Darboux and Binary Darboux Transformations for Discrete Integrable Systems. II. Discrete Potential mKdV Equation

The paper presents two results. First it is shown how the discrete potential modified KdV equation and its Lax pairs in matrix form arise from the Hirota-Miwa equation by a 2-periodic reduction. Then Darboux transformations and binary Darboux transformations are derived for the discrete potential modified KdV equation and it is shown how these may be used to construct exact solutions.


Introduction
The discrete version of the potential modified KdV equation that we want to investigate in this paper is the nonlinear partial difference equation Q(v, v 1 , v 2 , v 12 ; a 1 , a 2 ) ≡ a 1 (vv 2 − v 1 v 12 ) = a 2 (vv 1 − v 2 v 12 ). (1.1) The notation we adopted here and later is as follows, with forward shift operators T n 1 , T n 2 : v := v(n 1 , n 2 ), v 1 := T n 1 (v) = v(n 1 + 1, n 2 ), v 2 := T n 2 (v) = v(n 1 , n 2 + 1), v 12 := T n 1 T n 2 (v) = v(n 1 + 1, n 2 + 1), and a 1 , a 2 denote lattice parameters associated with the directions n 1 , n 2 respectively. Equation (1.1) was derived in [14] from the Cauchy matrix approach, and was originally found in [16,20] through the direct linearization approach. Up to a gauge transformation v → i n 1 +n 2 v and changing the lattice parameters as their reciprocals, equation (1.1) is equivalent to the equation H3 δ=0 in the Adler-Bobenko-Suris (ABS) classification [1], H3 δ ≡ a 1 (vv 1 + v 2 v 12 ) − a 2 (vv 2 + v 1 v 12 ) = δ a 2 2 − a 2 1 . (1.2) There are several papers dedicated to closed-form N -soliton solutions of the 'ABS list' [5,6,8,14]. So, in [14], based on a Cauchy matrix structure, the closed-form N -soliton solution of equation (1.1) was derived, in [8], following Hirota's method, the authors derive bilinear difference equations of equation (1.2) and its N -soliton solutions in terms of Casoratian determinants, and in [5], by the discrete inverse scattering transform, the authors point out that the soliton solutions of equation (1.2) derived from the Cauchy matrix approach are exactly the solutions obtained from reflectionless potentials. The Hirota-Miwa equation [9,13] is the three-dimensional discrete integrable system (a 1 − a 2 )τ 12 τ 3 + (a 2 − a 3 )τ 23 τ 1 + (a 3 − a 1 )τ 13 τ 2 = 0, (1.3) where lattice parameters a k are constants, k = 1, 2, 3, and for τ = τ (n 1 , n 2 , n 3 ) each subscript i denotes a forward shift in the corresponding discrete variable n i . It was discovered by Hirota [9] as a fully discrete analogue of the two-dimensional Toda equation and later Miwa [13] showed that it was intimately related to the KP (Kadomtsev-Petviashvili) hierarchy. In paper [10], Hirota gives the discretization of the potential modified KdV equation, which can be transformed into the form (1.1), and shows that it is a 4-reduction of the Hirota-Miwa equation (which Hirota named as the discrete analogue of a generalized Toda equation).
In this paper, we discuss in detail the Darboux and binary Darboux transformations and how these may be used to obtain exact solutions of the discrete potential modified Korteweg-de Vries (d-p-mKdV) equation (1.1). In contrast to the approaches presented in [4,14,16], we get (1.1) and its Lax pairs by reducing the Hirota-Miwa equation (1.3) and its Lax pairs. In fact, the 2-periodic reduction method studied here has already been investigated in [2] where authors present a multidimensionally consistent hierarchy of discrete systems whose first member is the equation (1.1). Otherwise, this was refined and extended to the non-commutative case in [7]. In [2,3,4,7,17], as we see that the integrability is understood in the sense of the multidimensional consistency property, which gives a Lax pair directly. We here, through a 2periodic reduction of the linear systems of the Hirota-Miwa equation (1.3), obtain the Lax pairs of the equation (1.1) which allows the application of the classical Darboux transformations [11,12]. However, up to gauge transformations, these Lax pairs are coincident with the ones given by the multidimensional consistency property [4]. This paper is part of the work which will explore the equations in the ABS list, their Lax pairs and Darboux transformations as reductions of the Hirota-Miwa equation.
The outline of this paper is as follows. In Section 2, we recall important results on Darboux transformations and binary Darboux transformations of the Hirota-Miwa equation. In particular, in a departure from the results in [18,19,21], we write the linear system of Hirota-Miwa equation in a different form, by the gauge transformation φ → 3 i=1 a −n i i φ, which is suitable for making the reduction. In Section 3, we show that how the d-p-mKdV equation and its Lax pairs in matrix form arise from the Hirota-Miwa equation by a 2-periodic reduction. Then its Darboux transformations and binary Darboux transformations are derived and it is shown how these may be used to construct exact solutions.

Hirota-Miwa equation
The Hirota-Miwa equation (1.3) arises as the compatibility conditions of the linear system where for φ = φ(n 1 , n 2 , n 3 ) each subscript i denotes a forward shift in the corresponding discrete variable n i , for example, φ 1 = T n 1 (φ) = φ(n 1 + 1, n 2 , n 3 ). This linear system (2.1) is compatible if and only if (a 1 − a 2 )u 12 + (a 2 − a 3 )u 23 + (a 3 − a 1 )u 13 = 0, Note that when one uses the formula u ij = τ ij τ /τ i τ j , (2.2a) gives (1.3) and (2.2b) is satisfied identically. A second way is to suppose u ij = (v j − v i + (a i − a j ))/(a i − a j ). This ansatz solves (2.2a) exactly and (2.2b) becomes the discrete potential KP (d-p-KP) equation [15]. In this paper, in particular we deal with the Hirota-Miwa equation (1.3) together with its linear system in the form (2.1). Using the reversal-invariance property of the Hirota-Miwa equation, i.e., it is invariant with respect to the reversal of all lattice directions n i → −n i , we have the linear system in formal adjoint form [18] 3) The subscript i denotes a backward shift with respect to n i , for example, ψ 1 = T −1 n 1 (ψ) = ψ(n 1 − 1, n 2 , n 3 ).

Darboux and binary Darboux transformations
The basic Darboux transformation for the Hirota-Miwa equation is stated in the following proposition.
Proposition 2.1. Let θ be a non-zero solution of the linear system (2.1) for some τ . Then the transformation is with respect to the variable n i .
Next we write down the closed form expression for the result of N applications of the above Darboux transformation, which give solutions in Casoratian determinant form. To do this we need to define the Casoratian of N solutions. Let θ = (θ 1 (n 1 , n 2 , n 3 ), θ 2 (n 1 , n 2 , n 3 ), . . . , θ N (n 1 , n 2 , n 3 )) T be an N -vector solution of (2.1). The Casoratian determinant (with forwardshifts) can be written as which may also be unambiguously defined in the following notation as where θ(k) denotes the N -vector θ 1 (n 1 , n 2 , n 3 ), θ 2 (n 1 , n 2 , n 3 ), . . . , θ N (n 1 , n 2 , n 3 ) T subject to the k times shift T k n i on n i which gives n i → n i + k, 0 ≤ k ≤ N − 1, and i = 1, 2 or 3, the same value being taken for i in each column in the determinant. Then we have the following. Proposition 2.2. Let θ 1 , θ 2 , . . . , θ N be non-zero, independent solutions of the linear system (2.1) for some τ , such that C [i] θ 1 , θ 2 , . . . , θ N = 0. Then the N -fold Darboux transformation Now we can apply the reflections n i → −n i , i = 1, 2, 3, to the above results to deduce adjoint Darboux transformation for the second linear system (2.3). Proposition 2.3. Let ρ be a non-zero solution of the linear system (2.3) for some τ . Then the transformation , using the subscript [i] to designate that the backward shifts of the determinant C [i] (ρ, ψ) is with respect to the variable n i . The N -fold adjoint Darboux transformation is expressed in terms of the Casoratian To construct a binary Darboux transformation, we introduce the potential ω = ω(φ, ψ), defined by the relations Proposition 2.5. For some τ , let θ and φ be two non-zero solutions of the linear system (2.1), ρ and ψ be two non-zero solutions of the linear system (2.3), then The N -fold iteration of these binary Darboux transformations are given below.

Explicit solutions obtained by Darboux transformations
Here we present explicit examples of the classes of solutions that may be obtained by means of the Darboux transformations derived above. We choose the seed solution of the Hirota-Miwa equation (1.3) as τ = τ 0 = 1. With this choice, the first linear system (2.1) reads and the basic eigenfunctions, depending on a single parameter p are found to be In a similar way the basic eigenfunctions of the adjoint linear system (2.3), depending on a single parameter q, are For these eigenfunctions above we may integrate (2.4) and obtain the potential (2.7) Given the above expression it is straightforward to write down the following explicit solution for the Hirota-Miwa equation (1.3) where θ k = α k θ(n 1 , n 2 , n 3 ; p k ) + θ(n 1 , n 2 , n 3 ; p k ) where θ(n 1 , n 2 , n 3 ; p k ) and θ(n 1 , n 2 , n 3 ; p k ) are given by (2.5) and p k , p k , p k = p k and α k are arbitrary constants; where ρ k = β k ρ(n 1 , n 2 , n 3 ; q k ) + ρ(n 1 , n 2 , n 3 ; q k ) where ρ(n 1 , n 2 , n 3 ; q k ) and ρ(n 1 , n 2 , n 3 ; q k ) are given by (2.6) and q k , q k , q k = q k and β k are arbitrary constants; where ω k,l is given by (2.7) with and p = p k , q = q l and c = c kl .
3 Discrete potential modif ied KdV equation (3.1) Introduce f (n 1 , n 2 , n 3 ) andf (n 1 , n 2 , n 3 ) and impose a 2-periodic property on the τ function (3.1) as below Note here that the reduction condition (3.2) gives a 3 = 0, p = −p, and Moreover (3.2) and (3.3) indicate the symmetric property between f and f , with respect to n 3 . By applying the reduction condition (3.2) to the Hirota-Miwa equation (1.3), together with parameter reduction a 3 = 0, we get There are two ways to obtain the equation (3.4b), one way is applying the symmetric property between f and f to the equation (3.4a), the another one is taking the shift operator T n 3 on the Hirota-Miwa equation (1.3), and using the reduction condition (3.3).
Eliminating u in (3.6) gives which is the d-p-mKdV equation (1.1) and is exactly same as the one first given by Nijhoff, cf. [14], through the Cauchy matrix approach. Moreover, the relation (3.6) serves as the discrete Miura transformation between the d-KdV equation in potential u (or more specifically, say u 2 [21]) and the d-p-mKdV equation (3.7) in potential v.
Another interesting result is that with the periodic property of f and f , we have the following formulae on the potentials u and v as follows So the potentials u and v also satisfy the 2-periodic property in the virtual variable n 3 . We observe that if v is a solution of the d-p-mKdV equation then, as in the continuous case, −v is a solution, but in the discrete case, v −1 is a yet another solution.
Under the reduction condition (3.2), from the τ function (3.1), we easily get the exact solution of (3.4) which directly gives the one soliton solution of the d-p-mKdV equation Note here that in the equation (3.7), there is no shift depends on the discrete variable n 3 . So treating the n 3 as a virtual variable for the d-p-mKdV equation is allowable. Next, we show the way of discovering the linear system in matrix form of the d-p-mKdV equation (3.7) from the linear system of the Hirota-Miwa equation (2.1) through the 2-periodic reduction technique.
Introduce eigenfunctions φ(n 1 , n 2 , n 3 ) and φ(n 1 , n 2 , n 3 ) and impose a 2-periodic condition on the eigenfunction φ(n 1 , n 2 , n 3 ) in the linear system (2.1) as below where the parameter λ serves as the spectral parameter. From (3.8), we have So (3.8) and (3.9) mean the symmetric property between φ and φ, with respect to n 3 . By applying the reduction conditions (3.2) and (3.8), together with a 3 = 0, to the linear system (2.1), we get Then by using the symmetric property (3.3) and (3.9) respectively between f and f , φ and φ, we get Substituting (3.5) into (3.10) and (3.11) gives

Darboux and binary Darboux transformations
In this section, we will see that through the reduction conditions (3.

2) and (3.8), it is easy to investigate the Darboux and binary Darboux transformations of d-p-mKdV equation.
Let v be a solution of the d-p-mKdV equation (3.7) and Φ = (φ, φ) T be a vector solution of its Lax pair (3.14). The fundamental Darboux transformation of the d-p-mKdV equation is given as below. (θ), θ = µ −1 T n 3 (θ), is a vector solution of the linear system (3.14) by taking λ = µ for some v, then leaves (3.14) invariant. Otherwise, We remark that may also write the gauge transformation of Φ = (φ, φ) T in (3.15) in matrix form as follows But for later convenience of the construction of the binary Darboux transformation, we here write in scalar form shown in (3.15). Next we write down the closed form expression for the result of N applications of the above Darboux transformation, which give solutions in Casoratian determinant form.
, be N non-zero independent vector solutions of the linear system (3.14) by taking λ = λ k , k = 1, 2, . . . , N , for some v, such that C [3] θ 1 , θ 2 , . . . , θ N = 0. Then The d-p-mKdV equation (3.7) is invariant with respect to the reversal of all lattice directions n i → −n i , i = 1, 2. But its linear system (3.14) does not have such invariance and so the reflections n i → −n i , i = 1, 2, acting on (3.14) give a second linear system on the vector eigenfunction Ψ = (ψ, ψ) T , which also satisfy 2-periodic reduction condition as follows One then finds that Now we apply the reflections n i → −n i , i = 1, 2, in order to deduce Darboux transformation for the second linear system as below.
, is a vector solution of the linear system (3.18) by taking λ = µ for some v, then Next we write down the closed form expression for the result of N applications of the above Darboux transformation, which give solutions in Casoratian determinant form.
Proposition 3.5. For some v, let (θ, θ) T and (φ, φ) T be two non-zero vector solutions of the linear system (3.14), respectively corresponding to spectrum parameters µ and λ; (ρ, ρ) T and (ψ, ψ) T be two non-zero vector solutions of the linear system (3.18), respectively corresponding to spectrum parameters µ and λ, then leave (3.14) and (3.18) respectively invariant. Otherwise, The N -fold iteration of these binary Darboux transformations are given below.

Conclusions
In this paper, we presents two main results. In the first we show how the d-p-mKdV equation and its Lax pairs in matrix form arise from the Hirota-Miwa equation by 2-periodic reduction. The second is that Darboux transformations and binary Darboux transformations are derived for the d-p-mKdV equation and we show how these may be used to construct exact solutions. In this paper, we have revisited the Darboux and binary transformations of the Hirota-Miwa equation but in a departure from the results in [18,19,21], by the gauge transformation φ → we write the linear system of Hirota-Miwa equation in a way which is suitable for obtaining the Lax pair of the d-pmKdV equation naturally by a 2-periodic reduction. Up to gauge transformations, these Lax pairs, which allow the application of the classical Darboux transformations, are coincident with the ones given by the multidimensional consistency property [4]. Hietarinta and Zhang [8] derived the N -soliton solutions to the d-p-mKdV equation using Hirota's direct method and the authors mention that the bilinear equations they get are similar to the Hirota-Miwa equation (1.3). The results in this paper, in which similar results are obtained by reduction of the Hirota-Miwa equation, give an explanation of the observations in [8].

A Some proofs
This section contains proofs of some of the propositions in the main text. For the d-p-mKdV equation, one each of the N -fold basic Darboux transformations and binary Darboux transformations is proved. The omitted proofs are very similar.
where α κ = a κ vv −1 κ and β κ = a κ v −1 v κ are scalars, κ = 1, 2. Then, for (A.1a), it follows that Substituting into the left-hand side of (A.1a), and using the Laplace theorem, we get In a similar way, we can prove that (A.1b) is also satisfied.

A.2 Proof of Proposition 3.6
The proof is by induction. Let Suppose for N = k, the Proposition 3.6 is right, i.e., we have