund transformation can be defined as follows.
Definition Given two evolution equations, u t = K(u) and v t = G(v), then B (u , v) = 0 represents a Bäcklund transformation between them whenever given two solutions of such equations, respectively, u(x, t) and v(x, t), such that B(u(x, t), v(x, t))| t=0 = 0 (2.4) then, it follows The connection between solutions of the two equations via the Bäcklund transformation B can be graphicallly represented as Among the many properties, note that hereditariness is preserved under Bäcklund transformations. That is, according to [22,53], let Φ(u) denote the hereditary recursion operator admitted by one of the nonlinear evolution equations (2.1), say u t = K(u), so that it can be written as then, also the other equation (2.1) admits a hereditary recursion operator which can be constructed via the Bäcklund transformation and the recursion operator Φ(u). Specifically, the recursion operator Ψ(v) admitted by the second equation is given by and B u and B v denote the Frechet derivatives of the Bäcklund transformation B(u, v). Then, according to [22], on subsequent applications of the admitted recursion operators, the result can be extendeed to the two generated hierarchies [23] their base members equations, which correspond to n = 1, coincide with equations (2.1). Fixed any m ∈ N, the two equations ] m are connected via the same Bäcklund Transformation which connects the two base members equations. This extension is graphically represented, ∀ n ∈ N, as

Burgers Bäcklund charts
This section is devoted to the Burgers equation. It represents the most simple example of nonlinear evolution equation whose linearisation via a Bäcklund transformation, the well known Cole-Hopf transformation [21,36], represented a crucial step. Indeed, the Cole-Hopf transformation opened the way to the study of many applicative problems, such as initial boundary value problems, see for instance [29,6]. This section is divided in three different subsections which, respectively, are concerned about the Abelian, the non-Abelian Burgers equation and, finally, a comparison between some properties in the two cases. Specifically, the first subsection, provides a brief survey of some properties enjoyed by the Burgers equation and the corresponding hierarchy, in the Abelian framework; accordingly, the unknown is assumed to be a real valued function 2 .
The next subsection is concerned about the non-Abelian Burgers equations and the corresponding hierarchies [11,14,16]. The first appearence of a non-commutative Burgers hierarchies is due to Levi, Ragnisco and Bruschi [42] who were concerned about matrix equations. The non-commutative Burgers equations and their properties were studied by Kupershmidt in [40], the hereditariness of the recursion operator of the non-commutative Burgers equation in [11] was proved in [14]. The same recursion operator was independently obtained, via a Lax pair method in [32], by Gürses, Karasu and Turhan [33]. Recently, in [16], where a comparison with the mentioned results is provided, also the mirror non-Abelian Burgers equation, together with the admitted recursion operator and, hence, the corresponding hierarchy are constructed.
In the last subsection there is a concise comparison between the two different cases Abelian versus non-Abelian.

Abelian Bäcklund chart
It is linked via the Cole-Hopf transformation [21,36], which can be represented as the special Bäcklund transformation to the linear heat equation The latter admits the (trivial) hereditary recursion operator Φ(u) = D, where D denotes derivation with respect to the spatial variable x. Hence, the heat equation reads and the corresponding hierarchy, respectively, can be written as As well known [23] the link (3.2) between the heat (3.3) and the Burgers (3.1) equations can be depicted via the following Bäcklund chart The link via Cole-Hopf transformation [21,36], on one side, allows to construct solutions of assigned initial boundary value problems modelled via Burgers equation, as testified by many results among which also [29,6]. On the other side, indicates how to obtain, on application of formula (2.8), from the recursion operator of the heat equation, the Burgers recursion operator Furthermore, the Bäcklund chart can extended to the corresponding hierarchies according to (2.11), wherein Φ(u) = D and Ψ(v) is given by (3.7), that is As a consequence, according to the wide literature, see for instance [7,23], also the infinitely many admitted conserved quantities can be constructed.

non-Abelian Bäcklund chart
This subsection is concerned about the non-Abelian Burgers equations which both represent the non-Abelian counterpart of the Abelian Burgers equation. For notational convenience, from here on, operator unknown are denoted via capital letters while lower case letters are used referring to real valued ones so that, for instance, the unknown functions u, in the linear heat and v in the Burgers equations in Subsection 3.1 are denoted by lower case letters, while, in the present subsection capital case letters are used since operator equations are considered. Consider the linear heat equation where, now, the unknown is an operator according to the general theory in [2,20,14]. The two different Cole-Hopf transformations [16], namely the and its mirror give, respectively, the non-Abelian Burgers equation 12) and the non-Abelian Burgers mirror equation The recursion operator admitted by the heat equation is represented by the trivial operator Φ(U ) = D, the non-Abelian Burgers and the mirror non-Abelian Burgers equations [14,16] are, respectively, the operators Ψ(S) and Φ(R) which, when L W and R W denote, in turn, left and right multiplication by W for any W , are given as follows which can also be written as [14] Ψ The corresponding hierarchies, namely, in turn, the heat, the non-Abelian Burgers and the mirror non-Abelian Burgers hierarchies are represented as The lowest order members of the first two hierarchies are, respectively (3.18) while, the first members of the mirror non-Abelian Burgers hierarchy, are Notably, in contrast with the non-Abelian Burgers hierarchy, each member of its mirror hierarchy can be obtained from the corresponding one by substitution of right multiplication by S with left multiplication by R.
The links among them can be summarised in the following Bäcklund chart which, again, can be extended to the whole corresponding hierarchies.

non-Abelian versus Abelian Bäcklund charts
This brief subsection is concerned about differences and similarities between the Abelian and non-Abelian Burgers hierarchies. Indeed, there are two different non-Abelian transformations which generalise the Abelian Cole-Hopf transformation. Hence, when the linear heat equation is considered, depending on whether the unknown is a real valued function, say scalar for simplicity, the usual (Abelian) Burgers equation (3.1) is obtained. Conversely, if the unknown in the linear heat equation is an operator, then the two different non-Abelian equations (3.12) and (3.13) are obtained. On the other hand, the Cole-Hopf transformation which links the Burgers to the heat equation, both in the non-Abelian as well as in the Abelian case, allows to construct the recursion operator (3.7) admitted by the (commutative) Burgers equation, or the two different recursion operators, in turn (3.14) and (3.16), admitted by the (non-commutative) Burgers and by its mirror equation, in the non-commutative case. Furthermore, the form of the latter ones is more complicated, however, as expected, when commutativity is assumed, both the Burgers and mirror Burgers recursion operators reduce to the usual (commutative) Burgers recursion operator. These operator allow to generate two hierarchies of nonlinear evolution equations: they are both the same ones obtained by Kupershmidt [40] who construct them via a recursive definition of the hierarchies themselves. Also Gürses, Karasu and Turhan [33] obtained the same recursion operator Ψ(S) and the hierarchy it generates via a method [32] based on the Lax pair formulation. The direct proof the hereditary property enjoyed by the recursion operators, in [16], requires more involved computation and ad hoc routines when proved via computer assisted computations to take into account the non-commutativity of products.
The following Section 4 shows that non-Abelian KdV-type equations are characterised by an even richer structure.

KdV-type Bäcklund charts
This Section is devoted to the two different Bäcklund charts connecting KdVtype equations, in turn, in the commutative case and in the non-commutative one. 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.

Abelian Bäcklund chart
In this subsection, the links among the various scalar nonlinear evolution equations are depicted in the Bäcklund chart in [9] wherein, inspired by the non-commutative results in [17], the previous Bäcklund chart constructed in [25] is extended to include also the KdV eigenfunction equation. The links, up to this stage, are summarised in the following Bäcklund chart. The third order nonlinear evolution equations in the Bäcklund chart are, respectively The Bäcklund transformations linking these equations among them, following the order in the Bäcklund chart itself, are: The latter exchanges the role played by the dependent and independent variables between them; hence,x =x(s, x) and ρ(x) := ρ(x(s, x)). Details on the reciprocal transformation (e) are analysed in [10,25] while a general introduction on reciprocal transformations together with various applications are illustrated in [59]. Notably, the Bäcklund chart represents a useful tool to reveal the nonlinear evolution equations it links enjoy invariances, which might be new or already known. In particular, the invariance M under the Möbius group of transformations exhibited by the KdV singularity manifold equation (KdV sing.), allows to further extend the Bäcklund chart as indicated in the following Fig.3. Indeed, the Möbius invariance allows to obtain the auto-Bäcklund --Dym(ρ) Figure 3: Abelian KdV-type hierarchies Bäcklund chart: induced invariances.
transformations AB k , k = 1 . . . 5. Thus, AB 1 and AB 2 are the well known auto-Bäcklund transformations admitted by the KdV and the mKdV hierarchies [49,7,25]. The auto-Bäcklund transformation AB 3 ≡ denotes an invariance enjoyed by the KdV eingenfunction equation [9]. Finally, AB 4 , and AB 5 , respectively, are auto-Bäcklund transformations, enjoyed by, respectively, the int. sol. KdV and Dym equations, according to [25]. Notably, the constructed Bäcklund chart not only can be can be extended in 2 + 1dimensions [51] but also can be regarded as a constrained version of such an extension [52]. Hence, both in 1 + 1 [27,30] as well as in 2 + 1 dimensions [55], solutions to Dym equations problems can be obtained. Furthermore, according to [25], the Hamiltonian and bi-Hamiltonian structure of all the nonlinear evolution equations in the Bäcklund chart can be constructed from those ones admitted by the KdV equation [44,24,26,28]. As discussed in [8], a Bäcklund chart connecting the Caudrey-Dodd-Gibbon-Sawata-Kotera and Kaup-Kupershmidt hierarchies [19,62,37] whose base member is a 5th order nonlinear evolution equations similar to the one connecting KdV-type equations can be constructed [57,10]. All the links in the Bäcklund chart, in Fig. 3, are valid for the whole hierarchies of nonlinear evolution equations generated via application of the admitted recursion operators. Hence, each element in the Fig. 3 can be interpreted as the whole corresponding hierarchy. The recursion operator, respectively, are

non-Abelian Bäcklund chart
This subsection is concerned about the Bäcklund chart connecting operator KdV-type equations. Accordingly, the results comprised in [12,15,17], are summarised in the following picture.
In Fig. 4, the third order nonlinear operator evolution equations 3 are, in turn, while the Bäcklund transformations linking the KdV with the mKdV, the amKdV with the int.sol KdV and the latter with the KdV sing. are, respectively Again, all the nonlinear evolution equations in the Bäcklund chart admit a recursion operator; hence, the Bäcklund chart can be extended to the whole corresponding hierarchies. Indeed, the recursion operator themselves can be constructed on application of formula (2.8) according to [22]. In this, way, in [15,17] the recursion operators of the newly inserted nonlinear evolution equations in the Bäcklund chart are obtained. The recursion operators, in turn are the following ones.
where D := D + C φ , C W := W, · , A W := W, · any arbitrary W , (4.5) All the recursion operators are hereditary [12,15,17], therefore, all the links in Fig. 4 hold true for the whole corresponding hierarchies. Furthermore, known invariances allow to prove new ones. This is the case of the non-Abelian Möbius invariance enjoyed by the non-Abelian KdV sing. equation and by the corresponding hierarchy which allow to prove nontrivial invariances admitted by hierarchies of nonlinear evolution equations in the same Bäcklund chart [17]. In particular, invariances enjoyed by the meta-mKdV and mirror meta-mKdV hierarchies can be proved [17].

non-Abelian versus Abelian Bäcklund charts
This subsection collects few remarks on the comparison between the Abelian and non-Abelian Bäcklund charts connecting KdV-type equations. They, respectively, are depicted in Fig. 2 and in Fig. 4. Notably, in the noncommutative case two different equations, namely meta-mKdV and mirror meta-mKdV, represent both the non-commutative counterpart of the nonlinear equation for the KdV eigenfunction, termed, for short, KdV eigenfunction equation The latter, inserted in the commutative Bäcklund chart in [9], finds its meaning in connection with the Lax pair formulation of the KdV equation [38,46,50,67]. Conversely, the non-commutative equations, denoted as meta-mKdV and mirror meta-mKdV, are introduced in the non-commutative Bäcklund chart in [17]. They both represent novel nonlinear evolution equations. Both these equations admit ahereditary recursion operator, constructed in [17] together with soliton solutions. Notably, the hereditariness of the admitted recursion operator follows from the hereditariness of the KdV recursion operator [66]. Furthermore, when commutativity is assumed, the mKdV and amKdV equations coincide, hence the Bäcklund chart as Fig. 4 reduces to the Abelian Bäcklund chart in Fig. 3 without the Dym equation.
On the other hand, also in the non-Abelian case, the invariance under the Möbius group of transformations, suitably defined, allows to double the Bäcklund chart. Hence, in both commutative as well as non-commutative cases new and/or well known invariances [17] exhibited by the nonlinear evolution equations which appear the Bäcklund chart can be obtained, or, if already known, recovered.
Furthermore, it should be noted that the Dym hierarchy is inserted in the Abelian Bäcklund chart. Conversely, at the moment, a non-commutative counterpart the reciprocal transformation remains to be investigated. Nevertheless, solutions to non-commutative problems can be obtained [63,64,65,18].