B\"acklund Transformations and Non-Abelian Nonlinear Evolution Equations: a Novel B\"acklund Chart

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 B\"acklund 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.

One of the main advantages of connecting non-Abelian equations via Bäcklund transformations is that solutions can be transferred from one equation to another. In [11] operator-valued solutions (which can be interpreted as operator analogs of solitons) to the pKdV, KdV and mKdV hierarchies are constructed. In [13] suitable projection techniques are exploited to derive solution formulae to the corresponding scalar and matrix hierarchies. Note that the study of non-Abelian nonlinear evolution equations found its original interest in the case of matrix equations [6,36]. The results in the present article are valid for operator-valued functions. This level of generality permits to construct solutions to scalar and matrix equations that can be viewed as countable superposition of solitons, see also [50,53] for the connection between countable nonlinear superposition and Banach space geometry, and [51,52] for further applications.
In [10], the term Bäcklund chart was introduced to indicate the net of Bäcklund transformations connecting different evolution equations. In [23] a wide Bäcklund chart which includes scalar 3rd order Abelian evolution equations is constructed. It connects, further to the KdV and the modified KdV equations, in particular the KdV singularity manifold equation, also known as as UrKdV or Schwarz-KdV [18,55,57,58], and the KdV interacting soliton equation [22]. The connections among the equations in this Abelian Bäcklund chart are applied to find, or recover, the recursion operators admitted by all these nonlinear evolution equations. Moreover, other structural properties such as the Hamiltonian, and bi-Hamiltonian, structure of these equations are also obtained via the Bäcklund chart [23]. The present study is concerned about the extension of the Bäcklund chart in [11] to obtain the non-Abelian analog of the links established in the scalar case [23]. Precisely, in addition to the pKdV, KdV and mKdV equations already connected in [11], a different version of the non-Abelian modified KdV and two non-Abelian equations, respectively, analogs of the KdV singularity manifold and of the KdV interacting soliton equation, are all linked together via Bäcklund transformations. New, in the present Bäcklund chart, are the inclusion of a second modified KdV equation, denoted as amKdV, for alternative mKdV equation [34,44], new are also the non-Abelian KdV singularity manifold equation and the non-Abelian KdV interacting soliton equation.
All the non-Abelian nonlinear evolution equations in the Bäcklund chart admit a recursion operator. The recursion operators of some of them, such as the pKdV, KdV and mKdV [11,44,54] are known. We recover, via the established connections, the recursion operators admitted by the non-Abelian amKdV, given in [44]. Then, the recursion operators of the non-Abelian KdV singularity manifold and of the KdV interacting soliton equations, both new, are constructed. Furthermore, the hereditariness of all the obtained recursion operators is proved combining links via Bäcklund transformations [19,20], with the hereditariness of non-Abelian KdV recursion operator [54]. Finally, since all the nonlinear third order non-Abelian evolution equations admit hereditary recursion operators, according to [19,20], all the links in the Bäcklund chart can be extended to the corresponding whole hierarchies.
The material is organized as follows. The opening Section 2 is devoted to the Bäcklund charts connecting nonlinear evolution equations which generalize, to the operator level, the pKdV, KdV and mKdV equations [11,13]. Sections 3 and 4 are devoted to the construction of a novel Bäcklund chart. In Section 3, the Bäcklund chart is extended to incorporate also the non-Abelian amKdV equation; its recursion operator is constructed from the known recursion operators of the mKdV equations; finally, the connection between the two different non-Abelian modified KdV equations is provided.
In Section 4, the novel Bäcklund chart is further enlarged. Then, the recursion operators of the KdV interacting soliton and of the KdV singularity manifold equations are constructed. In addition, in Section 4, a Möbius type invariance exhibited by the non-Abelian KdV singularity manifold equation is established.
The subsequent Section 5 is devoted to the proof of hereditariness of all the recursion operators in the previous sections. The hereditariness of all the recursion operators in the Bäcklund chart guarantees that the links can be transferred to whole hierarchies, relating their corresponding members. In the closing Section 6, further to some remarks on the interest of the present study, open problems and perspectives suggested by our new results are briefly outlined. The article is complemented with an Appendix where some needed definitions, such as the definition of Bäcklund transformation [19], adopted throughout, as well as known results obtained in the case of Abelian nonlinear evolution equations [23] are comprised.

Non-Abelian Bäcklund charts
In [11], the recursion operators admitted by the non-Abelian potential KdV, KdV and modified KdV equations were shown to be related by Bäcklund transformations. Consider first the operator potential KdV equation (pKdV) where the unknown W is a function whose values are bounded linear operators on some Banach space 1 . It admits the recursion operator where D denotes the derivative with respect to x, and C T , A T denote the commutator and anti-commutator with respect to T , namely, For earlier occurrences of recursion operators in the non-Abelian setting we refer to [5,25,32,44]. Consider next the operator Korteweg-de Vries equation (KdV) which admits the recursion operator The recursion operators (2.2) and (2.4) are linked via the Bäcklund transformation In fact, the transformation operator is Hence, the non-Abelian pKdV (2.1) and KdV (2.3) equations can also be written as Finally, the KdV equation (2.3) is related to the operator modified KdV equation (mKdV) via the Miura transformation Hence, the Miura transformation (2.6) allows to obtain the recursion operator Ψ(V ) of the mKdV (2.5) from the recursion operator Φ(U ) of the KdV (2.4) via The latter implies [11] for details), which can be equivalently written as The following Bäcklund chart summarizes the links among the non-Abelian pKdV (2.1), KdV (2.3), and mKdV (2.5) equations: When the respective recursion operators are applied to the above equations iteratively, the Bäcklund chart can be extended to the corresponding hierarchies, and the connections can be summarized in

On the non-Abelian mKdV equations
A distinguished feature of the Bäcklund chart studied in this article is that it proceeds via two versions of the non-Abelian mKdV equation. The link between those versions is considered in this section. The alternative non-Abelian mKdV equation (amKdV) was first described in [34], where the Lax pair formulation and the inverse scattering problem were studied. In contrast to the non-Abelian mKdV (2.5) studied in the previous section, (3.1) does not admit a Miura transformation [44]. Note also that, in the matrix case, (2.5) is invariant under both V → V * and V → −V , whereas (3.1) is only invariant under V → − V * (V * denoting the transpose of V ).
As already pointed out in [37], these two versions of the mKdV equation are linked by the Obviously, subsequent application of these Bäcklund transformations links (2.5) to (3.1). The recursion operator Ψ( V ) of (3.1) is It was first given in [32], where it was derived using Lax representation. Here we use the Bäcklund link between the mKdV and the amKdV equation to give an alternative derivation together with a more conceptual formulation of the operator itself.
To this end, we introduce the derivation Theorem 3.1. On use of the derivation (3.5), the recursion operator (3.4) of the amKdV equation can be written as Remark 3.2. Recall that two operators T and S are called related if they are of the form T = AB and S = BA. In [11], relatedness of the non-Abelian KdV and mKdV recursion operators on the image of the Miura transform was exploited to derive solutions of the non-Abelian mKdV hierarchy from solutions of the non-Abelian KdV hierarchy.
Here we observe that Ψ(V ) and Ψ( V ) are related in a generalized sense: Before proving Theorem 3.1, we observe some crucial properties of the derivation (3.5).
Proof . As an example, we verify Analogously, Proof of Theorem 3.1. On use of the Bäcklund transformations B 2 , B 3 , the amKdV recursion operator Ψ( V ) is related to the mKdV recursion operator Ψ(V ) via Let L V and R V denote left and right multiplication by V , then This leads to the transformation operator Hence, the amKdV recursion operator Ψ( V ) is obtained from the recursion operator Ψ(V ) of the mKdV equation as stated in (2.7) via where the last step requires Proposition 3.3.
To conclude this section, for the reader's convenience we give a direct verification that the mKdV and the amKdV equation Finally, evaluation of D = D + C V gives the right-hand side of the amKdV equation.

A new non-Abelian Bäcklund chart
In the last two sections a non-Abelian generalization of the Bäcklund chart linking pKdV, KdV and mKdV equations was presented, and extended to include the amKdV equation (3.1). The aim of the present section is a further extension to also include non-Abelian analogues of the KdV singularity manifold equation and the KdV interacting soliton equation. To the best of the authors' knowledge, the non-Abelian equations presented in this sections are new. Together with the links presented in the previous section, the resulting non-Abelian Bäcklund chart 2 we obtain can be summarized as follows: Moreover, the corresponding recursion operators are derived. Note that this chart extends to the respective hierarchies. In conclusion, it is shown that all recursion operators are hereditary. Finally, it is well-known that the KdV singularity manifold equation is invariant under the full Möbius group, and it is shown that a generalized property holds for its non-Abelian counterpart.
To begin with, we introduce a non-Abelian analogue of the KdV singularity manifold equation (KdV sing.) where {φ; x} denotes the following non-Abelian version of the Schwarzian derivative Remark 4.1.
a) To the best of the authors' knowledge, equation (4.2) is new. For earlier non-Abelian versions of (4.2) we refer to [56], see also [2]. The difference to our approach is the non-Abelian interpretation of terms of the form 1/u 2 x . b) In the scalar case there is a very satisfying group theoretic explanation of the relation between the KdV sing. (also called UrKdV), mKdV and KdV equations [58], see also [18,55]. It seems to be an interesting problem to find an appropriate extension of these ideas to the non-Abelian setting. Departing from the present results, a first major difficulty is that our link between (2.5) and (3.1) (which reduce to the same scalar equation) is not realized by an explicit mapping V = F ( V ).
The KdV sing. equation (4.2) exhibits invariances analogous to its scalar counterpart. Note that the following theorem implies invariance under the full Möbius group in the scalar case.   Proof . To start with we calculate the Schwarzian derivative of ψ := φ −1 . Since and therefore All terms with exception of the boxed ones cancel out, hence {ψ; x} = φ{φ, x}φ −1 . As a result, which completes the proof.
The following invariance is straightforward to verify. Both the recursion operators Υ(φ) of (4.2) as well as χ(S) of (4.3) can be constructed on use of the Bäcklund links (4.4), (4.5).
Proof . Indeed, from the Bäcklund links we have that the recursion operators satisfy It is straightforward to calculate (B 4 ) V = 2L φx , (B 4 ) φ = −(D − 2R V )D. Therefore, we get Similarly (B 5 ) S = −I, (B 5 ) φ = D, and hence Π 5 = D. Consequently, Using the identity (D − A V )L S = L S (D + C V ), which can be checked directly using only the product rule and V = 1 2 S −1 S x , we find From this and Theorem 3.1, we get The claim for χ(S) follows from observing that Finally, Since S holds, and thus which is the claim for Υ(φ) upon substituting S = φ x .
In the scalar case, the recursion operator for the KdV sing. equation (4.2) is well-known, see, e.g., [23], where it appears in the form To facilitate comparison to the non-Abelian case, we state the following reformulation of (4.6).
Proof . The corollary is an immediate consequence of Proposition 4.10, the fact that N (φ x ) = V and V x = D V . Then the following factorization holds: Proof . The proof of the proposition is completely analogous to the proof of the factorization of the non-Abelian KdV recursion operator on the image of a Miura transformation in [11,Proposition 14]. The crucial observation is that due to the product rule the identities hold for any derivation D.
There are close algebraic relations between the non-Abelian recursion operators of the amKdV, KdV sing. and int. soliton KdV equations. where

Hereditariness and hierarchies
The links obtained in the last two sections are summarized in mKdV(V ) As known from the scalar case, hereditariness is a crucial property of recursion operators [20,38]. Unfortunately its direct verification often requires involved computations, in particular in the non-Abelian case. The following proposition uses the Bäcklund links established in the present article to avoid computations by reducing the proof of hereditariness to the hereditariness of the non-Abelian KdV recursion operator, which is proved in [54]. Proof . In [54, (32)] it is verified that the non-Abelian KdV recursion operator (2.4) satisfies the identity implying that Φ(U ) is a strong symmetry (recursion operator in the sense of [43]) for the trivial member U t = U x of the non-Abelian KdV hierarchy [20]. Moreover, the main result in [54] is that Φ(U ) is hereditary. Hence Φ(U ) is a strong symmetry for all equations of the non-Abelian KdV hierarchy [20]. As shown in [26], the properties of a) and b) are preserved under Bäcklund transformations.
Each of the hierarchies in (5.1) is of the form We may rewrite the right-hand side of (5.2) as where X n is a vector field on the space of x-dependent operator-valued functions (see [54] for details). The main consequence of Proposition 5.1 is

Remarks, perspectives and open problems
This section collects some remarks on the results previously presented together with some perspectives study and open problems. The chain of Bäcklund transformations we obtained, represents a not at all trivial generalization to the operator level of the corresponding one [23] which links the scalar pKdV, KdV, mKdV, KdV interacting soliton and KdV singularity manifold hierarchies. Furthermore, it generalizes the non-Abelian Bäcklund chart in [11] since it connects further non-Abelian hierarchies. Specifically, new non-Abelian equations, and, then, the corresponding hierarchies, arise, such as the amKdV, in (3.1).

Remarks
• The two Bäcklund charts, respectively, in the Abelian [23] (see Appendix) and the non Abelian case, connect the pKdV, KdV, mKdV, KdV-singularity manifold and Interacting Soliton KdV equations or their non-Abelian analogs.
• Known the recursion operator of a nonlinear evolution equation then all the other nonlinear evolution equations linked to it via a Bäcklund chart admit a recursion operator. The latter can be constructed in the Abelian as well as in the non-Abelian case.
• All nonlinear evolution equations in the same Bäcklund chart share all the structural properties which are preserved under Bäcklund transformations as soon as a single equation, in it, enjoys them. Remarkable is the hereditariness of the recursion operators [54].
• Also in the non-Abelian operator case, given the hereditary recursion operators, the Bäcklund chart can be extended to the corresponding generated hierarchies. Again, the Bäcklund chart relates the corresponding members of each one of the involved hierarchies of nonlinear evolution equations.
• Even if there are similarities between the Abelian scalar case and the non-Abelian operator case, in the second case the structure is richer. Indeed, two distinct non-Abelian mKdV equations appear: the non-Abelian mKdV and an alternative mKdV equations which do coincide in the Abelian case. Correspondingly, when commutativity is assumed, combination of the Bäcklund transformations B 2 and B 3 produces the identity transformation and, hence, the Abelian Bäcklund chart [23] is recovered.
• A similar behavior can be observed also when the Cole-Hopf link connecting Burgers equation to linear heat equation is extended to the non-Abelian case [12,14]. The heat equation is connected to two different Burgers equations, termed Burgers and mirror Burgers equations in [9,35]. Then, recursion operators and the corresponding hierarchies follow from the Cole-Hopf link, which can be regarded as a special case of Bäcklund transformation.
Perspectives and open problems • We expect a similar situation when the 5th order nonlinear evolution equations which appear in the Bäcklund chart in [8,47] are extended to the non-Abelian case. This study is currently under investigation and we are devising also computer aided routines to check the algebraic properties of the recursion operators. Indeed, already in the Abelian case, the computations involved are very long and complicated.
• Furthermore, if a nonlinear evolution equation admits a Hamiltonian and bi-Hamiltonian structure, related to the recursion operator [21,24,27,38], then all nonlinear evolution equations in the same Bäcklund chart admit a Hamiltonian and bi-Hamiltonian structure.
• The approach can be extended also when nonlinear evolution equations in (2 + 1) dimensions are considered, namely, the unknown function is supposed to depend on two space variables rather than on a single space variable. Thus, in [40], the Kadomtsev-Petviashvili (KP), modified Kadomtsev-Petviashvili (mKP) and (2 + 1)-dimensional Harry Dym equations, which represent, in turn, the (2 + 1)-dimensional analog of KdV, mKdV, and Harry Dym equations are all connected via Bäcklund transformations. Notably, the connection among their (1 + 1)-dimensional corresponding equations [41] follows on imposing suitable constraints to the (2 + 1)-dimensional ones.

A Appendix
In this Appendix some background definitions which are of use throughout the whole article are briefly recalled in the opening subsection: this choice is due to the lack of uniqueness in many definitions in the literature. In the following subsection, some results strictly connected to the present investigation are retrieved.

A.1 Background notions
Here, to improve readability, some fundamental definitions are provided. First of all, we recall the notion of Bäcklund transformation, according to the definition of Fokas and Fuchssteiner [19] (see also the book by Rogers and Shadwick [49]). Consider non linear evolution equations of the type where the unknown function u depends on the independent variables x, t and, for fixed t, u(x, t) ∈ M , a manifold modeled on a linear topological space so that the generic fiber T u M , at u ∈ M , can be identified with M itself 3 , and K : M → T M , is an appropriate C ∞ vector field on a manifold M to its tangent manifold T M . Let, now As usual, when soliton solutions are considered, the further assumption M := M 1 ≡ M 2 is adopted. Then, the definition of Bäcklund transformation, according to [19], can be stated as follows.
Definition A.1. Given two evolution equations, u t = K(u) and v t = G(v), then B(u, v) = 0 represents a Bäcklund transformation between them if, whenever two solutions of these equations are given, let denote them u(x, t) and v(x, t) respectively, such that B(u, v) = 0 at the initial time t = 0, then this holds true for all times, namely, Therefore, the Bäcklund transformation B establishes a correspondence between solutions of the evolution equations it connects. This link is graphically represented as and, then, if the nonlinear evolution equation u t = K(u) admits a hereditary recursion operator Φ(u) [19], namely, The invariance, under the Möbius group of transformations of all the members of the KdV Singularity hierarchy allowed to recover the Invariance I, enjoyed by the Dym hierarchy [23] and to find a new invariance enjoyed by the Kawamoto equation [47] in the case of the 5th order Bäcklund chart.