Discretizations of the generalized AKNS scheme

We consider space discretizations of the matrix Zakharov-Shabat AKNS scheme, in particular the discrete matrix non-linear Scrhr\"odinger (DNLS) model, and the matrix generalization of the Ablowitz-Ladik (AL) model, which is the more widely acknowledged discretization. We focus on the derivation of solutions via local Darboux transforms for both discretizations, and we derive novel solutions via generic solutions of the associated discrete linear equations. The continuum analogue is also discussed. In this frame we also derive a discretization of the Burgers equation via the analogue of the Cole-Hopf transform. Using the basic Darboux transforms for each scheme we identify both matrix DNLS-like and AL hierarchies, i.e. we extract the associated Lax pairs, via the dressing process. We also discuss the global Darboux transform, which is the discrete analogue of the integral transform, through the discrete Gelfand-Levitan-Marchenko (GLM) equation. The derivation of the discrete matrix GLM equation and associated solutions are also presented together with explicit linearizations. Particular emphasis is given in the discretization schemes, i.e. forward/backward in the discrete matrix DNLS scheme versus symmetric in the discrete matrix AL model.


Introduction
We investigate two versions of the generalized semi-discrete NLS-type hierarchies [1]- [10]: 1) the discrete model introduced in [5,6] (DNLS), which is the natural discretization of the NLS model (AKNS more generally), as will be transparent below.
2) The generalized Ablowitz-Ladik model (see e.g. [3,11,12]), which is the widely studied version, although not the most natural disretization from the algebraic point of view. More precisely, our main aim is the study of the Gr(N |N + M) Grassmannian extension of the DNLS hierarchy and the Ablowitz-Ladik model (see also [13] on non-integrable NLS discretization). Generalized local [14] and global Darboux type [15,3] transformations are employed in order to identify solutions of the associated hierarchies of the nonlinear ODEs as well as produce the semi-discrete hierarchies, (regarding the continuous generalizations we refer the interested reader to [3,7,10]). The proposed DNLS-type hierarchy provides in fact a disretization scheme of the matrix AKNS scheme [1]; indeed discrete generalized versions of the mKdV and mKP naturally arise in this frame. It is also worth noting, that to our knowledge, this is the first time that generic solutions for the DNLS hierarchy are systematically derived, as the majority of the studies on the discretization of AKNS refer to the AL model.
The fundamental object in our approach is the Lax operator, i.e. given the Lax operator and implementing the dressing method we will be able to identify all the members of the two distinct discrete hierarchies. Algebraically speaking the Lax operators of the two schemes, DNLS and AL, are representations of two distinct deformed algebras: the DNLS model is associated to the classical Yangian (rational r-matrix), whereas the AL model is associated to a trigonometric r-matrix. Similarly, at the quantum level the first model corresponds to a representation of the Yangian [6], whereas the q-boson model (quantum AL [12]) is associated to the U q (sl 2 ) R matrix [16].
Let us focus here on the classical case, and recall the Hamiltonian description for both the scalar DNLS and AL models. The Lax operator for both models satisfies Sklyanin's bracket [17]: λ, µ are spectral parameters, r is the classical matrix that satisfies the classical Yang-Baxter equation. The r-matrix for the classical scalar DNLS models is the sl 2 Yangian matrix. In general, the gl k Yangian r-matrix [18] is given as e ij ⊗ e ji (1.2) where e ij are k × k matrices with entries: (e ij ) kl = δ ik δ jl . The r-matrix for AL model on the other hand is a trigonometric matrix, a variation of the classical sine Gordon r-matrix [12,17]: e jj ⊗ e jj + sinh (λ) where z = e λ is the multiplicative spectral parameter.
The Poisson structure (1.1) leads to two distinct algebras at the classical level: DNLS: X n , Y m = −δ nm , X n , X m = Y n , Y m = 0 (1.6) The Ablowitz-Ladik model is thus associated to a deformed harmonic oscillator classical algebra (q-bosons at the quantum level). The generalized DNLS-type hierarchy, is associated to a canonical algebra i.e. the harmonic oscillator model and is immediately associated to the NLS model for which the corresponding exchange relations satisfied by the continuous fieldsû, u and are the continuous analogues of (1.6): û(x), u(y) = −δ(x, y). The algebraic description allows the computation of the time components of the Lax pairs of the hierarchy via the fundamental Semenov-Tian-Shansky formula [19], that involves the r and L matrices. This universal formula has been extended in the case of open boundary conditions as well as at the quantum level [20,21]. It is worth noting that the Hamiltonian/algebraic description offers the strongest statement of integrability at both classical and quantum level, given that it naturally provides all the charges in involution. However, when deriving solutions of the associated integrable PDEs (ODEs) it is more efficient to consider the Lax pair picture and the dressing schemes, and this is precisely the formulation that we are adopting here.
Let us briefly outline what is achieved in the article: In section 2 we introduce the Grassmannian DNLS Lax operator as well as two fundamental Darboux matrices. Given the L-matrix and the local Darboux transforms [14], we compute the associated integrals of motion, and we also derive certain solution of the corresponding non-linear ODEs given the specific choice of the Darboux transform. Employing one of the fundamental Darboux matrices we also perform the dressing process and we identify the Lax pairs of the hierarchy; explicit expressions for the first few members are presented (up to the Hamiltonian of the generalized matrix mKdV). We derive two types of novel solitonic solutions, and we also produce explicit analytic expressions for the two-soliton solution using Bianchi's permutability theorem. More importantly, with the use of a Toda type Darboux matrix we identify generic new solutions of the non-linear ODEs in terms of solutions of the associated linear equations. Such general solutions are also discussed in the continuous case, and for the NLS case in particular we show that they are expressed in terns of the heat kernel. In this context we also derive a discrete version of the viscous Burgers equation via the discrete analogue of the Cole-Hopf transform. In section 3 we implement the dressing scheme for the Grassmannian AL model and we extract the Lax pairs for the discretization of NLS as well as solutions via certain local Darboux transforms. In this case too, we are able to derive general solutions of the non-linear ODEs in terms of solutions of the associated linear equations, via a local Darboux transform. In the last section we introduce the discrete Gelfand-Levitan-Marchenko equation (see also [3,14]) emerging from the upper/lower Borel matrix decomposition. We also introduce the linearizations of both DNLS and AL systems via a forward/backward discretization scheme associated to DNLS, and a symmetric scheme associated to AL. The corresponding discrete calculus and dispersion relations are also derived. General solutions of the discrete GLM equation in terms of solutions of the corresponding linear equations are then easily extracted. Some open fundamental questions for future investigations are also addressed.

2
The discrete Gr(N |N + M) NLS-type hierarchy The main aims in this section are: 1) the construction of the integrals of motion for the semi-discrete DNLS type hierarchy from the power series expansion of the monodromy matrix; 2) the derivation of the associated Lax pairs for the whole hierarchy via the dressing Darboux-Bäcklund scheme and 3) the identification of solutions of the emerging integrable non-linear ODEs via suitable choices of Darboux transformations. In addition to solitonic and multi-solitonic solutions we provide general solutions of the non-linear ODEs in terms of solutions of the associated linear equations, expressed as Fourier transforms.

The Lax pair & the Darboux-Backlund transform
The Lax pair in semi-discrete models (L, V ) depends on the fields (classical or quantum) and some spectral parameter and satisfies the auxiliary linear problem relations which lead to the discrete equations of motion: where the notation above denotes that the Lax pair (L, V (α) ) corresponds to the t α flow.
Our main input is the L-operator for the non-commutative version of the discrete NLS model (1.4), M n+1 (λ)L n (λ) = L n (λ) M n (λ), (2.6) whereL is of the same form as L but with fieldsX,Ŷ . Equation (2.6) can also be interpreted as a zero curvature condition in the frame of discrete space-time systems where L can be thought of as the function at some discrete time α andL as the function at α + 1, so the latter represents a discrete time map. Similarly, for the time components of the Lax pair the transformation (2.5) leads to Having at our disposal the two fundamental BT-Darboux equations we can solve them and explicitly derive the hierarchy as well as solutions of the associated integrable non-linear ODEs provided that the form of the transform M is given.
We consider below two types of Darboux transforms: The above will be used to provide the BT-Darboux relations and hence we can derive in straightforward manner the hierarchy as well as associated solutions.

NLS/Toda type Darboux
where ρ is a constant and for our proposes here will be set equal to zero (Todatype Darboux).

Conserved quantities
We first focus on the derivation of the associated conserved quantities, which are generated by the transfer matrix defines as It is straightforward to show using also the discrete zero curvature condition: The latter expression is zero if we impose periodic boundary conditions V N +1 = V 1 . Thus t(λ) = N k=0 λ k is the generating function of the integrals of motion. In particular, by expanding ln (t) = N k=0 H (k) λ k we find the local conserved quantities.
We keep here terms up to fourth order in the expansion of ln (t) and obtain the first few local integrals of motion of the DNLS model: The general recursion relation for the charges in involution is presented in the appendix.
Recall X n , Y n are matrices with entries X ij n , Y ij n respectively. Requiring Poisson commutativity for the conserved quantities leads to a gl N +M generalization of the canonical commutation relations of the scalar DNLS model (see also [10]): The above Poisson structure is further verified when the equations of motion are computed via the Lax pair representation. Indeed, comparison between the Hamiltonian approach i.e. the equations of motions obtained viaΦ = {H, Φ} (Φ ∈ {X ij , Y ij }) and the Lax pair description, which will be discussed later in the text, leads to agreement, provided that (2.13) are considered.

Dressing: Lax pairs and solutions
In this subsection we derive the "dressed" quantities V (α) of the hierarchy 1 , and also identify solutions of the tower of the emerging non-linear ODEs.

Basic Darboux
We focus now on the basic Darboux matrix (2.8) and on the time independent part of the transform (2.6) and obtain stationary solutions (solitons, traveling waves); the time dependence is dictated by (2.7). From (2.6) and obtain the following fundamental constraints: We moreover require that the Darboux matrix for λ = 0 satisfies the generic quadratic relation, M 2 n =ξM n + ζI, which leads to: This is a fundamental case to consider, and although simple it fully describes the dressing process for the construction of the Lax pairs. From the basic relations (2.14) we obtain (X =Ŷ = 0): Note that time dependence for the fields is always implied.
We now focus on the dressing process in order to derive the time components of the Lax pairs. Let L 0 , V (α),0 be the "bare" Lax pairs: where we define, I = I (N +M) and Also, the "dressed" time components of the Lax pairs can be expressed as formal series expansions n,k (t), (2.19) where the quantities w (α) k will be identified via the dressing transform. From the time part (2.7) we obtain a set of recursion relations, i.e.
We solve the latter recursion relations, and identify the first few time components of the Lax pairs: where the latter relations together with the constraints (2.16) suffice to provide w (α) 0 at each order.
We present below the first few members of the discrete hierarchy: 1. Discrete matrix DST-like equation The zero curvature conditions for t 1 leads to the equations of motion: 25) and the zero curvature condition for t 2 leads to: 3. Discrete matrix generalized mKdV equation n,0 is given by a rather lengthy expression in the appendix.
It is worth noting that the generalized zero curvature condition: , provides the matrix discrete mKP-like equation. This significant issue will be discussed in detail elsewhere together with the continuous case [10].

Solutions: 1-soliton
In order to obtain solutions of the non-linear ODEs emerging from the DNLS hierarchy we focus on the case whereX n =Ŷ n = 0. Then the basic relations (2.14) reduce to where we define ∆F n = F n+1 − F n . And the constraints (2.15) become We distinguish two cases below, which provide two types of solitons: 1. Let us solve the equations involving D n , X n , we first solve the equation for D n , and we consider ζ = 0, where ξ = 1 −ξ. Before we proceed with the solution we consider also the following ansatz for the matrix fields: and periodic conditions are ensured provided that ξ N = 1. Then from (2.29) and (2.33) and knowing the solutions for d n (2.35) we conclude Similarly, for Y n , A n we first identify the solution for A n and then we derive the one soliton solution for Y n . Focus on relations (2.29)-(2.31) for the pair A n , Y n : a n+1 − a n = κa n+1 a n −ξa n ⇒ a n+1 = a ñ κa n +ξ , (2.37) whereξ = ξ −1 andκ = −κξ −1 . The latter leads to the following solution: .
Periodic boundary conditions are valid for all the associated fields and this can be easily checked by inspection provided that ξ N = 1. Note that in the stationary solutions above the time dependence, for each time flow, is naturally introduced: ξ n → ξ n e Λ (α) tα , where Λ (α) = C α (ξ − 1) α , (see also next subsection and last section where a detailed discussion on the related dispersion relations is presented).

Toda type Darboux
We now present the BT-Darboux relations associated to the Toda type transform (2.9). After solving the set of equations provided by (2.6) for (2.9) we conclude To obtain solutions we focus on the special case whereŶ n = 0, we then observe that X n automatically satisfies the linearizations of the non-linear integrable ODEs (2.23), (2.26) and so on (for Y n = 0). We also consider the ansatz (2.33), (2.34) for the fields as well as:X n =x nB , wherex n then satisfies . . . The set of equations (2.43) reduce to y n − y n−1 = κy n a n , (2.49) a n+1 − a n = x n y n . Having the solutionx n at our disposal we can immediately solve for and hence obtains the explicit expressions for both fields: (2.54) Periodic boundary conditions (x N +1 = x 1 , y N +1 = y 1 ) are valid provided that x N +1 =x 1 . Also,x 2 (boundary term) in the expressions above is treated as a constant i.e. ∂ tx2 = 0. Expressions (2.54) are general new solutions of the non-linear ODEs for the DNLS hierarchy in terms of solutions of the associated linear equations. We describe below two simple solutions of the type (2.54), which reproduce the two types of solitons discussed in the previous subsection.
(2.57) 2. The second simple choice is: see also (2.48). After substituting the above in (2.54) we recover (2.42): Notice that we easily recover here the solutions of the previous subsection with a different, rather more convenient parametrization. We have not assumed any extra constraints for the fields contrary to the previous case, it is thus clear that the computation in this case is more straightforward, provided that the form of Fourier transform is available. The Fourier coefficients, and in turn the solutions of the non-linear ODEs, can be expressed in terms of some given initial profile, and different choices of initial profiles will provide distinct general solutions (for relevant discussion in the continuous case see for instance [22,15,23] and references therein). This is a very interesting problem, especially in the discrete case, and will be fully investigated elsewhere.
Remark 1: the continuous case. Let us briefly discuss the continuous case using the Toda type Darboux transformation (2.9). Recall the U -operator (see also [10]) of the continuous Lax pair (U, V ) We also consider the ansatz:V =ûB, V = uB and as in the discrete case B,B satisfy (2.34). The x-part of the Darboux transform (see e.g [10] and references therein) gives: where U 0 is also given by (2.61), but V → V 0 ,V →V 0 . If V 0 = 0, and alsoV 0 =û 0B , thenû 0 satisfies the linear equations (we will consider below the example of the NLSlike equations. For details on the V operators for the hierarchy we refer the interested reader in [10]): From the Darboux relations (2.62), and assuming A = aBB, we obtain: Solving the equations above leads to: which are the continuous analogues of (2.54). Choosing for instance the simple linear solutions:û 0 = c 1 + c 2 e −kx+Λ (α) s tα orû 0 = c 1 e −k1x+Λ (α) 1 tα + c 2 e −k2x+Λ (α) 2 tα we obtain one soliton solutions, i.e. the continuous analogues of (2.57), (2.60).
Let us also consider examples of more general solutions. For instance in the NLS case the fields satisfy: , substituting c(λ) in (2.64) and performing the Gaussian integral, due to the quadratic dispersion relation, we obtain: where we define ∆f n = f n+1 − f n , ∆ 2 f n = f n+2 − 2f n+1 + f n . We also set u n = ∆y n , and we obtain a discrete version of the Burgers equation Assuming the scaling ∆y n ∼ δ, we expand the exponentials and keep up to second order terms in (2.71), (2.72): The second of the equations above provides a good approximation for the discrete viscous Burgers equation in the thermodynamic limit.

Multiparticle-like solutions
The aim now is to identify more general solutions e.g. multi-solitons or bound states of two solitons i.e. breathers. This can be achieved through various paths, but we focus here on two distinct scenarios:

The general Darboux transform
The fundamental Darboux transforms introduced above provide easily the one soliton solution, however to obtain more general solutions of the non-linear ODEs above we implement the general Darboux expressed as a formal λ-series expansion, in the typical loop group expansion fashion [24], where g k are (N + M) × (N + M) matrices to be identified via the Darboux transformation relations (2.6). We focus on the fundamental relations arising from the discrete space part of the Darboux transform (2.6), whereL n = λΣ + + I, L n = λΣ + + I + U n (2.4), and we define Then the following set of recursion relations emerge g (m−1) Specifically, the m = 2 case provides for instance the two soliton solutions or the breather solution. Solving the above recursion relations provides all g k as well as solutions of the associated non-linear ODEs. Extra constraints are required in order to be able to solve explicitly the algebraic relations involved. This is particularly demanding computationally, therefore we focus below on a different, more practical methodology to obtain for instance the two-soliton solution.

Bianchi's permutability and the soliton lattice
It is practically convenient to derive the two soliton or breather solution via Bianchi's permutatbility theorem, i.e. from the commutativity of Darboux transforms. Let us consider the following two families of Darboux transforms: Via the fundamental relations (2.80) we then obtain: where we define in general,∆f n = f n . Now it is straightforward to solve the above algebraic relations and obtain the associated fields, i.e. the two-soliton solution. Indeed, from (2.82) and recalling (2.14) as well as the form of all solutions (2.33) we obtain explicit analytic expressions for the two-soliton solution: n )∆a n∆ x n ∆x n∆ y n−1 + κ∆a n∆ d n y n−1 = y (1) n−1 + κy (2) n−1 (∆d n ) 2 + x (2) n (∆y n−1 ) 2 + κ(a (2) n − d (2) n )∆d n∆ y n−1 ∆x n∆ y n−1 + κ∆a n∆ d n , where the one soliton solutions x n are given in (2.36), (2.39) and d n , a n are given in (2.35), (2.38) with corresponding parameters ξ i . The second type of solitons are given in (2.42) and d n = κd n − c, a n = κâ n − c, with parameters η i , ǫ i , see also (2.41) and definitions below.
The same Darboux transform M n → M (x) was also employed in the continuum case [10], thus the discussion above, and in particular relations (2.82), (2.83) are valid in the continuous case as well, given that: x n →û(x), y n → u(x), a n → a(x), d n → d(x) (2.84) whereû, u are the matrix AKNS fields (following the notation of [10]). The two soliton solutions are given by (2.83) in terms of one soliton solutionsû (i) (x), u (i) (x) (a (i) (x), d (i) (x)) reported in [10], subject to the "dictionary" (2.84).

The matrix Ablowitz-Ladik model
We come now to the study of the matrix AL model. As in the analysis presented in the previous section the main input is the L-operator, which for the matrix AL model is of the form z is the multiplicative spectral parameter andb, b are N × M and M × N matrices respectively. The conserved quantities can be obtained by expanding the monodromy matrix at z → ∞ or z → 0. By adding the first non trivial contributions in the z and z −1 powers series expansion we get the associated Hamiltonian as a linear combination of the z and z −1 expansion c is an arbitrary constant, and will be considered henceforth to be one.
We consider here, as in the previous section, two types of Darboux transforms: • The fundamental Darboux To construct the Lax pair as well as solitonic solutions we consider the fundamental Darboux matrix: where Q = e Θ is an arbitrary constant, A, D are N × N and M × M matrices respectively, and B, C are N × M and M × N matrices respectively.

• The deformed oscillator-type Darboux
A different Darboux that has been more widely used both at classical and quantum level [25,26,21] is given by

4)
A is an N × N matrix, and B, C are N × M and M × N matrices respectively.

Dressing: the Lax pairs
Our aim is to construct the Lax pair via the dressing scheme. To achieve this we consider the fundamental Darboux transform (3.3). Assuming specific forms for the linear time components (zero fields) of the Lax pairs of the hierarchy leads to two distinct models: 1) the matrix Ablowitz-Ladik model, 2) a generalization of the nonlinear network equations [14].
Before we proceed with the dressing it would be useful to obtain the associated BT-Darboux relations. From relations (2.6) for (3.1) and (3.3) we obtain the set of constraints: (3.5) The relations above will be used below forb (0) = b (0) = 0 to obtain the hierarchy via dressing, and also in the next subsection to derive solutions.
Let us consider the following general form for the V (α),0 operators: we could have considered in general V (α),0 = z α Σ + − κz −α Σ − . Let the dressed operators be of the form We focus on the derivation of the first few members of the hierarchy. The first observation emerging from our calculations is that the even powers in the spectral parameter expansion provide inconsistent results, i.e the only pairs that exist are the ones for odd powers. This remark is in agreement with the conserved quantities emerging from the transfer matrix, indeed from the expansion we only have conserved quantities for even powers of z or z −1 . For the first two members of the hierarchy in particular we obtain It is convenient to defineV (2) = V (2) − V (0) ; this corresponds to the addition of a term n log(1 − b nbn ) in the Hamiltonian (3.2). The semi-discrete zero curvature condition for the pair L,V (2) lead to the equations of motion for the generalized AL model in a matrix form: • Remark 2.
By assuming the following forms for the V -operators: n,k , (3.10) we arrive at a variation of the AL model, the matrix non-linear network equations [14]. Indeed, we find that V (0) = Σ + + Σ − , and (3.11) and the corresponding equations of motion read as:

Solutions
To find the one-soliton solution we consider the b (3.14) Suitably combining (3.13), (3.14): To solve the matrix relations above we implement, as in the preceding section, the following ansatz for the fields: The knowledge of a n , d n allows then the explicit computation of b,b.

The deformed oscillator-type Darboux
From the fundamental relation (2.6) we obtain the corresponding BT-Darboux relations for (3.4) We choose to consider the following, b (0) n = 0, then from (3.9) it is clear thatb Notice that this is a symmetric discretization as opposed to the forward discretization in the previous section (2.46). The general solution of symmetric discrete heat equation (3.19) is given bŷ 3.20) and the dispersion relation is obtained via (3.19): By imposing the extra constraint: A n = κb (0) n b n−1 + ζ (κ, ζ constants, see also [21]) we can explicitly solve equations (3.22) and express the fields in terms of the solutionŝ b Multi-soliton solutions can be also obtained via Bianchi's permutability theorem as in the previous section, however the computation is technically more demanding in this case. In general, the derivation of explicit expressions of solitonic as well as more general solutions via the fundamental Darboux transforms is more involved in this case compared to the derivation of the previous section for the generalized DNLS model. This is in fact one of the advantages of the DNLS model compared to the AL. In the subsequent section we are deriving a unifying frame by means of the discrete version of the Gelfand-Levian-Marchnko equation and appropriate linearizations in order to identify general solutions for both discretization schemes.

The discrete GLM equation
The main objective in this section is to construct and solve the discrete GLM equation and derive general solutions for the discrete ZS-AKNS hierarchy [15,3]. We consider below two discretization schemes, the forward/backward corresponding to the DNLS hierarchy, and the symmetric corresponding to the AL model (see also [3] on solutions of the generalized AL model via the inverse scattering transform).
Let us define where e ij are (2N + 1) × (2N + 1) matrices with entries: (e ij ) kl = δ ik δ jl , and also introduce the linear operators associated to the two distinct discretization schemes: 1. Forward/backward scheme where we define: Let also

Symmetric scheme
be a solution of the linear problem i.e.: and in addition require that f ∈ {f,f } are Hankel operators, i.e. f ij = f i+j . Moreover, the upper lower Borel decomposition for F is imposed: where K + , K − are upper, lower triangular matrices respectively. The factorization condition (4.6) leads to the discrete GLM equation. We drop henceforth the + in K + for simplicity, and via the factorization condition we obtain the discrete GLM equation, in the component form: where and also K − ij = F ij + N l≥i K il F lj , j ≤ i. From the discrete GLM equation two independent sets of algebraic equations are obtained, in analogy to the continuous case [10]: The solution of the sets of equations above leads to the simple formulas for the fields expressed solely in terms of the solutions of the linear problem: and in a compact form (see also e.g. [10] for the continuous analogue) where · denotes matrix multiplication according to (4.9)-(4.11).

Discrete Calculus & Solutions
Before we discuss the solutions of the GLM equation (4.12), we should first focus on the solution of the linear problem (4.5), which also provides the corresponding dispersion relations for each time flow. Let us first derive some preliminary results necessary for the solution of the linear problem. First, the powers of the difference operators D, D * (4.1) are expressed in a compact form as: Then one immediately obtains for any matrix f = i,j f ij e ij : (4.14) From the linear problem (4.5) and (4.2) , (4.3) we obtain the following fundamental relations: (4.15) which lead to: Note that the equations above are basically linearizations of the discetere NLStype hierarchy (see for instance (2.23), (2.24), (2.26), (2.27)), up to a time rescaling.

Symmetric
Focus on the a = 1 case associated to the AL model: where the latter equations lead to see also (3.21).
Similarly, for the continuous part: o s → −io, o ∈ {λ,λ}. This is in fact the scheme employed in [3] for the discrete matrix AL model.

Comments
Solutions of the non-linear integrable ODEs of the hierarchies under study can be expressed in terms of suitable local elements of B, C (4.11) (see also [3]). The latter statement is naturally the discrete analogue of the continuum case, where solutions of the non-linear integrable PDEs turn out to be "diagonal" solutions of the continuous GLM equation (see e.g. [15,22,10,27]). In this context the pertinent issue, which will be fully addressed elsewhere, is the systematic derivation of solutions of the nonlinear integrable ODEs in terms of local elements of B, C via the discrete analogue of the "dressing" scheme as described in [22] and [27] (see also recent generalizations on local and non-local PDEs in [23,28]). The derivation of general solutions of the nonlinear ODEs given any initial profile for the linear solutions is then possible. This also applies to the simple, but general expressions (2.54) emerging from the local Darboux transform (2.9).
We have only considered here periodic or vanishing boundary conditions for the solutions of the integrable ODEs. The significant point then is the implementation of integrable space and time integrable boundary conditions [29,30,31] in the discrete systems, and the effect of these boundary conditions on the behavior of the solutions. Having systematically studied the semi discrete version of the AKNS scheme the next natural step is to consider the full discrete space and time picture [32,33], along the lines described in [34], but also from the algebraic/Hamiltonian perspective. The fully discrete case represents various technical and conceptual difficulties, that are primarily associated with the consistent simultaneous discretizations of both space and time directions in such a way that integrabilty is ensured. These are all intriguing open question, that will be systematically addressed in future works.