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

Abstract

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 [14]. This branch has attracted considerable attention in recent years [511]. 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, 1214]. Furthermore, new types of SESCSs have also been studied, including the AKP-type and BKP-type equations [1, 11, 1517]. In this study, we consider the \(2+1\)-dimensional KP equation

$$ -4u_{t}+u_{xxx}+6uu_{x}+3 \int ^{x}u_{yy}\,dx=0 $$
(1)

as an example. Through the dependent-variable transformation

$$ u=2(\ln \tau )_{xx}, $$

the KP Eq. (1) can be represented in the bilinear form as

$$ \bigl(D^{4}_{x}-4D_{x}D_{t}+3D^{2}_{y} \bigr)\tau \cdot \tau =0, $$
(2)

where D is Hirota’s bilinear operator [18] given by

$$ D^{l}_{t}D^{m}_{y} a\cdot b= \biggl(\frac{\partial }{\partial t}-\frac{ \partial }{\partial t'} \biggr)^{l} \biggl( \frac{\partial }{\partial y}-\frac{\partial }{\partial y'} \biggr)^{m}a(t,y)b \bigl(t',y'\bigr) \bigg|_{t'=t,y'=y}. $$

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 differential–difference equations.

However, the following lattice equation was proposed by Blaszak and Szum [19] as an application of the “central extension procedure and operand formalism”:

$$\begin{aligned}& \frac{\partial u_{n}}{\partial t}=u_{n}{H}^{-1}p_{n-1}, \end{aligned}$$
(3)
$$\begin{aligned}& \frac{\partial v_{n}}{\partial t}=u_{n+1}-u_{n}+({E}+1)^{-1} \frac{\partial p_{n}}{\partial y}, \end{aligned}$$
(4)
$$\begin{aligned}& \frac{\partial p_{n}}{\partial t}=v_{n+1}-v_{n}-p_{n} {H}^{-1}p_{n}, \end{aligned}$$
(5)

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]

$$\begin{aligned}& \frac{\partial u_{n}}{\partial t}=u_{n}(w_{n}-w_{n-1}), \end{aligned}$$
(6)
$$\begin{aligned}& \frac{\partial v_{n}}{\partial t}=u_{n+1}-u_{n}+ \frac{\partial w_{n}}{\partial y}, \end{aligned}$$
(7)
$$\begin{aligned}& \frac{\partial w_{n+1}}{\partial t}+ \frac{\partial w_{n}}{\partial t}=v_{n+1}-v_{n}-w_{n+1}^{2}+w_{n}^{2}. \end{aligned}$$
(8)

Through the dependent-variable transformations

$$ u_{n}=\frac{\tau _{n+1}\tau _{n-1}}{\tau _{n}^{2}},\qquad v_{n}= \frac{D_{t}^{2}\tau _{n}\cdot \tau _{n+1}}{\tau _{n}\tau _{n+1}}, \qquad w_{n}=\biggl( \ln \frac{\tau _{n+1}}{\tau _{n}}\biggr)_{t}, $$

and by introducing the auxiliary variable z, Eqs. (6)–(8) can be transformed into the bilinear forms as

$$\begin{aligned}& \bigl(D_{z}e^{\frac{1}{2}D_{n}}-D_{t}^{2}e^{\frac{1}{2}D_{n}} \bigr)\tau _{n} \cdot \tau _{n}=0, \end{aligned}$$
(9)
$$\begin{aligned}& \biggl(D_{t}D_{z}-D_{t}D_{y}-4\sinh ^{2}\biggl(\frac{1}{2}D_{n}\biggr)\biggr)\tau _{n} \cdot \tau _{n}=0, \end{aligned}$$
(10)

where the difference operator \(e^{\delta D_{n}}\) is defined as [18]

$$ e^{\delta D_{n}}a\cdot b=a(n+\delta )b(n-\delta ). $$

We can see that \(D_{t}\) in Eq. (9) is a second-order operator. It is important that \(D_{t}\) acts on different functions \(\tau _{n+1}\) and \(\tau _{n}\), 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 one-soliton 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]:

$$ \tau _{n}=\operatorname{det} \biggl\vert {c_{ij}+ \int ^{t}\varphi _{i}(n)\psi _{j}(-n) \,dt} \biggr\vert _{1\leq i,j\leq N}, $$
(11)

where each \(c_{ij}\) is an arbitrary constant, and the functions \(\varphi _{i}(n)\) and \(\psi _{j}(-n)\) satisfy the following differential equations:

$$\begin{aligned}& \frac{\partial \varphi _{i}(n)}{\partial t}=\varphi _{i}(n+1),\qquad \frac{\partial \varphi _{i}(n)}{\partial y}= \varphi _{i}(n+2)+ \varphi _{i}(n-1), \end{aligned}$$
(12)
$$\begin{aligned}& \frac{\partial \varphi _{i}(n)}{\partial z}=\varphi _{i}(n+2),\qquad \frac{\partial \psi _{j}(-n)}{\partial y}=-\psi _{j}(-n+2)-\psi _{j}(-n-1), \end{aligned}$$
(13)
$$\begin{aligned}& \frac{\partial \psi _{i}(-n)}{\partial t}=-\psi _{i}(-n+1),\qquad \frac{\partial \psi _{j}(-n)}{\partial z}=-\psi _{j}(-n+2). \end{aligned}$$
(14)

Now we construct the special lattice ESCS by applying the SGP. First, the Grammian determinant function (11) is changed into the following form:

$$ f_{n}=\operatorname{det} \biggl\vert {C_{ij}(t)+ \int ^{t}\varphi _{i}(n)\psi _{j}(-n) \,dt} \biggr\vert _{1\leq i,j\leq N}, $$
(15)

wherein \(C_{ij}(t)\) are functions satisfying

$$ C_{ij}(t)= \textstyle\begin{cases} C_{i}(t),& i=j, 1\leqslant i\leqslant M\leqslant N, M\in Z^{+} , \\ c_{ij}& \text{otherwise}, 1\leqslant i,j\leqslant N. \end{cases} $$

Here each \(C_{i}(t)\) is a differentiable function with respect to t, \(c_{ij}\) are arbitrary constants, and the functions \(\varphi _{i}{(n)}\) and \(\psi _{i}{(-n)}\) satisfy the dispersion relations (12)–(14). For calculation, the function \(f_{n}\) can be rewritten in the Pfaffian form as follows:

$$ f_{n}=\bigl(1,2,\ldots ,N,N^{*},\ldots ,2^{*},1^{*}\bigr)_{n}\equiv (\circ )_{n}, $$
(16)

where the Pfaffian elements are defined as

$$ \bigl(i,j^{*}\bigr)_{n}=C_{ij}(t)+ \int ^{t}\varphi _{i}(n)\psi _{j}(-n) \,dt,\qquad (i,j)_{n}=\bigl(i^{*},j^{*} \bigr)_{n}=0,\quad 1\leq i,j\leq N. $$

We introduce other new functions expressed as

g i , n = C ˙ i ( t ) ( d 1 , 1 , 2 , , N , N , , i ˆ , , 1 ) n C ˙ i ( t ) ( d 1 , 1 ) n , i = 1 , 2 , , M ,
(17)
h i , n = C ˙ i ( t ) ( d 1 , 1 , , i ˆ , , N , N , , 2 , 1 ) n C ˙ i ( t ) ( d 1 , 2 ) n , i = 1 , 2 , , M ,
(18)

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

$$\begin{aligned}& \bigl(d_{m}^{*},i\bigr)_{n}=\varphi _{i}(n+m),\qquad \bigl(d_{m},j^{*} \bigr)_{n}=\psi _{j}(-n+m), \\& \bigl(d_{m},d_{l}^{*}\bigr)_{n}=(d_{m},i)_{n}= \bigl(d_{l}^{*},i^{*}\bigr)_{n}=0, \quad m,l \in \mathrm{Z}. \end{aligned}$$

In this condition, we introduce another set of auxiliary functions in the following expression:

$$\begin{aligned}& k_{i,n}=\dot{C}_{i}(t) \bigl(1,\ldots , \hat{i},\ldots ,N,N^{*},\ldots , \hat{i}^{*},\ldots ,1^{*}\bigr)_{n}, \end{aligned}$$
(19)
P i , n = C ¨ i ( t ) 2 C ˙ i ( t ) ( d 1 , 1 , 2 , , N , N , , i ˆ , , 1 ) n + C ˙ i ( t ) [ 1 j N j i ( d 1 , 1 , , j ˆ , , N , N , , i ˆ , , j ˆ , , 1 ) n ] ,
(20)
Q i , n = C ¨ i ( t ) 2 C ˙ i ( t ) ( d 1 , 1 , , i ˆ , , N , N , , 2 , 1 ) n + C ˙ i ( t ) [ 1 j N j i ( d 1 , 1 , , i ˆ , , j ˆ , , N , N , , j ˆ , , 1 ) n ] ,
(21)

where \(i = 1, 2,\ldots , M\). Thus the new functions in Eqs. (16)–(18) and auxiliary functions in Eqs. (19)–(21) satisfy the following bilinear expressions:

$$\begin{aligned}& \begin{aligned} \bigl(D_{z}-D_{t}^{2} \bigr)e^{\frac{D_{n}}{2}}f_{n}\cdot f_{n}&=-\sum _{i=1}^{M}D_{t}\bigl(e^{ \frac{D_{n}}{2}}+e^{-\frac{D_{n}}{2}} \bigr)k_{i,n}\cdot f_{n}+\sum_{i=1}^{M}D_{t}e^{\frac{D_{n}}{2}}g_{i,n} \cdot h_{i,n} \\ &\quad {}+\sum_{i=1}^{M}e^{\frac{D_{n}}{2}}(g_{i,n} \cdot Q_{i,n}+P_{i,n} \cdot h_{i,n}), \end{aligned} \end{aligned}$$
(22)
$$\begin{aligned}& \biggl(D_{t}D_{z}-D_{t}D_{y}-4\sinh ^{2}\biggl(\frac{1}{2}D_{n}\biggr) \biggr)f_{n}\cdot f_{n}=-2 \sum _{i=1}^{M}g_{i,n}\cdot h_{i,n}, \end{aligned}$$
(23)
$$\begin{aligned}& \bigl(e^{\frac{1}{2}D_{n}}-e^{-\frac{1}{2}D_{n}}\bigr)k_{i,n}\cdot f_{n}=-e^{ \frac{1}{2}D_{n}}g_{i,n}\cdot h_{i,n},\quad i=1,2,\ldots ,M, \end{aligned}$$
(24)
$$\begin{aligned}& \bigl(D_{t}+e^{-D_{n}}\bigr)f_{n}\cdot g_{i,n}=\Biggl(\sum_{j=1}^{M}k_{j,n} \Biggr)g_{i,n}-f_{n}P_{i,n}, \quad i=1,2,\ldots ,M, \end{aligned}$$
(25)
$$\begin{aligned}& \bigl(D_{t}+e^{-D_{n}}\bigr)h_{i,n}\cdot f_{n}=-h_{i,n}\Biggl(\sum_{j=1}^{M}k_{j,n} \Biggr)+f_{n}Q_{i,n}, \quad i=1,2,\ldots ,M, \end{aligned}$$
(26)
$$\begin{aligned}& (D_{z}-D_{y})e^{-\frac{1}{2}D_{n}}f_{n}\cdot g_{i,n}=e^{\frac{1}{2}D_{n}}f_{n} \cdot g_{i,n},\quad i=1,2,\ldots ,M, \end{aligned}$$
(27)
$$\begin{aligned}& (D_{z}-D_{y})e^{-\frac{1}{2}D_{n}}h_{i,n}\cdot f_{n}=e^{\frac{1}{2}D_{n}}h_{i,n} \cdot f_{n},\quad i=1,2,\ldots ,M. \end{aligned}$$
(28)

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 \(f_{n+1}\):

$$\begin{aligned}& f_{n+1}=f_{n}+\bigl(d_{-1},d_{0}^{*}, \circ \bigr)_{n},\qquad f_{n-1}=f_{n}- \bigl(d_{0},d_{-1}^{*},\circ \bigr)_{n}, \end{aligned}$$
(29)
$$\begin{aligned}& \frac{\partial f_{n}}{\partial z}=\bigl(d_{0},d_{1}^{*},\circ \bigr)_{n}+\bigl(d_{1},d_{0}^{*}, \circ \bigr)_{n}, \end{aligned}$$
(30)
$$\begin{aligned}& \frac{\partial f_{n}}{\partial y}=\bigl(d_{0},d_{1}^{*},\circ \bigr)_{n}+\bigl(d_{1},d_{0}^{*}, \circ \bigr)_{n}-\bigl(d_{-1},d_{-1}^{*}, \circ \bigr)_{n}, \end{aligned}$$
(31)
$$\begin{aligned}& \frac{\partial f_{n}}{\partial t}=\sum_{i=1}^{M}k_{i,n}+ \bigl(d_{0},d_{0}^{*}, \circ \bigr)_{n}, \end{aligned}$$
(32)
$$\begin{aligned}& \begin{aligned} \frac{\partial ^{2} f_{n}}{\partial t^{2}}&=\sum _{i=1}^{M} \frac{\partial k_{i,n}}{\partial t}+\bigl(d_{0},d_{1}^{*}, \circ \bigr)_{n}-\bigl(d_{1},d_{0}^{*}, \circ \bigr)_{n} \\ &\quad {} +\sum_{i=1}^{M} \dot{C}_{i}(t) \bigl(d_{0},d_{0}^{*},1, \ldots ,\hat{i},\ldots ,N,N^{*},\ldots ,\hat{i}^{*},\ldots ,1^{*}\bigr)_{n}, \end{aligned} \end{aligned}$$
(33)
$$\begin{aligned}& \begin{aligned} \frac{\partial ^{2} f_{n}}{\partial t\partial y}&=\sum _{i=1}^{M} \dot{C}_{i}(t)\bigl[ \bigl(d_{0},d_{1}^{*},1,\ldots ,\hat{i},\ldots ,N,N^{*}, \ldots ,\hat{i}^{*},\ldots ,1^{*} \bigr)_{n} \\ &\quad {} +\bigl(d_{1},d_{0}^{*},1,\ldots , \hat{i},\ldots ,N,N^{*},\ldots , \hat{i}^{*},\ldots ,1^{*}\bigr)_{n} \\ &\quad {} -\bigl(d_{-1},d_{-1}^{*},1,\ldots , \hat{i},\ldots ,N,N^{*}, \ldots ,\hat{i}^{*},\ldots ,1^{*}\bigr)_{n}\bigr] \\ &\quad {} +\bigl(d_{0},d_{2}^{*},\circ \bigr)_{n}+\bigl(d_{0},d_{-1}^{*},\circ \bigr)_{n}-\bigl(d_{2},d_{0}^{*}, \circ \bigr)_{n}-\bigl(d_{-1},d_{0}^{*},\circ \bigr)_{n} \\ &\quad {} -\bigl(d_{-1},d_{-1}^{*},d_{0},d_{0}^{*}, \circ \bigr)_{n}, \end{aligned} \end{aligned}$$
(34)
$$\begin{aligned}& \begin{aligned} \frac{\partial ^{2} f_{n}}{\partial t\partial z}&=\sum _{i=1}^{M} \dot{C}_{i}(t)[ \bigl(d_{0},d_{1}^{*},1,\ldots ,\hat{i},\ldots ,N,N^{*}, \ldots ,\hat{i}^{*},\ldots ,1^{*} \bigr)_{n} \\ &\quad {} +\bigl(d_{1},d_{0}^{*},1,\ldots , \hat{i},\ldots ,N,N^{*},\ldots , \hat{i}^{*},\ldots ,1^{*}\bigr)_{n} \\ &\quad {} +\bigl(d_{0},d_{2}^{*},\circ \bigr)_{n}-\bigl(d_{2},d_{0}^{*},\circ \bigr)_{n}, \end{aligned} \end{aligned}$$
(35)
$$\begin{aligned}& \frac{\partial f_{n+1}}{\partial t}=\sum_{i=1}^{M}k_{i,n+1}+ \bigl(d_{-1},d_{1}^{*}, \circ \bigr)_{n}, \end{aligned}$$
(36)
$$\begin{aligned}& \begin{aligned} \frac{\partial ^{2} f_{n+1}}{\partial t^{2}}&=\sum _{i=1}^{M} \frac{\partial k_{i,n+1}}{\partial t}+\bigl(d_{-1},d_{1}^{*},1, \ldots , \hat{i},\ldots ,N,N^{*},\ldots ,\hat{i}^{*},\ldots ,1^{*}\bigr)_{n} \\ &\quad {} +\bigl(d_{-1},d_{2}^{*},\circ \bigr)_{n}-\bigl(d_{0},d_{1}^{*},\circ \bigr)_{n}+\bigl(d_{0},d_{0}^{*},d_{-1},d_{1}^{*}, \circ \bigr)_{n}. \end{aligned} \end{aligned}$$
(37)

In addition, the derivatives of \(g_{i,n}\) and \(h_{i,n}\) are given as follows:

g i , n + 1 = C ˙ i ( t ) ( d 0 , 1 ) n ,
(38)
g i , n + 1 t = C ˙ i ( t ) ( d 1 , 1 ) n + P i , n + 1 ,
(39)
g i , n + 1 z = C ˙ i ( t ) [ ( d 2 , 1 ) n + ( d 0 , d 0 , d 1 , 1 ) n ] ,
(40)
g i , n + 1 y = C ˙ i ( t ) [ ( d 2 , 1 ) n + ( d 1 , 1 ) n + ( d 0 , d 0 , d 1 , 1 ) n ( d 0 , d 1 , d 1 , 1 ) n ] ,
(41)
h i , n 1 = C ˙ i ( t ) ( d 0 , 2 ) n ,
(42)
h i , n t = C ˙ i ( t ) [ ( d 0 , 2 ) n + ( d 1 , d 0 , d 0 , 2 ) n ] + Q i , n ,
(43)
h i , n 1 z = C ˙ i ( t ) [ ( d 2 , 2 ) n + ( d 0 , d 1 , d 0 , 2 ) n ] ,
(44)
h i , n 1 y = C ˙ i ( t ) [ ( d 0 , d 1 , d 0 , 2 ) n ( d 2 , 2 ) n ( d 1 , 2 ) n ( d 0 , d 1 , d 1 , 2 ) n ] ,
(45)

where \(i = 1, 2,\ldots , M\). Substituting expressions (29)–(33), (34)–(35), and (39) into Eq. (22), we obtain the Jacobi identities of the determinants

$$\begin{aligned}& 2\bigl[\bigl(d_{-1},d_{1}^{*},\circ \bigr)_{n}\bigl(d_{0},d_{0}^{*},\circ \bigr)_{n}-\bigl(d_{-1},d_{0}^{*}, \circ \bigr)_{n}\bigl(d_{0},d_{1}^{*},\circ \bigr)_{n}+\bigl(d_{-1},d_{0}^{*},d_{0},d_{1}^{*}, \circ \bigr)_{n}(\circ )_{n}\bigr] \\& \quad {}+\sum_{i=1}^{M} \dot{C}_{i}(t)\bigl[\bigl(d_{-1},d_{1}^{*},1, \ldots , \hat{i},\ldots ,N,N^{*},\ldots ,\hat{i}^{*},\ldots ,1^{*}\bigr)_{n}(\circ )_{n} \\& \quad {} -\bigl(1,\ldots ,\hat{i},\ldots ,N,N^{*},\ldots , \hat{i}^{*}, \ldots ,1^{*}\bigr)_{n} \bigl(d_{-1},d_{1}^{*},\circ \bigr)_{n}+ \bigl(d_{1}^{*},\star _{1}\bigr)_{n}(d_{-1}, \star _{2})_{n}\bigr] \\& \quad {}+\sum_{i=1}^{M} \dot{C}_{i}(t)\bigl[\bigl(d_{0},d_{0}^{*},1, \ldots , \hat{i},\ldots ,N,N^{*},\ldots ,\hat{i}^{*},\ldots ,1^{*}\bigr)_{n}(\circ )_{n} \\& \quad {} -\bigl(1,\ldots ,\hat{i},\ldots ,N,N^{*},\ldots , \hat{i}^{*},\ldots ,1^{*}\bigr)_{n} \bigl(d_{0},d_{0}^{*}, \circ \bigr)_{n}+\bigl(d_{0}^{*},\star _{1} \bigr)_{n}(d_{0},\star _{2})_{n}\bigr] \\& \quad {}+\sum_{i=1}^{M} \dot{C}_{i}(t)\bigl[\bigl(d_{-1},d_{0}^{*},1, \ldots , \hat{i},\ldots ,N,N^{*},\ldots ,\hat{i}^{*},\ldots ,1^{*}\bigr)_{n}\bigl(d_{0},d_{0}^{*}, \circ \bigr)_{n} \\& \quad {} -\bigl(d_{0},d_{0}^{*},1,\ldots , \hat{i},\ldots ,N,N^{*},\ldots ,\hat{i}^{*}, \ldots ,1^{*}\bigr)_{n}\bigl(d_{-1},d_{0}^{*}, \circ \bigr)_{n} \\& \quad {} +\bigl(d_{0}^{*},\star _{1} \bigr)_{n}\bigl(d_{-1},d_{0},d_{0}^{*}, \star _{2}\bigr)_{n}\bigr] \equiv 0, \end{aligned}$$

thereby indicating that Eq. (22) holds. Similarly, substitution of expressions (17)–(18) and (29)–(35) into Eq. (23) yields the following determinant identities:

$$\begin{aligned}& 2\bigl[\bigl(d_{-1},d_{0}^{*},\circ \bigr)_{n}\bigl(d_{0},d_{-1}^{*},\circ \bigr)_{n}-\bigl(d_{0},d_{0}^{*}, \circ \bigr)_{n}\bigl(d_{-1},d_{-1}^{*}, \circ \bigr)_{n}+\bigl(d_{-1},d_{-1}^{*},d_{0},d_{0}^{*}, \circ \bigr)_{n}(\circ )_{n}\bigr] \\& \quad {} +2\sum_{i=1}^{M} \dot{C}_{i}(t)\bigl[\bigl(d_{-1},d_{-1}^{*},1, \ldots ,\hat{i},\ldots ,N,N^{*},\ldots ,\hat{i}^{*},\ldots ,1^{*}\bigr)_{n}( \circ )_{n} \\& \quad {} -\bigl(1,\ldots ,\hat{i},\ldots ,N,N^{*},\ldots , \hat{i}^{*},\ldots ,1^{*}\bigr)_{n} \bigl(d_{-1},d_{-1}^{*}, \circ \bigr)_{n} \\& \quad {} +\bigl(d_{-1}^{*},\star _{1} \bigr)_{n}(d_{-1},\star _{2})_{n}\bigr] \equiv 0. \end{aligned}$$

Finally, we substitute Eqs. (16)–(17), (20), (32), and (38)–(39) into Eq. (25) to derive the following determinant identity:

$$\begin{aligned}& \sum_{i=1}^{M}\dot{C}_{i}(t) \bigl[\bigl(d_{0}^{*},\star _{1} \bigr)_{n}\bigl(d_{-1},d_{-1}^{*}, \circ \bigr)_{n}-\bigl(d_{-1}^{*},\star _{1} \bigr)_{n}\bigl(d_{-1},d_{0}^{*},\circ \bigr)_{n} \\& \quad{} +\bigl(d_{0}^{*},d_{-1},d_{-1}^{*}, \star _{1}\bigr)_{n}(\circ )_{n}\bigr]\equiv 0. \end{aligned}$$

The above results indicate that Eqs. (23) and (25) are true. In the same manner, we can prove the other bilinear equations in (22)–(28). Therefore the bilinear Eqs. (22)–(28) constitute the bilinear forms of the two-dimensional special lattice ESCS, and the functions in Eqs. (16)–(21) are the Grammian determinant solutions of the coupled system.

By the dependent-variable transformations

$$\begin{aligned}& u_{n}=\frac{f_{n+1}f_{n-1}}{f^{2}_{n}},\qquad w_{n}= \biggl(\ln \frac{f_{n+1}}{f_{n}} \biggr)_{t},\qquad v_{n}= \biggl(\ln \frac{f_{n+1}}{f_{n}} \biggr)_{z}, \end{aligned}$$
(46)
$$\begin{aligned}& \Phi _{i,n}=\frac{g_{i,n}}{f},\qquad \Psi _{i,n}= \frac{h_{i,n}}{f}, \end{aligned}$$
(47)

and auxiliary transformations

$$ \lambda _{i,n}=\frac{k_{i,n}}{f},\qquad \phi _{i,n}= \frac{P_{i,n}}{f},\qquad \theta _{i,n}=\frac{Q_{i,n}}{f}, $$
(48)

the bilinear system (22)–(28) can be transformed into the following differential–difference system:

$$\begin{aligned}& \begin{aligned} \frac{\partial w_{n+1}}{\partial t}+\frac{\partial w_{n}}{\partial t}-v_{n+1}+v_{n}+w^{2}_{n+1}-w^{2}_{n}+ \sum_{i=1}^{M}(\Phi _{i,n+2}\Psi _{i,n+1}+\Phi _{i,n+1}\Psi _{i,n})_{t} \\ \quad =\sum_{i=1}^{M}\bigl(\triangle ^{2}u_{n}\Phi _{i,n+1}\Psi _{i,n-1}-2 \triangle w_{n}\Phi _{i,n+1}\Psi _{i,n}\bigr)-\triangle \Biggl(\sum_{i=1}^{M} \Phi _{i,n+1} \Psi _{i,n}\Biggr)^{2}, \end{aligned} \end{aligned}$$
(49)
$$\begin{aligned}& \frac{\partial v_{n}}{\partial t}-\frac{\partial w_{n}}{\partial y}-u_{n+1}+u_{n}=- \sum _{i=1}^{M}\triangle (\Phi _{i,n} \Psi _{i,n}), \end{aligned}$$
(50)
$$\begin{aligned}& \frac{\partial \Phi _{i,n}}{\partial y}- \frac{\partial \Phi _{i,n}}{\partial z}=v_{n-1}\Phi _{i,n}+\Phi _{i,n-1}- \Phi _{i,n} \int ^{t}\frac{\partial w_{n}}{\partial y}\,\mathrm{d}t,\quad i=1,2, \ldots ,M, \end{aligned}$$
(51)
$$\begin{aligned}& \frac{\partial \Psi _{i,n}}{\partial y}- \frac{\partial \Psi _{i,n}}{\partial z}=-v_{n}\Phi _{i,n}- \Psi _{i,n+1}+ \Psi _{i,n} \int ^{t}\frac{\partial w_{n}}{\partial y}\,\mathrm{d}t,\quad i=1,2, \ldots ,M, \end{aligned}$$
(52)

where the operator is defined by \(\triangle u_{n}=u_{n+1}-u_{n}\).

Casoratian determinant solution to the bilinear SESCS (22)–(28)

The authors in [21] derived the Casoratian determinant solution for the bilinear two-dimensional special lattice equation. Herein we present the Casoratian determinant solution to the bilinear special lattice ESCS in Eqs. (22)–(28), which can be expressed in the following Pfaffian form:

$$\begin{aligned}& \begin{aligned}[b] f_{n}&=\operatorname{det} \bigl\vert \phi _{i}(n+j-1) \bigr\vert _{1\leqslant i,j\leqslant N} \\ & \triangleq (d_{0},d_{1},\ldots ,d_{N-1},N, \ldots ,2,1)_{n}, \end{aligned} \end{aligned}$$
(53)
g i , n = C ˙ i ( t ) ( d 1 , d 0 , , d N 1 , N , , 2 , 1 , b i ) n ,
(54)
h i , n = C ˙ i ( t ) ( d 1 , , d N 1 , N , , i ˆ , , 2 , 1 ) n .
(55)

Here the functions \(\phi _{i}(n)\) are given by the expression

$$ \phi _{i}(n)=\varphi _{i1}(n)+(-1)^{i-1}\varphi _{i2}(n),\quad 1\leq i \leq N, $$

where \(\varphi _{i1}(n)\) and \(\varphi _{i2}(n)\) satisfy the dispersion relations of Eqs. (12)–(13), and the functions \(C_{i}(t)\) have the form

$$ C_{i}(t)= \textstyle\begin{cases} C_{i}(t),& 1\leq i\leq M\leq N, \\ \text{constant},& \text{otherwise}. \end{cases} $$

At the same time the Pfaffian entries are defined by

$$\begin{aligned}& (d_{m},i)_{n}=\phi _{i}(n+m),\qquad (d_{m},d_{l})_{n}=(i,j)_{n}=0,\quad m \in \mathbf{Z}, \\& (d_{m},b_{i})_{n}=\varphi _{i2}(n+m), \qquad (b_{i},b_{j})_{n}=(b_{i},j)_{n}=0. \end{aligned}$$

Moreover, the other solutions \(k_{i}\), \(P_{i,n}\), and \(Q_{i,n} \) have the forms

$$\begin{aligned}& k_{i,n}=\dot{C}_{i}(t) (d_{0},d_{1}, \ldots ,d_{N-1},N,\ldots ,\hat{i}, \ldots ,1,b_{i})_{n}, \end{aligned}$$
(56)
P i , n = C ¨ i ( t ) 2 C ˙ i ( t ) ( d 1 , d 0 , , d N 1 , N , , 2 , 1 , b i ) n + C ˙ i ( t ) [ 1 j N j i ( d 1 , d 0 , , d N 1 , N , , j ˆ , , 1 , b j , b i ) n ] ,
(57)
Q i , n = C ¨ i ( t ) 2 C ˙ i ( t ) ( d 1 , , d N 1 , N , , i ˆ , , 1 ) n + C ˙ i ( t ) [ 1 j N j i ( d 1 , , d N 1 , N , , j ˆ , , i ˆ , , 1 , b j ) n ] ,
(58)

where \(i = 1, 2,\ldots , M\).

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 \(h_{i,n}\):

$$\begin{aligned}& f_{n}=(d_{0},d_{1},\ldots ,d_{N-1}, \bullet )_{n}, \end{aligned}$$
(59)
g i , n = C ˙ i ( t ) ( d 1 , d 0 , , d N 1 , , b i ) n ,
(60)
h i , n = C ˙ i ( t ) ( d 1 , , d N 1 , ) n .
(61)

Then we have the following formulas:

$$\begin{aligned}& \frac{\partial f_{n}}{\partial t}=\sum_{i=1}^{M}k_{i,n}+(d_{0}, \ldots ,d_{N-2},d_{N},\bullet )_{n}, \end{aligned}$$
(62)
$$\begin{aligned}& \begin{aligned} \frac{\partial ^{2} f_{n}}{\partial t^{2}}&=\sum _{i=1}^{M} \frac{\partial k_{i,n}}{\partial t}+\sum _{i=1}^{M}\dot{C}_{i}(t) (d_{0}, \ldots ,d_{N-2},d_{N},\star ,b_{i})_{n} \\ &\quad {}+(d_{0},\ldots ,d_{N-3},d_{N-1},d_{N}, \bullet )_{n}+(d_{0},\ldots ,d_{N-2},d_{N+1}, \bullet )_{n}, \end{aligned} \end{aligned}$$
(63)
$$\begin{aligned}& \begin{aligned} \frac{\partial ^{2} f_{n}}{\partial t\partial z}&=\sum _{i=1}^{M} \dot{C}_{i}(t) \bigl[(d_{0},\ldots ,d_{N-2},d_{N+1},\star ,b_{i})_{n} \\ &\quad {}-(d_{0},\ldots ,d_{N-3},d_{N-1},d_{N}, \star ,b_{i})_{n}\bigr]+(d_{0}, \ldots ,d_{N-2},d_{N+2},d_{N},\bullet )_{n} \\ &\quad {}-(d_{0},\ldots ,d_{N-4},d_{N-2},d_{N-1},d_{N}, \bullet )_{n}. \end{aligned} \end{aligned}$$
(64)

Further, we obtain the formulas for functions \(k_{i,n}\), \(g_{i,n}\), and \(h_{i,n}\) by

$$\begin{aligned}& k_{i,n+1}=\dot{C}_{i}(t) (d_{0},d_{1}, \ldots ,d_{N-1},\star ,b_{i})_{n}, \end{aligned}$$
(65)
g i , n + 1 = C ˙ i ( t ) ( d 0 , d 1 , , d N , , b i ) n ,
(66)
g i , n + 1 t = P i , n + 1 + C ˙ i ( t ) ( d 0 , , d N 1 , d N + 1 , , b i ) n ,
(67)
h i , n t = Q i , n + C ˙ i ( t ) ( d 1 , , d N 2 , d N , ) n .
(68)

Now substituting Eqs. (62)–(68) into Eq. (22) yields

$$\begin{aligned}& \dot{C}_{i}(t)\bigl[(d_{1},d_{2},\ldots ,d_{N},\star ,b_{i})_{n}(d_{0},d_{1}, \ldots ,d_{N-2},d_{N},\bullet )_{n} \\& \quad {}-(d_{0},d_{1},\ldots ,d_{N-2},d_{N}, \star ,b_{i})_{n}(d_{1},d_{2}, \ldots ,d_{N},\bullet )_{n} \\& \quad {}+(d_{0},d_{1},\ldots ,d_{N},\bullet ,b_{i})_{n}(d_{1},\ldots ,d_{N-2},d_{N}, \star )_{n}\bigr] \\& \quad {}+\dot{C}_{i}(t)\bigl[(d_{0},d_{1}, \ldots ,d_{N-1},\star ,b_{i})_{n}(d_{1},d_{2}, \ldots ,d_{N-1},d_{N+1},\bullet )_{n} \\& \quad {}-(d_{1},d_{2},\ldots ,d_{N-1},d_{N+1}, \star ,b_{i})_{n}(d_{0},d_{1}, \ldots ,d_{N-1},\bullet )_{n} \\& \quad {}-(d_{1},d_{2},\ldots ,d_{N-1},d_{N+1}, \star )_{n}(d_{0},d_{1},\ldots ,d_{N-1}, \bullet ,b_{i})_{n}\bigr]=0, \end{aligned}$$

which is the sum of the Plücker identities of the determinants. Then substituting Eqs. (60)–(61) and (65)–(66) into Eq. (24) yields

$$\begin{aligned}& (d_{1},d_{2},\ldots ,d_{N},\star ,b_{i})_{n}(d_{0},d_{1},\ldots ,d_{N-1}, \bullet )_{n} \\& \quad {}-(d_{0},d_{1},\ldots ,d_{N-1},\star ,b_{i})_{n}(d_{1},d_{2},\ldots ,d_{N}, \bullet )_{n} \\& \quad {}+(d_{0},d_{1},\ldots ,d_{N},\bullet ,b_{i})_{n}(d_{1},\ldots ,d_{N-2},d_{N}, \star )_{n}=0, \end{aligned}$$

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:

$$\begin{aligned}& (d_{-1},d_{0},\ldots ,d_{N-1},\bullet ,b_{i})_{n}(d_{0},d_{1},\ldots ,d_{N-2},d_{N}, \bullet )_{n} \\& \quad {}-(d_{-1},d_{0},\ldots ,d_{N-2},d_{N}, \bullet ,b_{i})_{n}(d_{0},d_{1}, \ldots ,d_{N-1},\bullet )_{n} \\& \quad {}+(d_{0},d_{1},\ldots ,d_{N},\bullet ,b_{i})_{n}(d_{-1},d_{0},\ldots ,d_{N-2}, \bullet )_{n}=0. \end{aligned}$$

These expressions show that Eqs. (22), (24), and (25) are valid. The other Eqs. (22)–(28) can be proved similarly. Therefore functions in (53)–(58) constitute the Casoratian determinant solutions of the bilinear ESCS in Eqs. (22)–(28).

One- and two-soliton solutions of the SESCS in (49)–(52)

Starting from the Grammian determinant solutions of Eqs. (15)–(18) and the transformations of Eqs. (46)–(47), we can obtain explicit solutions of the two-dimensional special lattice ESCS Eqs. (49)–(52). In this section, we take \(M = 1\), and the coupled system is read as

$$\begin{aligned}& \begin{aligned} &\frac{\partial w_{n+1}}{\partial t}+\frac{\partial w_{n}}{\partial t}-v_{n+1}+v_{n}+w^{2}_{n+1}-w^{2}_{n}+( \Phi _{n+2}\Psi _{n+1}+\Phi _{n+1}\Psi _{n})_{t} \\ &\quad =\triangle ^{2}(u_{n}\Phi _{n+1}\Psi _{n-1})-2\triangle (w_{n} \Phi _{n+1}\Psi _{n})-\triangle (\Phi _{n+1}\Psi _{n})^{2}, \end{aligned} \end{aligned}$$
(69)
$$\begin{aligned}& \frac{\partial v_{n}}{\partial t}-\frac{\partial w_{n}}{\partial y}-u_{n+1}+u_{n}=- \triangle (\Phi _{n}\Psi _{n}), \end{aligned}$$
(70)
$$\begin{aligned}& \frac{\partial \Phi _{n}}{\partial y}- \frac{\partial \Phi _{n}}{\partial z}=v_{n-1}\Phi _{n}+\Phi _{n-1}- \Phi _{n} \int ^{t}\frac{\partial w_{n}}{\partial y}\,\mathrm{d}t, \end{aligned}$$
(71)
$$\begin{aligned}& \frac{\partial \Psi _{n}}{\partial y}- \frac{\partial \Psi _{n}}{\partial z}=-v_{n}\Phi _{n}- \Psi _{n+1}+ \Psi _{n} \int ^{t}\frac{\partial w_{n}}{\partial y}\,\mathrm{d}t, \end{aligned}$$
(72)

which is the special lattice equation with one self-consistent source. Now we derive the one-soliton and two-soliton solutions of this system.

Example 1

We choose the parameter \(N = 1\) in Eqs. (15)–(18). Then the functions \(\varphi (n)\), \(\psi (-n)\), and \(f_{n}\) can be expressed in the following form:

$$\begin{aligned}& \varphi (n)=p^{n}e^{\xi },\qquad \xi =\bigl(p^{2}+p^{-1} \bigr)y+pt+p^{2}z, \\& \psi (-n)=q^{-n}e^{\eta },\qquad \eta =-\bigl(q^{2}+q^{-1} \bigr)y-qt-q^{2}z, \\& f_{n}=C_{1}(t)+\frac{1}{p-q} \biggl( \frac{p}{q} \biggr)^{n}e^{\xi + \eta },\qquad C_{1}(t)=e^{2a(t)}/(p-q), \end{aligned}$$
(73)

where p and q are arbitrary constants, and \(a(t)\) is a differentiable function of t. According to expressions (46)–(47), the explicit forms \(u_{n}\), \(w_{n}\), \(\Phi _{n}\), and \(\Psi _{n}\) are as follows:

$$\begin{aligned}& u_{n}= \frac{ [1+ (\frac{p}{q} )^{n+1}e^{\xi +\eta -2a(t)} ]\cdot [1+ (\frac{p}{q} )^{n-1}e^{\xi +\eta -2a(t)} ]}{ [1+ (\frac{p}{q} )^{n}e^{\xi +\eta -2a(t)} ]^{2}}, \end{aligned}$$
(74)
$$\begin{aligned}& w_{n}= \frac{(p-q-2\dot{a}(t)) (\frac{p-q}{q} ) (\frac{p}{q} )^{n}e^{\xi +\eta -2a(t)}}{ [1+ (\frac{p}{q} )^{n+1}e^{\xi +\eta -2a(t)} ]\cdot [1+ (\frac{p}{q} )^{n}e^{\xi +\eta -2a(t)} ]}, \end{aligned}$$
(75)
Φ n = 2 ( p q ) a ˙ ( t ) p n 1 e ξ a ( t ) 1 + ( p q ) n e ξ + η 2 a ( t ) ,
(76)
Ψ n = 2 ( p q ) a ˙ ( t ) q n 1 e η a ( t ) 1 + ( p q ) n e ξ + η 2 a ( t ) .
(77)

If we set \(a(t) = t\), \(p = 3\), \(q = 1/5\), \(t = 1\), and \(n = 2\), then the profiles of these solutions are as shown in Fig. 1.

Figure 1
figure1

One soliton of \(u_{n}\), \(w_{n}\), \(\Phi _{n}\), and \(\Psi _{n}\)

Example 2

We set \(N = 2\) in expressions (15)–(18), and the functions \(\varphi _{i}(n)\) and \(\psi _{i}(-n)\) possess the following structures:

$$\begin{aligned}& \varphi _{i}(n)=p_{i}^{n}e^{(p_{i}^{2}+p_{i}^{-1})y+p_{i}t+p_{i}^{2}z} \triangleq p_{i}^{n}e^{\xi _{i}},\quad i=1,2, \\& \psi _{i}(-n)=q_{i}^{-n}e^{-(q_{i}^{2}+q_{i}^{-1})y-q_{i}t-q_{i}^{2}z} \triangleq q_{i}^{-n}e^{\eta _{i}},\quad i=1,2, \end{aligned}$$

where \(p_{i}\) and \(q_{i}\) are real constants. In this case the function \(f_{n}\) is a second-order determinant, and \(C_{1}(t)\) included in \(f_{n}\) is chosen as \(e^{2\beta (t)}/(p_{1}-q_{1})\). Through computations we obtain the following form:

$$ f_{n}=\frac{e^{2\beta (t)}}{(p_{1}-q_{1})(p_{2}-q_{2})}\cdot \tilde{f}_{n}, $$
(78)

wherein the function \(\tilde{f}_{n}\) is defined by

$$\begin{aligned} \tilde{f}_{n}=1+\biggl(\frac{p_{1}}{q_{1}}\biggr)^{n}e^{\xi _{1}+\eta _{1}-2 \beta (t)}+ \biggl(\frac{p_{2}}{q_{2}}\biggr)^{n}e^{\xi _{2}+\eta _{2}} +A\biggl( \frac{p_{1}p_{2}}{q_{1}q_{2}}\biggr)^{n}e^{\xi _{1}+\eta _{1}+\xi _{2}+ \eta _{2}-2\beta (t)}, \end{aligned}$$
(79)

and \(A=\frac{(p_{1}-p_{2})(q_{1}-q_{2})}{(p_{1}-q_{2})(q_{1}-p_{2})}\). Subsequently, we obtain the expressions for the solutions \(u_{n}\) and \(w_{n}\) as

$$ u_{n}=\frac{\tilde{f}_{n+1}\tilde{f}_{n-1}}{\tilde{f}_{n}^{2}},\qquad w_{n}= \biggl(\ln \frac{\tilde{f}_{n+1}}{\tilde{f}_{n}}\biggr)_{t}. $$
(80)

Contrarily, the functions \(g_{n}\) and \(h_{n}\) in Eqs. (16)–(17) have the Pfaffian forms

g n = C ˙ 1 ( t ) ( d 1 , 1 , 2 , 2 ) n and h n = C ˙ 1 ( t ) ( d 1 , 2 , 2 , 1 ) n
(81)

and can be rewritten as

g n = 2 β ˙ ( t ) p 1 q 1 p 2 q 2 [ p 1 n 1 e ξ 1 + β ( t ) + ( p 1 p 2 ) ( p 1 q 2 ) ( p 1 p 2 q 2 ) n 1 e ξ 1 + ξ 2 + η 2 + β ( t ) ] ,
(82)
h n = 2 β ˙ ( t ) p 1 q 1 p 2 q 2 [ q 1 n 1 e η 1 + β ( t ) + ( q 1 q 2 ) ( q 1 p 2 ) ( p 2 q 1 q 2 ) n + 1 e ξ 2 + η 1 + η 2 + β ( t ) ] .
(83)

Finally, the solutions \(\Phi _{n}\) and \(\Psi _{n}\) have the following forms:

$$\begin{aligned}& \Phi _{n}=\gamma (t)\cdot \frac{p^{n-1}_{1}e^{\xi _{1}-\beta (t)}+\frac{(p_{1}-p_{2})}{(p_{1}-q_{2})}(\frac{p_{1}p_{2}}{q_{2}})^{n-1}e^{\xi _{1}+\xi _{2}+\eta _{2}-\beta (t)}}{\tilde{f}_{n}}, \end{aligned}$$
(84)
$$\begin{aligned}& \Psi _{n}=\gamma (t)\cdot \frac{q^{-n-1}_{1}e^{\eta _{1}-\beta (t)}+\frac{(q_{1}-q_{2})}{(q_{1}-p_{2})}(\frac{p_{2}}{q_{1}q_{2}})^{n+1}e^{\xi _{2}+\eta _{1}+\eta _{2}-\beta (t)}}{\tilde{f}_{n}}, \end{aligned}$$
(85)

where γ(t)= 2 ( p 1 q 1 ) β ˙ ( t ) . If we choose \(\beta (t) = t\), \(p_{1} = 1.5\), \(p_{2} = 0.25\), \(q_{1} = 1\), \(q_{2} = 0.5\), \(t = 1\), and \(n = -2\), then the profiles of the above solutions are as shown in Fig. 2.

Figure 2
figure2

Interactions of two solitons in (a) and (b) with the sources \(\Phi _{n}\) and \(\Psi _{n}\) as in (c) and (d)

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.

Availability of data and materials

The datasets used or analyzed during the current study are available from the corresponding author upon reasonable request.

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This work was partially supported by the National Natural Science Foundation of China (Grant no. 11271266).

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Wang, HY., Zhu, GQ. New type of source extension for a two-dimensional special lattice equation and determinant solutions. Adv Differ Equ 2021, 67 (2021). https://doi.org/10.1186/s13662-021-03219-w

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Keywords

  • Two-dimensional special lattice equation
  • Self-consistent sources
  • Determinant solution
  • Soliton solution
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