A Discrete Negative Order Potential Korteweg–de Vries Equation

1 Introduction

Studying negative order integrable equations is particularly interesting both from mathematical and a physical point of view. Actually, many physically meaningful systems, such as the Camassa–Holm equation [1], the Degasperis–Procesi equation [2], and the short pulse equation [3], are associated to negative order equations through reciprocal transformations [4]. Besides, negative order flows can be used to construct infinitely many symmetries for the nonisospectral Ablowitz–Ladik hierarchy [5]. A well-known example of the negative order equations is the sine-Gordon (sG) equation, which was originally introduced by Bour in the course of study of surfaces of constant negative curvature in ℝ³ [6] and rediscovered by Frenkel and Kontorova in their study of crystal dislocations [7]. Up to now, two negative order KdV (nKdV) equations have been proposed: Verosky [8] studied symmetries and negative powers of recursion operator and gave the following nKdV equation

and Lou [9] presented additional symmetries based on the invertible recursion operator of the KdV system and particularly provided the following nKdV equation

which can be reduced from (1 and 2) under the following transformation

Equation (4) is the famous Schrödinger equation, which can also be referred to as the Hill equation when α is a periodic function. Fuchssteiner [10] investigated the gauge-equivalent relation between the nKdV equation (1 and 2) and the following Camassa–Holm equation [1]

Recently, Fan and Li discussed solitons and kink wave solutions for the integrable equation [11]

and pointed out this equation can be viewed as a reduction form of (6). Moreover, Qiao and Fan systematically studied the nKdV equations (1 and 2) and (3 and 4), particularly including Hamiltonian structures, Lax pairs, conservation laws, and explicit multisoliton and multikink wave solutions [12]. It is remarkable to note that an x-integral form of (7):

can be transformed into the usual sG equation ϕ_xt=−2s in ϕ through introducing variable transformation v=eiϕ2 (cf. [13]).

Many integrable discretisation versions of the continuous integrable equations have been proposed in recent decades. The study of integrable partial difference equations dates back to the pioneering work of Ablowitz and Ladik [14], [15] and Hirota [16]. Subsequently, Sato’s approach [17] and direct linearisation method [18], [19] were also established to construct the discrete integrable system. Besides, with the help of property of multidimensional consistency [20], [21], several discrete systems were classified, including Adler–Bobenko–Suris lattice list [22], lattice Boussinesq type equations [23], and lattice Kadomtsev–Petviashvili type equations [24]. With regard to the solutions for discrete integrable system, some approaches have been developed in recent years, such as Cauchy matrix approach [25], generalised Cauchy matrix approach [26], bilinear method [27], Bäcklund transformation [28], Darboux transformation [29], inverse scattering transformation [30], and algebro-geometric method [31].

The generalised Cauchy matrix method is purely an algebraic procedure that enables us to obtain integrable equations and their various kinds of explicit solutions. Recently, the author (Zhao) of this article has successfully used this method to construct a discrete and a semidiscrete negative order Ablowitz–Kaup–Newell–Segur equation, where some exact solutions were obtained [32]. With the help of this approach, the purpose of this article is to construct the discrete version of the following npKdV equation

Equations (3 and 4) can be deduced from (9 and 10) under transformation u+t=∂⁻¹α (u_x=α), where ∂−1=12(∫−∞x−∫x∞). Besides we also consider some exact solutions and continuum limits for the resulting discrete npKdV equation. This article is organised as follows. In Section 2, we briefly review the Sylvester equation and some properties of master function S^(i,j) (defined as (15)), such as recurrence relations, invariance, and symmetry property. In Section 3, by imposing dispersion relations on r, a discrete npKdV equation is obtained in closed form. In Section 4, we solve determining equation set and construct several kinds of solutions more than multisoliton solutions. In Section 5, we discuss continuum limits of the obtained discrete npKdV equation. Section 6 is for conclusions. In addition, an Appendix is given as a complement to the article.

2 The Sylvester Equation and Master Function

3 The Discrete npKdV Equation

In order to construct discrete npKdV equation, the shifts of r appeared in (12) are set as

where p and q are continuous lattice parameters, associated with the grid size in the directions of the lattice given by the independent variables n, m, respectively. In (27 and 28), we have used the conventional notations to denote the lattice shifts by

r=rn,m↦r˜=rn + 1,m, r=rn,m↦r^=rn,m + 1, r=rn,m↦r˜^=rn + 1,m + 1.

For convenience, we call the gather of the Sylvester equation (12) and system (27 and 28) the determining equation set.

4 Exact Solutions

According to the analysis of Section 3, we know that solution for the discrete npKdV equation (53–54) is given by scalar functions

where ^tc, r, M, and K are defined by the determining equation set (12) and (27 and 28). Therefore, for deriving exact solutions to this equation, we just need to solve the determining equation set (12) and (27 and 28). In terms of the invariance of S⁽i^,j⁾ under transformation (22 and 23), we turn to solve the Jordan canonical form of equations (12) and (27 and 28), i.e.

where Γ is the canonical form of the matrix K. Corresponding to condition ℰ(K)∩ℰ(−K)=Ø, hereafter we assume ℰ(Γ)∩ℰ(−Γ)=Ø. The Sylvester equation (63) has been exactly solved by factorising M into FGH. For the detailed solving procedure, one can refer to [13], [26] and here we list out the main results.

5 Continuum Limit

The continuum limit is always an interesting topic to recognise the connections among discrete equations, semidiscrete equations, and continuous equations. To obtain the corresponding differential-difference equation, we consider the limit

in which t can be identified as the continuous temporal variable. Then for the discrete plane wave factor

ρ=(p+kp−k)n(q+k−1q−k−1)mρ0,

we find

and (53 and 54) give rise to (in terms of q⁰)

where u=u(n, t) and v=v(n, t). We call (92 and 93) the semidiscrete npKdV equation. In the full continuum limit,

in which x can be identified as the continuous spatial variable, we have

ρ→ekx − 2tkρ0.

Equations (92 and 93) yield (in terms of p⁻¹) the continuous npKdV equation (9 and 10).

The solutions for both semidiscrete npKdV equation (92 and 93) and continuous npKdV equation (9 and 10) are still expressed by (61 and 62) and possess three types of exact solutions given in Section 4. The slight difference is that the plane wave factor (70) is replaced by

for semidiscrete case, and

for continuous case.

6 Conclusion

Discrete integrable systems have received much attention in various fields, such as mathematical physics, numerical algorithm, statistical physics, discrete differential geometry, special functions, combinatorics, and cellular automata. Recent studies revealed that the mathematical structures of discrete integrable systems are richer than those of continuous integrable systems. Up to now, many works have been done to deal with the positive order discrete integrable equations, while there was only little work on the negative order discrete integrable equations. In this article, by defining the plain wave factor dispersion relation (27 and 28) we generalise the Cauchy matrix approach to the discrete negative order case. The Sylvester equation of our interest is (12) and it defines the master function S^(i,j)in (15), whose shift relations lead to the discrete npKdV equation (53 and 54). Multisoliton solutions, multiple-pole solutions, and the most general mixed solutions are obtained by taking different forms of Γ in the determining equation set (63–65). The continuum limits of the discrete npKdV equation are also considered. The continuous equations (3 and 4) have both bell-type soliton solutions and kink-soliton solutions [11], [12], which may be the first example has this property. We believe that the discrete equation (53 and 54) and the semidiscrete equation (92 and 93) also should possess both bell-type soliton solutions and kink-soliton solutions, which is left for further discussion. We have not touched in this article on the discrete version for Camassa–Holm equation, which might be done by searching the discrete gauge–equivalent relation between the discrete npKdV equation (53 and 54) and the discrete Camassa–Holm equation. Besides, many physically meaningful systems have been studied well by various methods [37], [38], [39], [40], [41], how to establish their discrete models and exact solutions by generalised Cauchy matrix approach is also an interesting topic and worthy to be considered. We hope that the results given in this article will be helpful to study the negative order discrete integrable system.