New type of source extension for a two-dimensional special lattice equation and determinant solutions

We present a new type of two-dimensional special lattice equations with self-consistent sources using the source generation procedure. Then we obtain the Grammy-type and Casorati-type determinant solutions of the coupled system. Further, we present the one-soliton and two-soliton solutions.


Introduction
Soliton equations with self-consistent sources (SESCSs) are integrable coupled generalizations of the soliton equations. These coupled systems are usually relevant to problems in certain areas of physics, such as hydrodynamics, solid-state physics, and plasma physics [1][2][3][4]. This branch has attracted considerable attention in recent years [5][6][7][8][9][10][11]. In [10] the authors proposed a method, termed the source generation procedure (SGP), to construct SESCSs, which has been applied to study different kinds of SESCSs [7,8,[12][13][14]. Furthermore, new types of SESCSs have also been studied, including the AKP-type and BKP-type equations [1,11,[15][16][17]. In this study, we consider the 2 + 1-dimensional KP equation as an example. Through the dependent-variable transformation u = 2(ln τ ) xx , the KP Eq. (1) can be represented in the bilinear form as where D is Hirota's bilinear operator [18] given by The application of the SGP is closely related to the bilinear form of the soliton equation. For example, applying the SGP to the operator D t in (2) produces the first type of KPESCS [2,5,10], whereas a new type of KPESCS [1,11] is obtained by applying the SGP method to the operator D y in (2). In terms of the bilinear form, the operator D t is of the first order, and D y is of the second order, which results in different types of SESCSs. Therefore it is natural to determine if there are any other SESCSs of this new type, especially in differentialdifference equations. However, the following lattice equation was proposed by Blaszak and Szum [19] as an application of the "central extension procedure and operand formalism": where E is the shift operator, that is, Eu n = u n+1 , and H = (E + 1)/(E -1). By setting w n = (E + 1) -1 p n Eqs. (3)-(5) can be rewritten as [20] ∂u n ∂t = u n (w nw n-1 ), ∂v n ∂t = u n+1u n + ∂w n ∂y , Through the dependent-variable transformations and by introducing the auxiliary variable z, Eqs. (6)-(8) can be transformed into the bilinear forms as where the difference operator e δD n is defined as [18] e δD n a · b = a(n + δ)b(nδ).
We can see that D t in Eq. (9) is a second-order operator. It is important that D t acts on different functions τ n+1 and τ n , which is different from that for operator D y in the bilinear KP Eq. (2). The purpose of this study is to construct and solve a new type of special lattice ESCS. The remainder of this paper is organized as follows. In Sect. 2, we propose a new type of special lattice ESCS using the SGP, and obtain its Grammian determinant solution. In Sect. 3, we derive the Casorati determinant solution. In Sect. 4, we describe the onesoliton and two-soliton solutions of the coupled system. Finally, we present conclusions in Sect. 5.

New type of special lattice ESCS and Grammian determinant solution
The Grammian determinant solutions of bilinear Eqs. (9)-(10) have the following forms [21]: where each c ij is an arbitrary constant, and the functions ϕ i (n) and ψ j (-n) satisfy the following differential equations: Now we construct the special lattice ESCS by applying the SGP. First, the Grammian determinant function (11) is changed into the following form: wherein C ij (t) are functions satisfying Here each C i (t) is a differentiable function with respect to t, c ij are arbitrary constants, and the functions ϕ i (n) and ψ i (-n) satisfy the dispersion relations (12)- (14). For calculation, the function f n can be rewritten in the Pfaffian form as follows: where the Pfaffian elements are defined as We introduce other new functions expressed as where the dot and hat symbols above a variable respectively denote the derivative with respect to the variable t and the deletion of the letter under it. Here the above Pfaffian entries refer to In this condition, we introduce another set of auxiliary functions in the following expression: In the following section, we consider Eqs. (22), (23), and (25) as examples for verification.
The key to the proof is in the derivatives of functions f n , f n+1 , g i,n , and h i,n . According to Eqs. (16)- (21), we have the following differential formulas concerning f n and f n+1 : In addition, the derivatives of g i,n and h i,n are given as follows: The above results indicate that Eqs. (23) and (25) and auxiliary transformations the bilinear system (22)-(28) can be transformed into the following differential-difference system: where the operator is defined by u n = u n+1u n .
Herein we only provide the proof of Eqs. (22), (24), and (25). We use the following notation for the functions f n , g i,n , and h i,n : Then we have the following formulas: Further, we obtain the formulas for functions k i,n , g i,n , and h i,n by Now substituting Eqs. (62)-(68) into Eq. (22) yieldṡ which is the sum of the Plücker identities of the determinants. Then substituting Eqs. which are again the Plücker identities of the determinants. Finally, substituting f n , k i,n , g i,n , and P i,n into Eq. (25) gives the following determinant identity: These expressions show that Eqs. In this section, we take M = 1, and the coupled system is read as ∂ n ∂y -∂ n ∂z = v n-1 n + n-1n t ∂w n ∂y dt, which is the special lattice equation with one self-consistent source. Now we derive the one-soliton and two-soliton solutions of this system.

Discussion and conclusion
In this study, we applied SGP to the bilinear form of the two-dimensional special lattice equation and presented a new type of special lattice ESCS given by Eqs. (49)-(52). Additionally, we obtained the Grammian and Casoratian determinant solutions to the coupled system. According to the Grammian determinant solution, we considered the special lattice with one self-consistent source as an example to examine its one-soliton and two-soliton solutions. For further study of the integrability of the coupled system, we can examine the commutativity of the SGP and bilinear Bäcklund transformation, which will enable deriving the bilinear Bäcklund transformation for the coupled system.