Symmetry, Integrability and Geometry: Methods and Applications Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions

We express AKNS hierarchies, admitting reductions to matrix NLS and matrix mKdV hierarchies, in terms of a bidifferential graded algebra. Application of a universal result in this framework quickly generates an infinite family of exact solutions, including e.g. the matrix solitons in the focusing NLS case. Exploiting a general Miura transformation, we recover the generalized Heisenberg magnet hierarchy and establish a corresponding solution formula for it. Simply by exchanging the roles of the two derivations of the bidifferential graded algebra, we recover"negative flows", leading to an extension of the respective hierarchy. In this way we also meet a matrix and vector version of the short pulse equation and also the sine-Gordon equation. For these equations corresponding solution formulas are also derived. In all these cases the solutions are parametrized in terms of matrix data that have to satisfy a certain Sylvester equation.


Introduction
A unification of some integrability aspects and solution generating techniques has recently been achieved for a wide class of "integrable" partial differential or difference equations (PDEs) in the framework of bidifferential graded algebras [1]. The hurdle to take is to find a bidifferential calculus (i.e. bidifferential graded algebra) associated with the respective PDE. In particular, a surprisingly simple result (Theorem 3.1 in [1] and Theorem 1 below) then generates a (typically large) class of exact solutions. This has been elaborated in detail for matrix NLS systems in a recent work [2]. The present work extends some of these results to a corresponding hierarchy and moreover to related hierarchies. It demonstrates how to deal with whole hierarchies instead of only single equations or systems in the bidifferential calculus approach and shows moreover that certain relations between hierarchies find a nice explanation in this framework. Except for certain specializations, we deal with "non-commutative equations", i.e. we treat the dependent variables as non-commutative matrices, and the solution formulas that we present respect this fact.
In Section 2 we introduce some basic structures needed in the sequel. Section 3 presents a bidifferential calculus for a matrix AKNS hierarchy. We derive a class of solutions of the latter and address some reductions of the hierarchy. In Sections 4 and 5 we show that, in the bidifferential calculus framework, a "reciprocal" [3] or "negative" extension of the hierarchy naturally appears. "Negative flows" have been considered previously via negative powers of a recursion operator (see e.g. [4,5,6,7] and also [8,9,10,11] for other aspects). In our picture, these rather emerge as "mixed equations", bridging between the ordinary hierarchy and a "purely negative" counterpart. Section 6 elaborates this program for a "dual hierarchy". Here we recover in the bidifferential calculus framework in particular a well-known duality or gauge equivalence between the (matrix) NLS and (generalized) Heisenberg magnet hierarchies [12,13,14,15,16,17,18,19]. Section 7 contains some concluding remarks.
2 Basic structures Definition 1. A graded algebra is an associative algebra Ω over C with a direct sum decomposition Ω = r≥0 Ω r into a subalgebra A = Ω 0 and A-bimodules Ω r , such that Ω r Ω s ⊆ Ω r+s .
For any algebra A, a corresponding graded algebra is given by where (C N ) denotes the exterior algebra of C N , N > 1. Defining graded derivations d,d on A, they extend in an obvious way to Ω such that the Leibniz rule holds and elements of (C N ) are treated as constants with respect to d andd. Given a bidifferential calculus, it turns out that the equation has various integrability properties [1,2]. By choosing a suitable bidifferential calculus, this equation covers in particular the familiar selfdual Yang-Mills equation (in one of its gaugereduced potential versions), but also e.g. discrete integrable equations [1,2]. In the next section we demonstrate that, by choosing an appropriate bidifferential calculus, (

AKNS hierarchies
Let B 0 be the algebra of complex smooth functions of independent variables t 1 , t 2 , t 3 , . . ., B an extension by certain operators (specified below), and A = Mat(m, m, B), m > 1. Let m = m 1 + m 2 with m i ∈ N, and let P be the projection where I m denotes the m × m identity matrix. If the dimension is obvious from the context, we will simply denote it by I. It will also be convenient to introduce the matrix

NLS system
A particular bidifferential calculus on A is determined by where ζ 1 , ζ 2 is a basis of 1 (C 2 ), and we set x = t 1 . Here B is the extension of B 0 by the partial derivative operator ∂ x . Evaluation of (2.2) yields 3) using the familiar notation for commutator and anti-commutator. The block-decomposition together with where we set a "constant" of integration to zero. As a consequence of the form of P, there is no equation forp. Though at this point we could simply set it to zero, this would be inconsistent with further methods used in this work (cf. Remark 3).
1 Introducing B = [dg −(dg)∆]g −1 , this equation reads dB = 0, and as a consequence, taking alsod∆ = (d∆)∆ into account, we find thatdB = B ∧ B. If, as in familiar cases, this (partial) zero curvature equation implies B = (dg ′ )g ′−1 , then (2.4) is gauge-equivalent to d[(dg ′ )g ′−1 ] = 0, so that the term involving ∆ in (2.4) can be generated by a gauge transformation. It is nevertheless helpful to consider the modified equation (2.4) in order to accommodate more easily certain examples of integrable equations in this formalism [1].

Extension to a hierarchy
Another bidifferential calculus on A is determined by (3.7) Here E λ and E µ are commuting invertible operators, which also commute with P, and B is the extension of B 0 by these operators. Introducing (3.8) In the following, E λ will be chosen as the Miwa shift operator, hence with an arbitrary constant λ (see e.g. [25]). The generating equation 3 (3.8) already appeared in [25] and we recall some consequences from this reference. Expanding (3.8) in powers of the arbitrary constants (indeterminates) λ and µ, we recover (3.3) as the coefficient of λ 1 µ 0 . Decomposing the matrix φ into blocks according to (3.4), this results in and Again, there is no equation forp. Since the last equations separate with respect to λ and µ, they imply Multiplying the first equation from the right byq −[λ] , the second from the left by q, using (3.6) and adding the resulting equations, we obtain If we set integration constants to zero, this leads to (3.14) Using this equation, (3.12) becomes which are generating equations for a hierarchy that contains the (matrix) NLS system. Together with (3.6), this leads to which is a generating equation for the potential KP hierarchy [26,25] 4 . If we think of p as determined via (3.6) in terms of q andq, then the last equation is a consequence of (3.15). Furthermore, the two equations (3.12) are the linear respectively adjoint linear system of the KP hierarchy (cf. [27]) in the form of generating equations. Let us recall that For n = 2 we recover (3.5). For n = 3, and after elimination of t 2 -derivatives using (3.5), we obtain the system which admits reductions to matrix KdV and matrix mKdV equations (see also e.g. [21,22,28]). In the same way, any pair of equations in (3.18) can be expressed in the form q tn = Q n (q,q, q x ,q x , . . . , q x n ,q x n ),q tn =Q n (q,q, q x ,q x , . . . , q x n ,q x n ), (3.20) by use of the equations for q t k ,q t k , with k = 2, . . . , n − 1. For n = 4, we find Remark 1. Expansion of (3.7) leads to Selecting the terms with the same powers of λ and µ, this suggests to define For any choice of non-negative integers m, n, this determines a bidifferential calculus. With m = 0 and n = 1, we recover (3.2). for an m × m matrix Ψ, where ν is a constant (cf. [1]). If Ψ is invertible, this is (2.3) with ∆ = νI. Evaluation for the above bidifferential calculus leads to In terms of ψ given by this takes the form which is a generating equation for all Lax pairs of the hierarchy. Expanding in powers of λ, the first two members of this family of linear equations are This constitutes a Lax pair for the NLS system (3.5). In order to obtain a more common Lax pair for the NLS system, we have to eliminate p x via (3.6) and add a constant times the identity matrix to L and M , together with a redefinition of the "spectral parameter". See also [2].

A class of solutions
So far we defined a bidifferential calculus on Mat(m, m, B). In the following we need to extend it to a larger algebra. The space of all matrices over B, with size greater or equal to that of n 0 × n 0 matrices, attains the structure of a complex algebra A with the usual matrix product extended trivially by setting AB = 0 whenever the sizes of A and B do not match. For the example introduced in the preceding subsection, we set n 0 = 2. Let Ω be the corresponding graded algebra (2.1). For each n ≥ 2 and a split n = n 1 + n 2 , we choose a projection matrix P (n 1 ,n 2 ) of the form (3.1). Then we can extend the bidifferential calculus defined in (3.7) to A by simply defining the commutators appearing there appropriately, e.g. for an n × m matrix f we set For the following general result, see [1,2]. Theorem 1. Let (Ω, d,d) be a bidifferential calculus with Ω = A ⊗ (C N ) and A = Mat n 0 (B), for some n 0 ∈ N. For fixed n ≥ n 0 , let X, Y ∈ Mat(n, n, B) be solutions of the linear equations dX = (dX)S,dY = (dY )S, with some m × m matrix θ. By application of d, this then implies that φ solves (2.2).
Now we apply this theorem to the bidifferential calculus associated with the hierarchy introduced in Section 3. We fix n 1 , n 2 , and write P for P (n 1 ,n 2 ) . The linear equationdX = (dX)S is then equivalent to d andd-constancy of S means that S is constant in the usual sense (i.e. does not depend on the independent variables t 1 , t 2 , . . .) and satisfies [P, S] = 0, which restricts it to a block-diagonal matrix, i.e. S = block-diag(S 1 , S 2 ). Decomposing X = X d + X o into a block-diagonal and an off-block-diagonal part, and using [P, whereP denotes the projection P − J . The first equation implies that X d is constant. We write X d = A d . Noting that PS andPS commute, the solution of the second equation is A corresponding expression holds for Y ,

Now (3.22) splits into the two parts
Assuming that A d is invertible, we can solve the first of these equations for R and use the resulting formula to eliminate R from the second. This results in Using the identity ( and All this leads to the following result, which generalizes Proposition 5.1 in [2]. be constant complex matrices, and let K (of size n 1 × n 2 ) andK (of size n 2 × n 1 ) be solutions of the Sylvester equations solve the hierarchy (3.15). Furthermore, solves (3.14) and then also the potential KP hierarchy (3.16).
Proof . The expressions (3.29), (3.30) and (3.31) follow, respectively, from (3.25), (3.28) and (3.27), by writing , and using (3.4). From Theorem 1 we know that (3.30) and (3.31) solve the hierarchy equations (3.9), (3.10) and (3.11). It should be noticed, however, that on the way to the hierarchy (3.15) the step to (3.14) involved a restriction. Hence we have to verify that p given by (3.31) actually solves (3.14). Noting that where I stands for the respective identity matrix, where we used the second of equations (3.29). Now we find that p = UΞKZ −1 V satisfies There are matrix data for which the Sylvester equations (3.29) have no solution. But if S and −S have no common eigenvalue, they admit a solution, irrespective of the right hand side, and this solution is then unique (see e.g. Theorem 4.4.6 in [29]). Remark 3. Evaluated for the bidifferential calculus (3.7), (3.24) reads Using (3.4) and a corresponding block-decomposition for θ, this equation splits into the system which implies q x = −qp + r,q x = −qp −r, and p x = −qq. With the help of these equations we recover (3.12) and thus also (3.13). We observe that the solution generating method based on Theorem 1 also imposes differential equations onp. The corresponding generating equation is actually the direct counterpart of (3.14), which the solutions determined by Proposition 1 satisfy.
Remark 4. If θ in (3.24) is not restricted and if we have trivial d-cohomology, then (3.24) is equivalent to (2.2). But d given by (3.7) has non-trivial cohomology. Indeed, the following 1-form is d-closed but not d-exact, Here a, b only depend on the parameter λ, respectively µ, and a star stands for an arbitrary entry (not restricted in the dependence on the independent variables and the respective parameter). Adding ρ on the r.h.s. of (3.24) would achieve equivalence with (2.2).
which has the same effect as where ǫ = ±1, and ω is the involution on the algebra of matrices either given by the identity map or by complex conjugation (ω = * ), or the anti-involution either given by transposition (ω = ⊺) or by Hermitian conjugation (ω = †) (see also [2]) 5 . It follows that the odd-time part of the hierarchy, expressed in the form (3.20), is consistent with the reduction condition q = ǫq ω , (3.33) which reduces any of its pairs to a single member. In particular, (3.19) becomes the matrix mKdV equation The reduced hierarchy is therefore a matrix mKdV hierarchy. After the replacement , also the even-time equations of the hierarchy are consistent with the above reduction (3.33), provided we choose for ω complex conjugation ( * ) or Hermitian conjugation ( †), so that i ω = −i. Then (3.5) becomes the matrix NLS equation This matrix version of the NLS equation apparently first appeared in [20]. The corresponding reduced hierarchy is a matrix NLS hierarchy.
Since Proposition 1 provides a class of solutions of the original hierarchy in terms of matrix data, we should address the question what kind of constraints a reduction imposes on the latter.

Reduction using an involution
If ω is one of the involutions specified above, settinḡ with ǫ ′ = ±1, and arranging thatΞ = Ξ ω , i.e. ξ(−S ω ) = ξ(−S) = −ξ(S) ω , achieves the reduction condition (3.33). This forces us to set Renaming m i to m and n i to n, this leads to the following consequence of Proposition 1.
Proposition 2. Let S, U, V be constant n × n, m × n, respectively n × m matrices, and let K be a solution of the Sylvester equation solves the m × m matrix "real" mKdV, respectively NLS-mKdV hierarchy.
If the involution is given by complex conjugation, in the focusing NLS case the solutions obtained from this proposition include matrix (multiple) solitons. See [2] for corresponding results for the respective NLS equation, the first member of the hierarchy 6 .

Reduction using an anti-involution
In this case the reduction condition (3.33) can be implemented on the solutions determined by Proposition 1 by settinḡ and arranging again thatΞ = Ξ ω . For the anti-involutions specified above, we are forced to set n 1 = n 2 , which we rename to n. Then we have the following result.
Proposition 3. Let S, V ,V be constant matrices of size n × n, n × m 1 and n × m 2 , respectively. Let K,K be (with respect to ω) Hermitian solutions of the Sylvester equations 7 Then solves the m 1 × m 2 matrix "real" mKdV, respectively NLS-mKdV hierarchy.
If the involution is given by Hermitian conjugation, in the focusing NLS case the solutions obtained from the last proposition include matrix (multiple) solitons. Corresponding results for the respective NLS equation, the first member of the hierarchy, have been obtained in [2].

The reciprocal AKNS hierarchy
Exchanging the roles of d andd in (3.7), we have (4.1) Here the Miwa shift operator is defined in terms of a new set of independent variables,t i , i = 1, 2, . . ., and f ∈ Mat(m, m,B), whereB is the algebra of smooth functions of these variables, extended by the Miwa shifts. Now (2.2) results in In terms of wherex =t 1 , the above equation reduces to this takes the form which is a generating equation for all Lax pairs of the reciprocal hierarchy. The first two members of this family of linear equations are ψx =Lψ,

A class of solutions
We apply again Theorem 1. Using (4.1),dX = (dX)S takes the form assuming that S is invertible. Decomposition as in Section 3.3 leads to These are the same formulas we obtained in Section 3.3, but with S replaced by S −1 . From (3.22) we obtain, however, the same Sylvester equation, SK − KS = V U , using the same definitions as in (3.26). Furthermore, we obtain again (3.27) and (3.28), and thus the following counterpart of Proposition 1.

The combined hierarchy
The bidifferential calculus determined by To order λ 0 , respectively µ 0 , this yields To order λ 0 µ 0 we have
To order λ, (5.2) leads to which decomposes into Furthermore, to order µ (5.3) yields which leads to In the same way as in the preceding sections, we arrive at the following result.  This simply extends Propositions 1 and 4 by adding the respective expressions for ξ(S).

A reduction
Let q,q, p,p be square matrices of the same size. Settinḡ q = ǫq,p = p, where ǫ = ±1, (5.6) the system (5.4), (5.5) reduces to This reduction corresponds to the choice ω = id in Section 3.4 and extends more generally to the "odd flows" of the combined hierarchy. Imposing the conditions (3.34) on the matrix data of Proposition 5, then leads to the following result.
Proposition 6. Let S, U , V be constant n × n, m × n, respectively n × m matrices, and let K be a solution of the Sylvester equation SK + KS = V U . Then solve the odd part of the combined hierarchy with the reduction condition (5.6).

Short pulse equation
Let us impose the additional condition that p is a scalar times the identity matrix. Then we have with a new dependent scalar variable z, the last system is turned into In terms of u(x, z) given by u(x, z(x,x)) = 2q(x,x), (5.9) we obtain The change of independent variables requires zx = 0. The last equation is then equivalent to which is a matrix version of the short pulse equation. The latter apparently first appeared in [32] (see also [33,34]) and was later derived as an approximation for the propagation of ultra-short pulses in nonlinear media [35]. It was further studied in particular in [36,37,38,39,40,41,42,43]. Of course, we have to take the additional condition into account that u 2 has to be a scalar times the identity matrix. This is achieved if where e i e j + e j e i = 2η ij I with η ij = ±δ ij (Clifford algebra), since then u 2 = u, u I, where u = (u 1 , . . . , u r ) ⊺ and u, u = r i=1 η ij u i u j . In this case (5.10) becomes This vector version of the short pulse equation is different from those considered in [42,44]. In the following example we obtain an infinite set of exact solutions of the 2-component system via Proposition 6. Example 1. We can alternatively express the solution given in Proposition 6 in the a priori more redundant form where C is any constant n × n matrix that commutes with S. More precisely, the introduction of C allows us to fix some of the freedom in the choice of the coefficients of the matrices U , V .
The following choices involve further restrictions, however. We consider the case m = 2, n = 2N , and choose S = diag(s 1 I 2 , . . . , s N I 2 ), where σ 1 is the respective Pauli matrix. The Sylvester equation is then solved by Choosing C block-diagonal where the 2 × 2 blocks on the diagonal are a linear combination of I 2 and the Pauli matrix σ 3 = diag(1, −1), then C commutes with S. Furthermore, it follows that KΞ consists of 2 × 2 blocks (KΞ) ij which are linear combinations of the off-diagonal Pauli matrices σ 1 and σ 2 . It further follows that ((KΞ) 2 ) ij is diagonal. As a consequence, the inverse of I 2N − ǫ(KΞ) 2 also consists of diagonal 2 × 2 blocks 11 . Because U consists of off-diagonal 2 × 2 blocks, we conclude that q given by the above formula is an off-diagonal 2 × 2 matrix. Hence its square is proportional to the identity matrix I 2 . Since N ∈ N is arbitrary, we thus have an infinite family of exact solutions of the system (5.11) with r = 2, where the components of the vector u = (u 1 , u 2 ) ⊺ are given by u = u 1 σ 1 + u 2 σ 2 . Fig. 1 shows a plot of a 2-soliton solution.
Remark 7. In order to obtain a Lax pair for the short pulse equation, we start with 11 This is quite obvious if we regard the matrices as N × N matrices over the commutative algebra of diagonal 2 × 2 matrices.
with L andL taken from Remarks 2 and 6, respectively. Without imposing a reduction, the integrability conditions are (5.4) and (5.5) (modulo an integration with respect tox). Writing ψ(x,x) = χ(x, z(x,x)), we find where we applied the reduction conditions (5.6) and used (5.7), (5.8) and (5.9). Using the symmetry u → −u of the short pulse equation, we recover the Lax pair given in [36] for the scalar case and with ǫ = −1.

Dual AKNS hierarchies
For the bidifferential calculus determined by (3.7), the Miura transformation (2.3) with ∆ = 0 takes the form Multiplying from the left by g −1 −[λ]−[µ] and from the right by g, this can be written as To order µ 0 this yields Applying a Miwa shift with −[µ] and subtracting the result from this equation, we obtain Since the r.h.s. is symmetric in λ, µ, this implies (6.2) up to an x-independent "constant of integration". Hence (6.2) reduces to (6.3). The first non-trivial equation resulting from an expansion of (6.3) in powers of the indeterminate λ is obtained as the term linear in λ, (see also [18]), we have the identities S tn = [S, g −1 g tn ], n = 1, 2, . . . , (6.5) and The dual hierarchy equation (6.4) can now be expressed as follows, In order to express this solely in terms of S we use the Miura transformation (6.1), which to order λ 0 reads This imposes the following condition on g, J, g x g −1 = 0, (6.8) which can be expressed as {S, g −1 g x } = 0, hence together with (6.5) we have Inserting this in the expression for S t 2 , and using the above identities, leads to which is the (generalized) Heisenberg magnet equation (see also Remark 5).
Remark 8. The ordinary Heisenberg magnet equation S t = S × S xx is obtained from the 2 × 2 matrix case by writing S = 3 k=1 S k σ k , where σ 1 , σ 2 , σ 3 are the Pauli matrices, and setting t 2 = −it.
More generally, with the help of the Miura transformation (6.1) the hierarchy equations resulting from (6.3) can be expressed solely in terms of S. (6.1) implies where we assumed that (3.14) holds and used (6.7) in the last step. Multiplication from the left by g −1 and from the right by g leads to where we used g −1 Pg = 1 2 (I + S), (6.9), and S 2 = I. At order λ 1 we obtain hence, using (6.6) and (6.9), With the help of (6.10), (6.5) together with (6.12) implies At order λ 2 , (6.3) yields With the help of (6.9) and (6.13), this can be arranged into the form In a similar way, from (6.3) and (6.11) we obtain and S tn = 2F (n−1) x , n = 2, 3, 4, 5, which likely extends to all higher n ∈ N. Of course, the expression for S tn can be recovered by inserting the corresponding expression for g −1 g tn in (6.5).
Remark 9. Conditions like (6.8) 12 originated from the use of the Miura transformation, and they are in fact needed to express the original hierarchy for the matrix variable g in terms of S. The "mismatch" in the Miura transformation, leading to the restriction of the form of g, can be traced back to the fact that in the step from (2.4) to (2.3) we are dropping cohomological terms (see Remark 4).
Example 2. If m = 2, we write and thus The condition (6.8) amounts to κ xκ − σ xσ = 0 and κκ x − σσ x = 0. By adding these two equations, we find that (κκ − σσ) x = 0. Evaluation of the linear equation for the bidifferential calculus given by (3.7) leads to we obtain Expansion in powers of λ yields The first two equations constitute a Lax pair for the generalized Heisenberg magnet equation (with S 2 = I).

A class of solutions
The following result is an analog to that in Section 3.3 (see also Remark 3 in [1]). It allows to generate solutions of (2.4) from solutions of a linear system. Proof . Using the Leibniz rule and the assumptions, we havē and thus Remark 11. The assumptions in Theorem 2 give rise to integrability conditions. The latter are satisfied if dS = (dS)S,d∆ = (d∆)∆,dC = (d∆)C,dZ = (dZ)S.
In the following we exploit Theorem 2 for the bidifferential calculus given by (3.7) with some simplifications. We set ∆ = 0 and make the further assumption that S is d-andd-constant, and we write Z =Ũ Y , whereŨ ∈ Mat(m, n, B) is d-andd-constant and Y ∈ Mat(n, n, B) solves dY = (dY )S. This is motivated by the fact that then the first of conditions (6.15) reduces to (3.25), i.e. SK − KS = V U , (6.19) assuming that A d is invertible and using results from Section 3.3, in particular the definitions (3.26) for K, U , V . Furthermore, we obtain and the second of conditions (6.15) yields C = W d SA −1 d (which simply determines C) and, assuming that S is invertible, The solution (6.16) of (6.18) (with ∆ = 0) is then given by

Rewriting this as
it can easily be decomposed into a part that commutes with J, and a part that anti-commutes with J, Using our concrete form of J and J , the matrices K, S, U , V have the form given in (3.32), and we have This leads to

A. Dimakis and F. Müller-Hoissen
The only restrictions that have to be imposed on the matrices K,K, S,S, U ,Ū , V ,V result from (6.19). They are The solutions of the hierarchy for g obtained in this way also determine solutions of the generalized Heisenberg hierarchy. This is so because the solutions constructed above via Theorem 2 are actually solutions of the Miura transformation and our choice of matrix data via Proposition 1 ensures that (3.14) holds (which we used in Section 6.1).

Reciprocal dual and combined dual AKNS hierarchies
Elaborating the dual equation (2.4) with the "reciprocal" bidifferential calculus determined by (4.1), instead of using that determined by (3.7), we simply obtain (6.2) with g replaced by g −1 . Again, we can combine the dual AKNS hierarchy and its reciprocal version, adopting the procedure in Section 5. New equations arise from the mixed parts, hence from evaluating (2.4) using the bidifferential calculus given by which is a constituent of the calculus determined by (5.1). This results in To order λ 0 µ 0 this is The Miura transformation between the combined hierarchies consists of a pair of Miura transformations, one for the original hierarchy and another one for the reciprocal. It results in the following two generating equations, In particular, this yields g x g −1 = [P, φ], φx = [P, g]g −1 . (6.22) The formulas in Section 6.2 still generate solutions of the combined dual hierarchy and also of the Miura transformation (cf. (6.17)), provided we extend the expression for ξ(S) used there to ξ(S) = k≥1 S k t k + k≥1 S −kt k . (6.23)
The function f drops out of equation (6.21). As a consequence of the form of g, the condition (6.8), which arose from the Miura transformation, is satisfied. (6.22) requires the reduction conditions (5.6) with ǫ = −1 and then reads In order to generate solutions of the sine-Gordon equation (and more generally of the corresponding hierarchy), we have to choose the matrix data in such a way that g has the above form. We set where α = W V . We still have to ensure that κ 2 + σ 2 = f 2 , with some function f that does not depend on x. But since our procedure actually solves the Miura transformation (recall (6.17)), we already know that (6.8) is satisfied, hence (6.14) holds, which shows that κ 2 + σ 2 indeed does not depend on x.

Conclusions
We have shown in particular how a large family of solutions of matrix NLS equations, obtained in [2] with the help of general results of [1], extends to solutions of the corresponding hierarchies.
Moreover, by a simple exchange of the roles of d andd, we obtained a "reciprocal" or "purely negative" counterpart of the AKNS hierarchy, which turned out to be the nonlinear part of the potential KP hierarchy. Combining the two hierarchies then gives rise to additional "mixed flows". In this way we recovered in particular the short pulse equation and obtained an apparently new vector version of it (different from those considered in [42,44]), for which we presented soliton solutions in the 2-component case.
Via specialization of the general Miura transformation to the bidifferential calculus studied in this work, we recovered a relation between the AKNS hierarchy and the "dual" hierarchy of the generalized Heisenberg magnet model. As the first "mixed flow" of the dual hierarchy combined with its negative counterpart, with a certain reduction the sine-Gordon equation showed up.
In this work we concentrated on a simple method, introduced in [1], to generate a class of solutions, parametrized by certain matrix data (essentially of arbitrary size) subject to a Sylvester equation. The largest part of the work in [2] concentrated on narrowing down a remaining redundancy in the matrix data that determine a matrix NLS solution. We expect that most of these results can be carried over to the cases treated in the present work.