B\"acklund-Darboux Transformations and Discretizations of Super KdV Equation

For a generalized super KdV equation, three Darboux transformations and the corresponding B\"acklund transformations are constructed. The compatibility of these Darboux transformations leads to three discrete systems and their Lax representations. The reduction of one of the B\"acklund-Darboux transformations and the corresponding discrete system are considered for Kupershmidt's super KdV equation. When all the odd variables vanish, a nonlinear superposition formula is obtained for Levi's B\"acklund transformation for the KdV equation.


Introduction
It is well known that the modern theory of integrable systems or soliton theory begins with the study of the celebrated KdV equation by Kruskal and his collaborators [4]. Various types of extensions of this equation exist in literature (see [1] for example) and one of them is the super extensions. The first such extension was proposed by Kupershmidt [7], which reads as where subscripts denote partial derivatives, t and x are the temporal variable and spatial variable respectively. u is a bosonic (even or commuting) variable and ξ is a fermionic (odd or anticommuting) variable which fulfill where ( ) (i) = ∂ x i ( ). For ξ = 0, (1) becomes the KdV equation. Like the KdV equation itself, the super KdV equation (1), being a bi-Hamiltonian system and possessing Lax representation, is integrable in the conventional sense.
A different super KdV equation was proposed slightly later by Manin and Radul [11] in their study of the supersymmetric KP hierarchy. This system, being the simplest and most important reduction of the supersymmetric KP hierarchy, reads as u t = u xxx + 6uu x + 3ξ xx ξ, Even though the above systems (1) and (2) are similar in appearance, they are very different. In fact, as observed by Mathieu [12,13], the latter is invariant under the following transformation where is a fermionic parameter. Then one may introduce a new independent fermionic variable θ and super field α = ξ + θu, together with the corresponding super derivative D = ∂ θ + θ∂ x . In this way, the system (2) may be reformulated as a single equation For this reason, the system (1) is often referred as the super or fermionic KdV equation, while the system (2) is known as the supersymmetric KdV equation. Nowadays, discrete integrable systems are very hot topic in the soliton theory, and to construct the discrete versions of the non-commuting extensions of integrable equations is very interesting. Most recently, Grahovski and Mikhailov [5] proposed integrable discretizations for a class of nonlinear Schrödinger equations on Grassmann algebras. Also, with Levi we succeeded in discretizing the supersymmetric KdV equation (2) and both semi-discrete and fully discrete supersymmetric KdV equations are given [19]. The aim of this paper is to study Kupershmidt's super KdV equation (1) in the same spirits. In the following discussion, we will assume that u and ξ depend on not only continuous variables x and t, but also are functions of integer-valued variables n and m. The subscripts [1] and [2] used in the following denote the shifts of the discrete variables, for example, ξ [1] = ξ(x, t, n + 1, m), ξ [2] = ξ(x, t, n, m + 1).
The outline of this paper is as follows. In Section 2, we recall a generalized super KdV system and its Lax representation. In Section 3, three different Darboux and Bäcklund transformations are worked out for the generalized super KdV system. Then in Section 4, we employ these transformations to construct discrete integrable super systems and the relevant reductions are discussed. Using two kinds of elementary Darboux transformations, we obtain two differencedifference equations. And by a pair of binary Darboux transformations, we get a differentialdifference equation. The final section summarizes the results.

A generalized super KdV system
We aim to construct Darboux and Bäcklund transformations for the super KdV equation (1). To this end, our strategy is to consider a more general super system where u = u(x, t) is a bosonic variable, ξ = ξ(x, t) and η = η(x, t) are fermionic variables. To the best of our knowledge, above system was studied first by Holod and Pakuliak [6]. The associated spectral problem is Introducing σ x = ηψ and χ = (ψ, ψ x , σ) T , then we may rewrite (4) in matrix form, that is, A direct calculation shows that the Lax equation or the zero curvature condition gives the generalized super KdV system (3).

Remark. For all fermionic variables disappear, (3) reduces to the KdV equation with linear spectral problem
In the following, we will use (5) to construct Bäcklund and Darboux transformations.

Darboux and Bäcklund transformations
Now we manage to construct Darboux and Bäcklund transformations for the generalized super KdV system (3). For convenience, we introduce the potentials w and w [i] such that u = w x , T is a solution of (5) for λ = p 1 , then we find three Darboux transformations and their corresponding Bäcklund transformations, which are listed below. and then χ [1] satisfies The compatibility of the two linear systems (6) and (7) yields which leads to a Bäcklund transformation Remark. During the 5th International Workshop on Nonlinear Mathematical Physics and the 12th National Conference on Integrable Systems, held in Hangzhou last summer, we learnt that professor R.G. Zhou also considered such Darboux transformation [20].
then χ [1] satisfies The compatibility of the linear systems (9) and (10) supplies which gives the following Bäcklund transformation Remark. For all fermionic variables vanish, it is easy to see that above Darboux transformations and Bäcklund transformations reduce to the well-known results for the KdV equation.
Similarly, the compatibility of the above linear systems leads to or the Bäcklund transformation It is remarked that in the last case if the fermionic variables ξ and η vanish, we recover a Darboux transformation and related Bäcklund transformation for the KdV equation, which are nothing but the ones obtained by Levi [9]. Such a Darboux transformation is also known as binary Darboux transformation in literature [14]. Explicitly, the related Bäcklund transformation [9] reads as Darboux transformations of binary type may be regarded as the composition of elementary Darboux transformations [16] (see also [3]). Thus it is natural to expect that Levi's Bäcklund transformation (13) is also the composition of elementary Bäcklund transformations and this is indeed the case. To see this we consider two copies of elementary Bäcklund transformation for the KdV equation, namely (w [1] then, eliminatingw leads to which reduces to (13) (1), the reduction is feasible and easy to implement for the last case. Indeed, for ξ = η, define and then it is straightforward to check that χ [1] satisfies [1] .
The corresponding Bäcklund transformation is Thus, we obtain a Bäcklund transformation for Kupershmidt's super KdV equation (1).

Discrete systems
Integrable discretizations have been studied extensively since the seventies of last century and various approaches have been proposed (see [15,18]). Among them, the method, based on Darboux and Bäcklund transformations, which first appeared in [8,10], has been proved to be very fruitful. This idea is also applicable to super integrable systems and supersymmetric integrable systems [5,19]. We now adopt this idea for the super KdV equation (3) and its reduction -Kupershmidt's super KdV equation to construct their discrete counterparts. We start our consideration with Darboux transformations presented in the first two cases of last section.
Remark. For all fermionic fields vanish, both (17) and (20) reduce to which is the potential KdV lattice or H1 in Adler-Bobenko-Suris's classification [2] or the classical nonlinear superposition formula for the KdV equation.
By a direct calculation, one can check that (25) is satisfied if we substitute (26) into it and take account of the corresponding Bäcklund transformations (14) into consideration.
Also, a direct calculation shows that (27) is satisfied if we substitute (28) into it and take the corresponding Bäcklund transformations (13) into consideration. Thus (28) may be regarded as the nonlinear superposition formula for the Bäcklund transformation (13) for the KdV equation.
To the best of our knowledge, this result is also new.

Conclusion
In this paper, we have constructed three types of Bäcklund and Darboux transformations for a generalized super KdV equation. By means of these transformations, the super KdV equation has been discretized. In particular, by considering the reductions, we have succeeded in obtaining a Bäcklund transformation and a discrete version for Kupershmidt's super KdV equation. As a by-product, we have found a nonlinear superposition formula for the Bäcklund transformation obtained by Levi early. The discretization for the supersymmetric Schrödinger equation [17] is under investigated and will appear elsewhere.