1 Introduction

The two-dimensional Toda lattice, which can be regarded as a spatial discretization of the KP equation, takes the following form:

$$ \frac{\partial^{2}}{\partial x \partial s} \ln(V_{n}+1)=V_{n+1}+V_{n-1}-2V _{n}, $$
(1)

where \(V_{n}\) denotes \(V(n,x,s)\). We use the above notation throughout the paper. Under the dependent variable transformation

$$ V_{n}=\frac{\partial^{2}}{\partial x \partial s}\ln f_{n}, $$
(2)

equation (1) is transformed into the bilinear form [1, 2]:

$$ D_{x}D_{s}f_{n}\cdot f_{n}=2\bigl(e^{D_{n}}f_{n}\cdot f_{n}-f^{2}_{n} \bigr), $$
(3)

where the bilinear operators are defined by [2]

$$\begin{aligned}& D_{x}^{m}D_{t}^{n}f\cdot g= \frac{\partial^{m}}{\partial y^{m}}\frac{ \partial^{n}}{\partial s^{n}}f(x+y,t+s)g(x-y,t-s)\bigg|_{s=0,y=0}, \\& e^{D_{n}}f_{n}\cdot g_{n}=f_{n+1}g_{n-1}. \end{aligned}$$

It is shown in [2, 3] that the two-dimensional Toda lattice equation possesses the following bilinear Bäcklund transformation:

$$\begin{aligned}& D_{x}f_{n+1}\cdot f'_{n}=- \frac{1}{\lambda}f_{n}f'_{n+1}+\nu f _{n+1}f'_{n}, \end{aligned}$$
(4)
$$\begin{aligned}& D_{s}f_{n}\cdot f'_{n}=\lambda f_{n+1}f'_{n-1}-\mu f_{n}f'_{n}, \end{aligned}$$
(5)

where λ, μ, ν are arbitrary constants. Equations (4)-(5) are transformed into the following nonlinear form:

$$\begin{aligned}& \frac{\partial}{\partial x}u_{n}=(\mu+u_{n}) (v_{n}-v_{n+1}), \end{aligned}$$
(6)
$$\begin{aligned}& \frac{\partial}{\partial s}v_{n}=(\nu+v_{n}) (u_{n-1}-u_{n}), \end{aligned}$$
(7)

through the dependent variable transformation \(u_{n}=\frac{\partial}{ \partial s}\ln(\frac{f_{n}}{f'_{n}})\), \(v_{n}=-\frac{\partial}{\partial x}\ln(\frac{f_{n}}{f'_{n-1}})\). Equations (4)-(5) or (6)-(7) are called the modified two-dimensional Toda lattice [2, 3]. The solutions \(V_{n}\) of the two-dimensional Toda lattice (1) and \(u_{n}\), \(v_{n}\) of the modified two-dimensional Toda lattice (6)-(7) are connected through a Miura transformation [2].

The soliton equations with self-consistent sources can model a lot of important physical processes. For example, the KdV equation with self-consistent sources describes the interaction of long and short capillary-gravity waves [4]. The KP equation with self-consistent sources describes the interaction of a long wave with a short-wave packet propagating on the \(x,y\) plane at an angle to each other [5, 6]. Since the pioneering work of Mel’nikov [7], lots of soliton equations with self-consistent sources have been studied via inverse scattering methods [7–11], Darboux transformation methods [12–17], Hirota’s bilinear method and the Wronskian technique [18–24].

In [25], a new algebraic method, called the source generation procedure, is proposed to construct and solve the soliton equations with self-consistent sources both in continuous and discrete cases. The source generation procedure has been successfully applied to many \((2+1)\)-dimensional continuous and discrete soliton equations such as the Ishimori-I equation [26], the semi-discrete Toda equation [27], the modified discrete KP equation [28], and others. The purpose of this paper is to construct the modified two-dimensional Toda lattice with self-consistent sources via the source generation procedure and clarify the determinant structure of N-soliton solution for the modified two-dimensional Toda lattice with self-consistent sources.

The paper is organized as follows. In Section 2, we derive the Grammian solution to the modified two-dimensional Toda lattice equation and then construct the two-dimensional Toda lattice equations with self-consistent sources. In Section 3, the Casoratian formulation of N-soliton solution for the modified two-dimensional Toda lattice with self-consistent is given. Section 4 is devoted to showing that the commutativity of the source generation procedure and Bäcklund transformation is valid for two-dimensional Toda lattice. We end this paper with a conclusion and discussion in Section 5.

2 The modified two-dimensional Toda lattice equation with self-consistent sources

The N-soliton solution in Casoratian form for the modified two-dimensional Toda lattice equation (4)-(5) is given in [2] and [29]. In this section, we first derive the Grammian formulation of the N-soliton solution for the modified two-dimensional Toda lattice equation, and then we construct the modified two-dimensional Toda lattice equation with self-consistent sources via the source generation procedure.

If we choose \(\lambda=1\), \(\nu=\mu=0\), then the modified two-dimensional Toda lattice (4)-(5) becomes

$$\begin{aligned}& \bigl(D_{x}e^{\frac{1}{2}D_{n}}+e^{-\frac{1}{2}D_{n}}\bigr)f_{n} \cdot f'_{n}=0, \end{aligned}$$
(8)
$$\begin{aligned}& \bigl(D_{s}-e^{D_{n}}\bigr)f_{n}\cdot f'_{n}=0. \end{aligned}$$
(9)

Proposition 1

The modified two-dimensional Toda lattice (8)-(9) has the following Grammian determinant solution:

$$\begin{aligned}& f_{n}=\det \biggl\vert c_{ij}+(-1)^{n} \int_{-\infty}^{x}\phi_{i}(n)\psi _{j}(-n)\,dx \biggr\vert _{1\leq i,j \leq N}= \vert M \vert , \end{aligned}$$
(10)
$$\begin{aligned}& f'_{n}(n,x,s)= \left \vert \textstyle\begin{array}{@{}c@{\quad}c@{}} M & \Phi(n) \\ \Psi(n)^{T} & -\phi_{N+1}(n) \end{array}\displaystyle \right \vert , \end{aligned}$$
(11)

where

$$\begin{aligned}& \Phi(n)=\bigl(-\phi_{1}(n),\ldots,-\phi_{N}(n) \bigr)^{T}, \end{aligned}$$
(12)
$$\begin{aligned}& \begin{aligned}[b] \Psi(n)&=\biggl(c_{N+1,1}+(-1)^{n} \int_{-\infty}^{x}\phi_{N+1}(n)\psi _{1}(-n)\,dx,\ldots, \\ &\quad c_{N+1,N}+ \int_{-\infty}^{x}(-1)^{n}\phi_{N+1}(n) \psi_{N}(-n)\,dx\biggr)^{T}, \end{aligned} \end{aligned}$$
(13)

in which the \(\phi_{i}(n)\) denote \(\phi_{i}(n,x,s)\) and the \(\psi_{i}(-n)\) denote \(\psi_{i}(-n,x,s)\) for \(i=1,\ldots,N+1\). In addition, \(c_{ij}\) (\(1\leq i,j \leq N+1\)) are arbitrary constants and \(\phi_{i}(n)\), \(\psi_{i}(-n)\) (\(i=1,\ldots,N+1\)) satisfy the following dispersion relations:

$$\begin{aligned}& \frac{\partial\phi_{i}(n)}{\partial x}= \phi_{i}(n+1),\quad \quad\frac{ \partial\psi_{i}(-n)}{\partial x}= \psi_{i}(-n+1), \end{aligned}$$
(14)
$$\begin{aligned}& \frac{\partial\phi_{i}(n)}{\partial s}= -\phi_{i}(n-1),\quad \quad\frac{ \partial\psi_{i}(-n)}{\partial s}= - \psi_{i}(-n-1). \end{aligned}$$
(15)

Proof

The Grammian determinants \(f_{n}\) in (10) and \(f'_{n}\) in (11) can be expressed in terms of the following Pfaffians:

$$\begin{aligned}& f_{n}=\bigl(a_{1},\ldots,a_{N},a^{*}_{N}, \ldots,a^{*}_{1}\bigr)=(\star), \end{aligned}$$
(16)
$$\begin{aligned}& f'_{n}=\bigl(a_{1},\ldots,a_{N+1},d^{*}_{0},a_{N}^{*}, \ldots,a^{*} _{1}\bigr)=\bigl(a_{N+1},d^{*}_{0}, \star\bigr), \end{aligned}$$
(17)

where the Pfaffian elements are defined by

$$\begin{aligned}& \bigl(a_{i},a^{*}_{j}\bigr)_{n}=c_{ij}+(-1)^{n} \int_{-\infty}^{x}(-1)^{n} \phi_{i}(n)\psi_{j}(-n)\,dx, \end{aligned}$$
(18)
$$\begin{aligned}& \bigl(d^{*}_{m},a_{i}\bigr)= \phi_{i}(n+m),\bigl(d_{m},a^{*}_{j} \bigr)=(-1)^{n+m}\psi _{j}(-n+m), \end{aligned}$$
(19)
$$\begin{aligned}& (a_{i},a_{j})_{n}=\bigl(a^{*}_{i},a^{*}_{j} \bigr)_{n}=(d_{m},d_{k})=\bigl(d_{m},d ^{*}_{k}\bigr)=\bigl(d^{*}_{m},d^{*}_{k} \bigr)=0, \end{aligned}$$
(20)

in which \(i,j=1,\ldots,N+1\) and k, m are integers.

Using the dispersion relations (14)-(15), we can compute the following differential and difference formula for the Pfaffians (16)-(17):

$$\begin{aligned}& f_{n+1,x}=\bigl(d_{-1},d^{*}_{1}, \star\bigr), \qquad f_{n+1}=(\star)+\bigl(d_{-1},d^{*}_{0}, \star\bigr), \end{aligned}$$
(21)
$$\begin{aligned}& f_{ns}=\bigl(d_{-1},d^{*}_{-1}, \star\bigr),\quad\quad f'_{nx}=\bigl(a_{N+1},d^{*}_{1}, \star \bigr), \quad\quad f'_{n-1}=\bigl(a_{N+1},d^{*}_{-1}, \star\bigr) \end{aligned}$$
(22)
$$\begin{aligned}& f'_{n+1}=\bigl(a_{N+1},d^{*}_{1}, \star\bigr)+ \bigl(a_{N+1},d_{-1},d^{*}_{o},d ^{*}_{1},\star\bigr), \end{aligned}$$
(23)
$$\begin{aligned}& f'_{ns}=\bigl(a_{N+1},d_{-1},d^{*}_{-1},d^{*}_{0}, \star\bigr)-\bigl(a_{N+1},d^{*} _{-1},\star\bigr). \end{aligned}$$
(24)

Substituting equations (21)-(24) into the modified two-dimensional Toda lattice (8)-(9) gives the following two Pfaffian identities:

$$\begin{aligned}& \bigl(d_{-1},d^{*}_{1},\star\bigr) \bigl(a_{N+1},d^{*}_{0},\star\bigr)- \bigl(d_{-1},d^{*} _{0},\star\bigr) \bigl(a_{N+1},d^{*}_{1},\star\bigr)+(\star) \bigl(a_{N+1},d_{-1},d^{*} _{0},d^{*}_{1}, \star\bigr)=0, \\& \bigl(d_{-1},d^{*}_{0},\star\bigr) \bigl(a_{N+1},d^{*}_{-1},\star\bigr)- \bigl(d_{-1},d^{*} _{-1},\star\bigr) \bigl(a_{N+1},d^{*}_{0},\star\bigr)+(\star) \bigl(a_{N+1},d_{-1},d^{*} _{-1},d^{*}_{0}, \star\bigr)=0. \end{aligned}$$

 □

In order to construct the modified two-dimensional Toda lattice with self-consistent sources, we change the Grammian determinant solutions (10)-(11) into the following form:

$$\begin{aligned}& f(n,x,s)=\det \biggl\vert \gamma_{ij}(s)+(-1)^{n} \int_{-\infty}^{x}(-1)^{n} \phi_{i}(n)\psi_{j}(-n)\,dx \biggr\vert _{1\leq i,j \leq N}= \vert F \vert , \end{aligned}$$
(25)
$$\begin{aligned}& f'_{n}(n,x,s)= \left \vert \textstyle\begin{array}{@{}c@{\quad}c@{}} F & \Phi(n) \\ \Psi(n)^{T} & -\phi_{N+1}(n) \end{array}\displaystyle \right \vert , \end{aligned}$$
(26)

where Nth column vectors \(\Phi(n)\), \(\Psi(n)\) are given in (12)-(13) and \(\phi_{i}(n)\), \(\psi_{i}(-n)\) (\(i=1,\ldots, {N+1}\)) also satisfy the dispersion relations (14)-(15). In addition, \(\gamma_{ij}(s)\) satisfies

$$\begin{aligned} \gamma_{ij}(s) = \textstyle\begin{cases} \gamma_{i}(s), & i=j\text{ and } 1\leq i \leq K \leq N, \\ c_{ij}, & \text{otherwise}, \end{cases}\displaystyle \end{aligned}$$
(27)

with \(\gamma_{i}(s)\) being an arbitrary function of s and K being a positive integer.

The Grammian determinants \(f_{n}\) in (25) and \(f'_{n}\) in (26) can be expressed by means of the following Pfaffians:

$$\begin{aligned}& f_{n}=\bigl(1,\ldots,N,N^{*},\ldots,1^{*} \bigr)=(\cdot), \end{aligned}$$
(28)
$$\begin{aligned}& f'_{n}=\bigl(1,\ldots,N+1,d^{*}_{0},N^{*}, \ldots,1^{*}\bigr)=\bigl(N+1,d^{*} _{0},\cdot \bigr), \end{aligned}$$
(29)

where the Pfaffian elements are defined by

$$\begin{aligned}& \bigl(i,j^{*}\bigr)_{n}=\gamma_{ij}(s)+(-1)^{n} \int_{-\infty}^{x}(-1)^{n} \phi_{i}(n)\psi_{j}(-n)\,dx,\quad\quad \bigl(i^{*},j^{*} \bigr)_{n}=0, \end{aligned}$$
(30)
$$\begin{aligned}& \bigl(d^{*}_{m},i\bigr)=\phi_{i}(n+m),\quad\quad \bigl(d_{m},j^{*}\bigr)=(-1)^{n+m}\psi _{j}(-n+m),\quad\quad (i,j)_{n}=0, \end{aligned}$$
(31)
$$\begin{aligned}& (d_{m},i)=\bigl(d^{*}_{m},j^{*} \bigr)=(d_{m},d_{k})=\bigl(d_{m},d^{*}_{k} \bigr)=\bigl(d^{*} _{m},d^{*}_{k} \bigr)=0, \end{aligned}$$
(32)

in which \(i,j=1,\ldots,N+1\) and k, m are integers.

It is easy to show that the functions \(f(n,x,s)\), \(f'(n,x,s)\) given in (28)-(29) still satisfy equation (8). However, they will not satisfy (9), and they satisfy the following new equation:

$$ D_{s}f_{n}\cdot f'_{n}-f_{n+1}f'_{n-1}=- \sum_{j=1}^{K}g_{n}^{(j)}h _{n}^{(j)}, $$
(33)

where the new functions \(g_{n}^{(j)}\), \(h_{n}^{(j)}\) are given by

$$\begin{aligned}& g_{n}^{(j)}=\sqrt{\dot{\gamma}_{j}(t)}\bigl(1, \ldots,N,d^{*}_{0},N ^{*},\ldots, \hat{j^{*}},\ldots,1^{*}\bigr), \end{aligned}$$
(34)
$$\begin{aligned}& h_{n}^{(j)}=\sqrt{\dot{\gamma}_{j}(t)}\bigl(1, \ldots,\hat{j},\ldots ,N+1,N^{*},\ldots,1^{*}\bigr), \end{aligned}$$
(35)

where \(j=1,\ldots,K\) and the dot denotes the derivative of \(\gamma_{j}(t)\) with respect to t. Furthermore, we can show that \(f_{n}\), \(f'_{n}\), \(g_{n}^{(j)}\), \(h_{n}^{(j)}\) (\(j=1,\ldots,K\)) satisfy the following 2K equations:

$$\begin{aligned}& \bigl(D_{x}e^{\frac{1}{2}D_{n}}+e^{-\frac{1}{2}D_{n}}\bigr)f\cdot g_{n}^{(j)}=0, \quad j=1,\ldots,K, \end{aligned}$$
(36)
$$\begin{aligned}& \bigl(D_{x}e^{\frac{1}{2}D_{n}}+e^{-\frac{1}{2}D_{n}}\bigr)h_{n}^{(j)} \cdot f'_{n}=0, \quad j=1,\ldots,K. \end{aligned}$$
(37)

In fact, we have the following differential and difference formula for \(f_{n}\) in (28), \(f'_{n}\) in (29) and \(g_{n}^{(j)}\), \(h _{n}^{(j)}\) (\(j=1,\ldots,K\)) by employing the dispersion relations (14)-(15):

$$\begin{aligned}& \begin{aligned}[b] f_{ns}&=\bigl(d_{-1},d^{*}_{-1}, \cdot\bigr) \\ &\quad{} +\sum_{j=1}^{K}\dot{ \gamma}_{j}(s) \bigl(1,\ldots,\hat{i},\ldots,N,N^{*}, \ldots,\hat{i^{*}},\ldots,1^{*}\bigr), \end{aligned} \end{aligned}$$
(38)
$$\begin{aligned}& \begin{aligned}[b] f'_{ns}&=\bigl(N+1,d_{-1},d^{*}_{-1},d^{*}_{0}, \cdot\bigr)-\bigl(N+1,d^{*}_{-1}, \cdot\bigr) \\ &\quad {}+\sum_{i=1}^{K}\dot{ \gamma}_{i}(s) \bigl(N+1,d^{*}_{0},1,\ldots, \hat{i}, \ldots,N,N^{*},\ldots,\hat{i^{*}}, \ldots,1^{*}\bigr), \end{aligned} \end{aligned}$$
(39)
$$\begin{aligned}& f_{n+1}=(\cdot)+\bigl(d_{-1},d^{*}_{0}, \cdot\bigr),\quad\quad f'_{n-1}=\bigl(N+1,d^{*}_{-1}, \cdot\bigr), \end{aligned}$$
(40)
$$\begin{aligned}& g^{(j)}_{n-1}=\sqrt{\dot{\gamma}_{j}(t)}\bigl(1, \ldots,N,d^{*}_{-1},N ^{*},\ldots, \hat{j^{*}},\ldots,1^{*}\bigr), \end{aligned}$$
(41)
$$\begin{aligned}& \begin{aligned}[b] g_{n-1,x}^{(j)}&=\sqrt{\dot{\gamma}_{j}(t)}\bigl[ \bigl(1,\ldots,N,d^{*} _{0},N^{*},\ldots, \hat{j^{*}},\ldots,1^{*}\bigr) \\ &\quad{} +\bigl(1,\ldots,N,d_{0},d^{*}_{0},d^{*}_{-1},N^{*}, \ldots,\hat{j^{*}}, \ldots,1^{*}\bigr)\bigr], \end{aligned} \end{aligned}$$
(42)
$$\begin{aligned}& f_{n-1}=(\cdot)-\bigl(d_{0},d^{*}_{-1}, \cdot\bigr),\quad\quad f_{nx}=\bigl(d_{0},d^{*}_{0}, \ldots\bigr), \end{aligned}$$
(43)
$$\begin{aligned}& \begin{aligned}[b] h^{(j)}_{n+1}&=\sqrt{\dot{\gamma}_{j}(t)}\bigl[ \bigl(1,\ldots,\hat{j}, \ldots,N+1,N^{*},\ldots,1^{*}\bigr) \\ &\quad{} +\bigl(1,\ldots,\hat{j},\ldots,N+1,d_{-1},d^{*}_{0}N^{*}, \ldots,1^{*}\bigr)\bigr], \end{aligned} \end{aligned}$$
(44)
$$\begin{aligned}& h^{(j)}_{n+1,x}=\sqrt{\dot{\gamma}_{j}(t)} \bigl(1,\ldots,\hat{j}, \ldots,N+1,d_{-1},d^{*}_{1},N^{*}, \ldots,1^{*}\bigr), \end{aligned}$$
(45)
$$\begin{aligned}& f'_{nx}=\bigl(N+1,d^{*}_{1}, \cdot\bigr), \end{aligned}$$
(46)
$$\begin{aligned}& f'_{n+1}=\bigl(N+1,d^{*}_{1}, \cdot\bigr)+ \bigl(N+1,d_{-1},d^{*}_{0},d^{*}_{1}, \cdot\bigr), \end{aligned}$$
(47)

where \(\hat{\ }\) indicates deletion of the letter under it.

Substitution of equations (38)-(47) into equations (33), (36)-(37) gives the following Pfaffian identities:

$$\begin{aligned}& \bigl[\bigl(d_{-1},d^{*}_{-1},\cdot\bigr) \bigl(N+1,d^{*}_{0},\cdot\bigr)-(\cdot) \bigl(N+1,d _{-1},d^{*}_{-1},d^{*}_{0}, \cdot\bigr)-\bigl(d_{-1},d^{*}_{0},\cdot\bigr) \bigl(N+1,d ^{*}_{-1},\cdot\bigr)\bigr], \\& \quad{} +\sum_{j=1}^{K}\dot{ \gamma}_{j}(s)\bigl[\bigl(1,\ldots,N+1,d^{*}_{0},N^{*}, \ldots,1^{*}\bigr) \bigl(1,\ldots,\hat{i},\ldots,N,N^{*}, \ldots,\hat{i^{*}}, \ldots,1^{*}\bigr) \\& \quad{} -(\cdot) \bigl(1,\ldots,\hat{i},\ldots,N+1,d^{*}_{0},N^{*}, \ldots,i^{*}, \ldots,1^{*}\bigr) \\& \quad{} +\bigl(1,\ldots,N,d^{*}_{0},N^{*}, \ldots,\hat{i^{*}},\ldots,1^{*}\bigr) \bigl(1, \ldots, \hat{i},\ldots,N+1,N^{*},\ldots,1^{*}\bigr)\bigr]=0, \\& \bigl(d_{0},d^{*}_{0},\cdot\bigr) \bigl(1, \ldots,N,d^{*}_{-1},N^{*},\cdot, \hat{j^{*}},\ldots,1^{*}\bigr) \\& \quad{} -(\cdot) \bigl(1,\ldots,N,d_{0},d^{*}_{0},d^{*}_{-1},N^{*}, \cdot, \hat{j^{*}},\ldots,1^{*}\bigr) \\& \quad{} -\bigl(d_{0},d^{*}_{-1},\cdot \bigr) \bigl(1,\ldots,N,d^{*}_{0},N^{*},\cdot, \hat{j^{*}},\ldots,1^{*}\bigr)=0, \end{aligned}$$

and

$$\begin{aligned}& \bigl(N+1,d^{*}_{0},\cdot\bigr) \bigl(1,\ldots,\hat{i}, \ldots,N+1,d_{-1},d^{*} _{1},N^{*}, \ldots,1^{*}\bigr) \\& \quad{} -\bigl(N+1,d^{*}_{1},\cdot\bigr) \bigl(1, \ldots,\hat{i},\ldots,N+1,d_{-1},d^{*} _{0},N^{*}, \ldots,1^{*}\bigr) \\& \quad{} +\bigl(N+1,d_{-1},d^{*}_{0},d^{*}_{1}, \cdot\bigr) \bigl(1,\ldots,\hat{i},\ldots,N+1,N ^{*}, \ldots,1^{*}\bigr)=0, \end{aligned}$$

respectively. Therefore, equations (8), (33), (36)-(37) constitute the modified two-dimensional Toda lattice with self-consistent sources, and it possesses the Grammian determinant solution (28)-(29), (34)-(35).

Through the dependent variable transformation

$$ u_{n}=\frac{f_{n+1}f'_{n-1}}{f_{n}f'_{n}}, \quad\quad v_{n}=-\frac{\partial}{ \partial x}\ln \biggl(\frac{f_{n}}{f'_{n-1}}\biggr),\quad\quad G_{n}^{(j)}= \frac{g_{n}^{(j)}}{f _{n}},\quad\quad H_{n}^{(j)}=\frac{h_{n}^{(j)}}{f'_{n}}, $$
(48)

the bilinear modified two-dimensional Toda lattice with self-consistent sources (8, 33, 36)-(37) can be transformed into the following nonlinear form:

$$\begin{aligned}& \frac{\partial}{\partial x}u_{n}=u_{n}(v_{n}-v_{n+1}), \end{aligned}$$
(49)
$$\begin{aligned}& \frac{\partial}{\partial s}v_{n}=v_{n}(u_{n-1}-u_{n})+v_{n} \sum_{j=1}^{K}\bigl[u_{n}G_{n}^{(j)}H_{n}^{(j)}-u_{n-1}G_{n-1}^{(j)}H_{n-1} ^{(j)}\bigr], \end{aligned}$$
(50)
$$\begin{aligned}& \frac{\partial}{\partial x}G_{n-1}^{(j)}+G_{n}^{(j)}u_{n}v_{n}=0, \quad j=1,\ldots,K, \end{aligned}$$
(51)
$$\begin{aligned}& \frac{\partial}{\partial x}H_{n+1}^{(j)}+H_{n}^{(j)}u_{n}v_{n+1}=0, \quad j=1,\ldots,K. \end{aligned}$$
(52)

When we take \(G_{n}^{(j)}=H_{n}^{(j)}=0\), \(j=1,\ldots,K\) in (49)-(52), the nonlinear modified two-dimensional Toda lattice with self-consistent sources (49)-(52) is reduced to the nonlinear modified two-dimensional Toda lattice (6)-(7) with \(\lambda=1\), \(\nu=\mu=0\).

If we choose

$$ \begin{gathered} \phi_{i}(n)=e^{\xi_{i}},\quad\quad \psi_{i}(-n)=(-1)^{n}e^{\eta_{i}}, \\ \xi _{i}=e^{q_{i}}x+q_{i}n-e^{-q_{i}}t,\quad\quad \eta_{i}=-e^{Q_{i}}x-Q_{i}n+e^{-Q _{i}}t, \end{gathered} $$
(53)

where \(i=1,2,\ldots,N+1\) in the Grammian determinants (25)-(26), (34)-(35), then we obtain the N-soliton solution of the modified two-dimensional Toda lattice with self-consistent sources (8), (33), (36)-(37). Here \(q_{i}\), \(Q_{i}\) (\(i=1,2, \ldots,N+1\)) are arbitrary constants.

For example, if we take \(K=1\), \(N=1\) and

$$ \phi_{1}(n)=e^{\xi_{1}},\quad\quad \phi_{2}(n)=e^{\xi_{2}},\quad\quad \psi_{1}(n)=e^{\eta _{1}},\quad\quad \gamma_{1}(t)= \frac{e^{2a(t)}}{e^{q_{1}}-e^{Q_{1}}},\quad\quad c_{21}=0, $$
(54)

where \(a(t)\) is an arbitrary function of t, then we have

$$\begin{aligned}& f_{n}(x,n,t)=\frac{e^{2a(t)}}{e^{q_{1}}-e^{Q_{1}}}\bigl(1+e^{\xi_{1}+\eta _{1}-2a(t)}\bigr), \end{aligned}$$
(55)
$$\begin{aligned}& f'_{n}(x,n,t)=-\frac{e^{2a(t)+\xi_{2}}}{e^{q_{1}}-e^{Q_{1}}}\biggl(1+ \frac{e ^{q_{2}}-e^{q_{1}}}{e^{q_{2}}-e^{Q_{1}}}e^{\xi_{1}+\eta_{1}-2a(t)}\biggr), \end{aligned}$$
(56)
$$\begin{aligned}& g^{(1)}_{n}(x,n,t)=-\sqrt{ \frac{e^{2\dot{a}(t)}}{e^{q_{1}}-e^{Q1}}}e^{\xi_{1}+a(t)}, \end{aligned}$$
(57)
$$\begin{aligned}& h^{(1)}_{n}(x,n,t)= \sqrt{\frac{e^{2\dot{a}(t)}}{e^{q_{1}}-e^{Q1}}} \frac{1}{e^{q_{2}}-e ^{Q_{1}}}e^{\xi_{2}-\eta_{1}+a(t)}. \end{aligned}$$
(58)

Therefore, the one-soliton solution of the nonlinear modified two-dimensional Toda lattice with self-consistent sources (49)-(52) is given by

$$\begin{aligned}& u_{n}(x,n,t)=\frac{e^{-q_{2}}(1+e^{q_{1}-Q_{1}}e^{\xi_{1}+\eta _{1}-2a(t)})(1+\frac{e ^{q_{2}}-e^{q_{1}}}{e^{q_{2}}-e^{Q_{1}}}e^{Q_{1}-q_{1}}e^{\xi_{1}+\eta _{1}-2a(t)})}{(1+e^{\xi_{1}+\eta_{1}-2a(t)})(1+\frac{e^{q_{2}}-e^{q _{1}}}{e^{q_{2}}-e^{Q_{1}}}e^{\xi_{1}+\eta_{1}-2a(t)})}, \end{aligned}$$
(59)
$$\begin{aligned}& v_{n}(x,n,t)=-\frac{\partial}{\partial x}\ln\biggl(\frac{1+e^{\xi_{1}+ \eta_{1}-2a(t)}}{-e^{\xi_{2}}(1+\frac{e^{q_{2}}-e^{q_{1}}}{e^{q_{2}}-e ^{Q_{1}}}e^{Q_{1}-q_{1}}e^{\xi_{1}+\eta_{1}-2a(t)})}\biggr), \end{aligned}$$
(60)
$$\begin{aligned}& G^{(1)}_{n}(x,n,t)=-\sqrt{2\dot{a}(t) \bigl(e^{q_{1}}-e^{Q1} \bigr)}\frac{e ^{\xi_{1}-a(t)}}{1+e^{\xi_{1}+\eta_{1}-2a(t)}}, \end{aligned}$$
(61)
$$\begin{aligned}& H^{(1)}_{n}(x,n,t)=\frac{-\sqrt{2\dot{a}(t)(e^{q_{1}}-e^{Q1})}}{e ^{q_{2}}-e^{Q_{1}}}\frac{e^{-\eta_{1}-a(t)}}{1+\frac{e^{q_{2}}-e^{q _{1}}}{e^{q_{2}}-e^{Q_{1}}}e^{\xi_{1}+\eta_{1}-2a(t)}}. \end{aligned}$$
(62)

If we take \(K=1\), \(N=2\) and

$$\begin{aligned}& \phi_{1}(n)=e^{\xi_{1}},\quad\quad \phi_{2}(n)=e^{\xi_{2}},\quad\quad \phi_{3}(n)=e^{\xi _{3}},\quad\quad \psi_{1}(n)=e^{\eta_{1}},\quad\quad \psi_{2}(n)=e^{\eta_{2}}, \\& \gamma_{1}(t)=\frac{e^{2a(t)}}{e^{q_{1}}-e^{Q_{1}}},\quad\quad \gamma _{2}(t)= \frac{1}{e ^{q_{2}}-e^{Q_{2}}},\quad\quad c_{12}=0,\quad\quad c_{21}=0,\quad\quad c_{31}=0, \\& c_{32}=0, \end{aligned}$$

we derive

$$\begin{aligned}& \begin{aligned}[b] f_{n}(x,n,t)&=\frac{e^{2a(t)}}{(e^{q_{1}}-e^{Q_{1}})(e^{q_{2}}-e^{Q _{2}})}\biggl(1+e^{\xi_{1}+\eta_{1}-2a(t)}+e^{\xi_{2}+\eta_{2}} \\ &\quad{} +\frac{(e^{q_{1}}-e^{q_{2}})(e^{Q_{1}}-e^{Q_{2}})}{(e^{q_{1}}-e^{Q _{2}})(e^{Q_{1}}-e^{q_{2}})}e^{\xi_{1}+\eta_{1}+\xi_{2}+\eta_{2}-2a(t)}\biggr), \end{aligned} \end{aligned}$$
(63)
$$\begin{aligned}& \begin{aligned}[b] f'_{n}(x,n,t)&=-\frac{e^{\xi_{3}+2a(t)}}{(e^{q_{1}}-e^{Q_{1}})(e^{q _{2}}-e^{Q_{2}})}\biggl(1+ \frac{e^{q_{3}}-e^{q_{1}}}{e^{q_{3}}-e^{Q_{1}}}e ^{\xi_{1}+\eta_{1}-2a(t)}+\frac{e^{q_{3}}-e^{q_{2}}}{e^{q_{3}}-e^{Q _{2}}}e^{\xi_{2}+\eta_{2}} \\ &\quad{}+\frac{(e^{q_{1}}-e^{q_{2}})(e^{Q_{2}}-e^{Q_{1}})(e^{q_{3}}-e^{q_{2}})(e ^{q_{3}}-e^{q_{1}})}{(e^{q_{1}}-e^{Q_{2}})(e^{q_{2}}-e^{Q_{1}})(e^{q _{3}}-e^{Q_{2}})(e^{q_{3}}-e^{Q_{1}})}e^{\xi_{1}+\eta_{1}+\xi_{2}+\eta _{2}-2a(t)}\biggr), \end{aligned} \end{aligned}$$
(64)
$$\begin{aligned}& g^{(1)}_{n}(x,n,t)=\sqrt{\frac{e^{2\dot{a}(t)}}{e^{q_{1}}-e^{Q _{1}}}} \frac{e^{\xi_{1}+a(t)}}{e^{q_{2}}-e^{Q_{2}}}\biggl(1+\frac{e^{q_{1}}-e ^{q_{2}}}{e^{q_{1}}-e^{Q_{2}}}e^{\xi_{2}+\eta_{2}}\biggr), \end{aligned}$$
(65)
$$\begin{aligned}& \begin{aligned}[b] h^{(1)}_{n}(x,n,t)&=-\sqrt{\frac{e^{2\dot{a}(t)}}{e^{q_{1}}-e^{Q _{1}}}} \frac{e^{\xi_{3}+\eta_{1}+a(t)}}{(e^{q_{2}}-e^{Q_{2}})(e^{q _{3}}-e^{Q_{1}})} \\ &\quad{}\times \biggl(1+ \frac{(e^{q_{2}}-e^{q_{3}})(e^{Q_{1}}-e^{Q_{2}})}{(e^{Q_{2}}-e^{q _{3}})(e^{Q_{1}}-e^{q_{2}})}e^{\xi_{2}+\eta_{2}}\biggr) \end{aligned} . \end{aligned}$$
(66)

Substituting functions (63)-(66) into the dependent variable transformations (48), we obtain two-soliton solution of the nonlinear modified two-dimensional Toda lattice with self-consistent sources (49)-(52).

3 Casorati determinant solution to the modified two-dimensional Toda lattice equation with self-consistent sources

In Section 2, we derived that the modified two-dimensional Toda lattice with self-consistent sources (8), (33), (36)-(37) possess the Grammian determinant solution (25), (26), (34), (35). In this section, we derive the Casoratian formulation of the N-soliton for the modified two-dimensional Toda lattice with self-consistent sources (8), (33), (36)-(37).

Proposition 2

The modified two-dimensional Toda lattice with self-consistent sources (8), (33), (36)-(37) has the following Casorati determinant solution:

$$\begin{aligned}& f_{n}=\det \bigl\vert \psi_{i}(n+j-1) \bigr\vert _{1\leq i,j \leq N}=(d_{0}, \ldots,d_{N-1},N,\ldots,1), \end{aligned}$$
(67)
$$\begin{aligned}& f'_{n}=\det \bigl\vert \psi_{i}(n+j-1) \bigr\vert _{1\leq i,j \leq N+1}=(d _{0},\ldots,d_{N},N+1, \ldots,1), \end{aligned}$$
(68)
$$\begin{aligned}& g^{(j)}_{n}=\sqrt{\dot{\gamma}_{j}(t)}(d_{0}, \ldots,d_{N},N, \ldots,1,\alpha_{j}), \end{aligned}$$
(69)
$$\begin{aligned}& h^{(j)}_{n}=\sqrt{\dot{\gamma}_{j}(t)}(d_{0}, \ldots,d_{N-1},N+1, \ldots,\hat{j},\ldots,1), \end{aligned}$$
(70)

where \(\psi_{i}(n+m)=\phi_{i1}(n+m)+(-1)^{i-1}C_{i}(s)\phi_{i2}(n+m)\) (\(m=0, \ldots,N\)) and

$$\begin{aligned} C_{i}(s) = \textstyle\begin{cases} \gamma_{i}(s), & 1\leq i \leq K \leq N+1, \\ \gamma_{i}, & \textit{otherwise}, \end{cases}\displaystyle \end{aligned}$$
(71)

with \(\gamma_{i}(s)\) being an arbitrary function of s and K, N being positive integers. In addition, \(\phi_{i1}(n)\), \(\phi_{i2}(n)\) satisfy the following dispersion relations:

$$\begin{aligned} \frac{\partial\phi_{ij}(n)}{\partial x}= \phi_{ij}(n+1),\qquad\frac{ \partial\phi_{ij}(n)}{\partial s}= - \phi_{ij}(n-1), \quad j=1,2, \end{aligned}$$
(72)

and the Pfaffian elements are defined by

$$\begin{aligned}& (d_{m},i)=\psi_{i}(n+m), \qquad (d_{m}, \alpha_{i})=\phi_{i2}(n+m), \end{aligned}$$
(73)
$$\begin{aligned}& (d_{m},d_{l})=(i,j)=0, \qquad (\alpha_{i},j)=( \alpha_{i},\alpha_{j})=0, \end{aligned}$$
(74)

in which \(i,j=1,\ldots,N+1\) and m, l are integers.

Proof

We can derive the following dispersion relation for \(\psi_{i}(n)\) (\(i=1, \ldots,N+1\)) from equations (72):

$$\begin{aligned}& \frac{\partial\psi_{i}(n)}{\partial x}= \phi_{i}(n+1), \end{aligned}$$
(75)
$$\begin{aligned}& \frac{\partial\psi_{i}(n)}{\partial s}= -\psi_{i}(n-1)+(-1)^{i-1} \dot{C_{i}(t)}\phi_{i2}(n). \end{aligned}$$
(76)

Applying the dispersion relation (75)-(76), we can calculate the following differential and difference formula for the Casorati determinants (67)-(70):

$$\begin{aligned}& f_{n+1,x}=(d_{1},\ldots,d_{N-1},d_{N+1},N, \ldots,1), \end{aligned}$$
(77)
$$\begin{aligned}& f_{n+1}=(d_{1},\ldots,d_{N},N, \ldots,1),\quad\quad f_{n-1}=(d_{-1},\ldots,d _{N-2},N, \ldots,1) \end{aligned}$$
(78)
$$\begin{aligned}& f'_{nx}=(d_{0},\ldots,d_{N-1},d_{N+1},N+1, \ldots,1), \end{aligned}$$
(79)
$$\begin{aligned}& \begin{aligned}[b] f_{n,s}&=-(d_{-1},d_{1},\ldots,d_{N-1},N, \ldots,1) \\ &\quad{} +\sum_{j=1}^{K}\dot{ \gamma}_{j}(t) (d_{0},\ldots,d_{N-1},N,\ldots, \hat{j},\ldots,1,\alpha_{j}), \end{aligned} \end{aligned}$$
(80)
$$\begin{aligned}& \begin{aligned}[b] f'_{n,s}&=-(d_{-1},d_{1}, \ldots,d_{N},N+1,\ldots,1) \\ &\quad{} +\sum_{j=1}^{K}\dot{ \gamma}_{j}(t) (d_{0},\ldots,d_{N},N+1,\ldots, \hat{j},\ldots,1,\alpha_{j}), \end{aligned} \end{aligned}$$
(81)
$$\begin{aligned}& f'_{n+1}=(d_{1},\ldots,d_{N+1},N+1, \ldots,1), \end{aligned}$$
(82)
$$\begin{aligned}& f'_{n-1}=(d_{-1},d_{1}, \ldots,d_{N-1},N+1,\ldots,1), \end{aligned}$$
(83)
$$\begin{aligned}& g^{(j)}_{n}=\sqrt{\dot{\gamma}_{j}(t)}(d_{-1}, \ldots,d_{N},N, \ldots,1,\alpha_{j}), \end{aligned}$$
(84)
$$\begin{aligned}& h^{(j)}_{n+1}=\sqrt{\dot{\gamma}_{j}(t)}(d_{1}, \ldots,d_{N},N+1, \ldots,\hat{j},\ldots,1), \end{aligned}$$
(85)
$$\begin{aligned}& f_{nx}=(d_{0},\ldots,d_{N-2},d_{N},N, \ldots,1), \end{aligned}$$
(86)
$$\begin{aligned}& g^{(j)}_{n,x}=\sqrt{\dot{\gamma}_{j}(t)}(d_{-1}, \ldots,d_{N-2},d _{N},N,\ldots,1,\alpha_{j}), \end{aligned}$$
(87)
$$\begin{aligned}& h^{(j)}_{n+1,x}=\sqrt{\dot{\gamma}_{j}(t)}(d_{1}, \ldots,d_{N-1},d _{N+1},N+1,\ldots,\hat{j},\ldots,1). \end{aligned}$$
(88)

By substituting equations (77)-(88) into the modified two-dimensional Toda lattice with self-consistent sources (8), (33), (36)-(37), we obtain the following Pfaffian identities, respectively:

$$\begin{aligned}& (d_{1},\ldots,d_{N-1},d_{N+1},N,\ldots,1) (d_{0},\ldots,d_{N},N+1, \ldots,1) \\& \quad{} -(d_{1},\ldots,d_{N},N,\ldots,1) (d_{0},\ldots,d_{N-1},d_{N+1},N+1, \ldots,1) \\& \quad{} +(d_{0},\ldots,d_{N-1},N,\ldots,1) (d_{1},\ldots,d_{N+1},N+1,\ldots ,1)=0, \\& \bigl[-(d_{-1},d_{1},\ldots,d_{N-1},N, \ldots,1) (d_{0},\ldots,d_{N},N+1, \ldots,1) \\& \quad{} +(d_{0},\ldots,d_{N-1},N,\ldots,1) (d_{-1},d_{1},\ldots,d_{N},N+1, \ldots,1) \\& \quad{} -(d_{1},\ldots,d_{N},N,\ldots,1) (d_{-1},\ldots,d_{N-1},N+1,\ldots ,1)\bigr] \\& \quad{} +\sum_{j=1}^{K}\dot{ \gamma}_{j}(s)\bigl[(d_{0},\ldots,d_{N-1},N, \ldots ,\hat{j},\ldots,1,\alpha_{j}) (d_{0}, \ldots,d_{N},N+1,\ldots,1) \\& \quad{} -(d_{0},\ldots,d_{N},N+1,\ldots, \hat{j},\ldots,1,\alpha_{j}) (d_{0}, \ldots,d_{N-1},N, \ldots,1) \\& \quad{} +(d_{0},\ldots,d_{N},N,\ldots,1, \alpha_{j}) (d_{0},\ldots,d_{N-1},N+1, \ldots, \hat{j},\ldots,1)\bigr]=0, \\& (d_{0},\ldots,d_{N-2},d_{N},N,\ldots,1) (d_{-1},\ldots,d_{N-1},N, \ldots,1,\alpha_{j}) \\& \quad{} -(d_{0},\ldots,d_{N-1},N,\ldots,1) (d_{-1},\ldots,d_{N-2},d_{N},N, \ldots,1, \alpha_{j}) \\& \quad{} +(d_{-1},\ldots,d_{N-2},N,\ldots,1) (d_{0},\ldots,d_{N},N,\ldots,1, \alpha_{j})=0, \end{aligned}$$

and

$$\begin{aligned}& (d_{1},\ldots,d_{N-1},d_{N+1},N+1,\ldots, \hat{j},\ldots,1) (d_{0}, \ldots,d_{N},N+1,\ldots,1) \\& \quad{} -(d_{1},\ldots,d_{N},N+1,\ldots, \hat{j},\ldots,1) (d_{0},\ldots,d _{N-1},d_{N+1},N+1, \ldots,1) \\& \quad{} +(d_{0},\ldots,d_{N-1},N+1,\ldots, \hat{j},\ldots,1) (d_{1},\ldots,d _{N+1},N+1,\ldots,1)=0, \end{aligned}$$

respectively. □

In order to obtain the one-soliton solution of the nonlinear modified two-dimensional Toda lattice with self-consistent sources (49)-(52), we take \(N=1\), \(K=1\) and

$$\begin{aligned}& \phi_{11}=\frac{e^{\xi_{1}}}{e^{q_{1}}-e^{Q_{1}}},\quad\quad \phi_{12}=e^{-\eta _{1}},\quad\quad \phi_{21}=-\frac{e^{\xi_{2}}}{e^{q_{2}}-e^{Q_{1}}}, \\& \gamma_{1}(t)=\frac{e^{a(t)}}{e^{q_{1}}-e^{Q_{1}}},\quad\quad \gamma_{2}=0, \end{aligned}$$

in the Casoratian determinants (67)-(70). Here \(\xi_{i}\), \(\eta_{i}\) (\(i=1,2\)) are given in (53) and \(a(t)\) is an arbitrary function of t. Hence we obtain

$$\begin{aligned}& f_{n}(x,n,t)=\frac{e^{2a(t)-\eta1}}{e^{q_{1}}-e^{Q_{1}}}\bigl(1+e^{\xi _{1}+\eta_{1}-2a(t)}\bigr), \end{aligned}$$
(89)
$$\begin{aligned}& f'_{n}(x,n,t)=- \frac{e^{2a(t)+\xi_{2}-\eta_{1}}}{e^{q_{1}}-e^{Q_{1}}}\biggl(1+ \frac{e^{q _{2}}-e^{q_{1}}}{e^{q_{2}}-e^{Q_{1}}}e^{\xi_{1}+\eta_{1}-2a(t)}\biggr), \end{aligned}$$
(90)
$$\begin{aligned}& g^{(1)}_{n}(x,n,t)= \sqrt{\frac{e^{2\dot{a}(t)}}{e^{q_{1}}-e^{Q1}}}e^{\xi_{1}-\eta_{1}+a(t)}, \end{aligned}$$
(91)
$$\begin{aligned}& h^{(1)}_{n}(x,n,t)=-\sqrt{ \frac{e^{2\dot{a}(t)}}{e^{q_{1}}-e^{Q1}}} \frac{e^{\xi_{2}+a(t)}}{e ^{q_{2}}-e^{Q_{1}}}. \end{aligned}$$
(92)

Substituting functions (89)-(92) into the dependent variable transformations (48), we get a one-soliton solution of the nonlinear modified two-dimensional Toda lattice with self-consistent sources (49)-(52) given in (59)-(62).

If we take \(N=2\), \(K=1\) and

$$\begin{aligned}& \phi_{11}=\frac{e^{\xi_{1}}}{e^{q_{1}}-e^{Q_{1}}},\quad\quad \phi_{12}=e^{-\eta _{1}},\quad\quad \phi_{21}=-\frac{e^{\xi_{2}}}{e^{q_{2}}-e^{Q_{1}}},\\& \phi_{22}=e ^{\eta_{2}},\quad\quad \phi_{31}=\frac{e^{\xi_{3}}}{e^{q_{3}}-e^{Q_{1}}}, \\& \gamma_{1}(t)=\frac{e^{a(t)}}{e^{q_{1}}-e^{Q_{1}}},\quad\quad \gamma_{2}=- \frac{1}{e ^{q_{2}}-e^{Q_{1}}},\quad\quad \gamma_{3}=0, \end{aligned}$$

in the Casoratian determinants (67)-(70), we get

$$\begin{aligned}& \begin{aligned}[b] f_{n}(x,n,t)&=\frac{(e^{Q_{1}}-e^{Q_{2}})e^{2a(t)-\eta1-\eta2}}{(e ^{q_{2}}-e^{Q_{1}})(e^{q_{1}}-e^{Q_{1}})}\biggl(1+\frac{e^{q_{1}}-e^{Q_{2}}}{e ^{Q_{1}}-e^{Q_{2}}}e^{\xi_{1}+\eta_{1}-2a(t)}+ \frac{e^{Q_{1}}-e^{q _{2}}}{e^{Q_{1}}-e^{Q_{2}}}e^{\xi_{2}+\eta_{2}} \\ &\quad{} +\frac{e^{q_{1}}-e^{q_{2}}}{e^{Q_{1}}-e^{Q_{2}}}e^{\xi_{1}+\eta_{1}+ \xi_{2}+\eta_{2}-2a(t)}\biggr), \end{aligned} \end{aligned}$$
(93)
$$\begin{aligned}& \begin{aligned}[b] f'_{n}(x,n,t)&=\frac{(e^{Q_{1}}-e^{Q_{2}})(e^{q_{3}}-e^{Q_{2}})e^{2a(t)+ \xi_{3}-\eta_{1}-\eta _{2}}}{(e^{q_{2}}-e^{Q_{1}})(e^{q_{1}}-e^{Q_{1}})}\biggl(1+ \frac{(e ^{q_{1}}-e^{Q_{2}})(e^{q_{1}}-e^{q_{3}})}{(e^{Q_{1}}-e^{Q_{2}})(e^{Q _{1}}-e^{q_{3}})}e^{\xi_{1}+\eta_{1}-2a(t)}\hspace{-20pt} \\ &\quad{}+ \frac{(e^{q_{2}}-e^{Q_{1}})(e^{q_{2}}-e^{q_{3}})}{(e^{Q_{1}}-e^{Q _{2}})(e^{q_{3}}-e^{Q_{2}})}e^{\xi_{2}+\eta_{2}} \\ &\quad{} +\frac{(e^{q_{1}}-e ^{q_{2}})(e^{q_{1}}-e^{q_{3}})(e^{q_{3}}-e^{q_{2}})}{(e^{Q_{1}}-e^{Q _{2}})(e^{Q_{1}}-e^{q_{3}})(e^{q_{3}}-e^{Q_{2}}))}e^{\xi_{1}+\eta_{1}+ \xi_{2}+\eta_{2}-2a(t)}\biggr), \end{aligned} \end{aligned}$$
(94)
$$\begin{aligned}& g^{(1)}_{n}(x,n,t)=\sqrt{\frac{e^{2\dot{a}(t)}}{e^{q_{1}}-e^{Q _{1}}}}e^{a(t)+\xi_{1}-\eta_{1}-\eta_{2}} \biggl(\bigl(e^{q_{1}}-e^{q_{2}}\bigr)e^{\xi _{2}+\eta_{2}}+ \frac{(e^{Q_{2}}-e^{Q_{1}})(e^{q_{1}}-e^{Q_{2}})}{e ^{q_{2}}-e^{Q_{1}}}\biggr), \end{aligned}$$
(95)
$$\begin{aligned}& \begin{aligned}[b] h^{(1)}_{n}(x,n,t)&=\sqrt{\frac{e^{2\dot{a}(t)}}{e^{q_{1}}-e^{Q _{1}}}}e^{a(t)-\eta_{1}-\eta_{2}} \biggl(\frac{e^{Q_{2}}-e^{q_{3}}}{(e^{q _{3}}-e^{Q_{1}})(e^{q_{2}}-e^{Q_{1}})}e^{\xi_{3}+\eta_{1}} \\ &\quad{} + \frac{e^{q_{2}}-e^{q_{3}}}{(e^{q_{3}}-e^{Q_{1}})(e^{q_{2}}-e^{Q_{1}})}e ^{\xi_{2}+\xi_{3}+\eta_{1}+\eta_{2}}\biggr). \end{aligned} \end{aligned}$$
(96)

We introduce five constants \(\delta_{1}\), \(\delta_{2}\), \(\delta_{3}\), \(\epsilon _{1}\), \(\epsilon_{2}\) satisfying

$$\begin{aligned}& e^{\delta_{1}}=e^{Q_{2}}-e^{q_{1}},\quad\quad e^{\epsilon_{1}}= \frac{1}{e^{Q _{2}}-e^{Q_{1}}},\quad\quad e^{\delta_{3}}=e^{Q_{2}}-e^{q_{3}},\quad\quad e^{\delta_{2}+ \epsilon_{2}}= \frac{e^{Q_{1}}-e^{q_{2}}}{e^{Q_{1}}-e^{Q_{2}}}, \end{aligned}$$

and take

$$\begin{aligned}& \tilde{\xi}_{1}=\xi_{1}+\delta_{1},\quad\quad \tilde{ \xi}_{2}=\xi_{2}+\delta _{2},\quad\quad \tilde{ \xi}_{3}=\xi_{3}+\delta_{3},\quad\quad \tilde{ \eta}_{1}=\eta_{1}+ \epsilon_{1},\quad\quad \tilde{ \eta}_{2}=\eta_{2}+\epsilon_{2}, \end{aligned}$$

then equations (93)-(96) become

$$\begin{aligned}& \begin{aligned}[b] f_{n}(x,n,t)&=\frac{(e^{Q_{1}}-e^{Q_{2}})e^{\epsilon_{1}+\epsilon _{2}}e^{-\tilde{\eta}_{1}-\tilde{\eta}_{2}+2a(t)}}{(e^{q_{2}}-e^{Q _{1}})(e^{q_{1}}-e^{Q_{1}})}\biggl(1+e^{\tilde{\xi}_{1}+\tilde{\eta}_{1}-2a(t)}+e ^{\tilde{\xi}_{2}+\tilde{\eta}_{2}} \\ &\quad{} +\frac{(e^{q_{1}}-e^{q_{2}})(e^{Q_{1}}-e^{Q_{2}})}{(e^{q_{1}}-e^{Q _{2}})(e^{Q_{1}}-e^{q_{2}})}e^{\tilde{\xi}_{1}+\tilde{\eta}_{1}+ \tilde{\xi}_{2}+\tilde{\eta}_{2}-2a(t)}\biggr), \end{aligned} \end{aligned}$$
(97)
$$\begin{aligned}& \begin{aligned}[b] f'_{n}(x,n,t)&=-\frac{(e^{Q_{1}}-e^{Q_{2}})e^{\epsilon_{1}+\epsilon _{2}}e^{\tilde{\xi}_{3}-\tilde{\eta}_{1}-\tilde{\eta}_{2}+2a(t)}}{(e ^{q_{2}}-e^{Q_{1}})(e^{q_{1}}-e^{Q_{1}})}\biggl(1+ \frac{e^{q_{3}}-e^{q_{1}}}{e ^{q_{3}}-e^{Q_{1}}}e^{\tilde{\xi}_{1}+\tilde{\eta}_{1}-2a(t)}+\frac{e ^{q_{3}}-e^{q_{2}}}{e^{q_{3}}-e^{Q_{2}}}e^{\tilde{\xi}_{2}+ \tilde{\eta}_{2}}\hspace{-20pt} \\ &\quad{} +\frac{(e^{q_{1}}-e^{q_{2}})(e^{Q_{2}}-e^{Q_{1}})(e^{q_{3}}-e^{q_{2}})(e ^{q_{3}}-e^{q_{1}})}{(e^{q_{1}}-e^{Q_{2}})(e^{q_{2}}-e^{Q_{1}})(e^{q _{3}}-e^{Q_{2}})(e^{q_{3}}-e^{Q_{1}})}e^{\tilde{\xi}_{1}+ \tilde{\eta}_{1}+\tilde{\xi}_{2}+\tilde{\eta}_{2}-2a(t)}\biggr), \end{aligned} \end{aligned}$$
(98)
$$\begin{aligned}& g^{(1)}_{n}(x,n,t)=\sqrt{\frac{e^{2\dot{a}(t)}}{e^{q_{1}}-e^{Q _{1}}}} \frac{e^{\epsilon_{1}+\epsilon_{2}}(e^{Q_{1}}-e^{Q_{2}})e^{ \tilde{\xi}_{1}-\tilde{\eta}_{1}-\tilde{\eta}_{2}+a(t)}}{e^{q_{2}}-e ^{Q_{1}}}\biggl(1+\frac{e^{q_{1}}-e^{q_{2}}}{e^{q_{1}}-e^{Q_{2}}}e^{ \tilde{\xi}_{2}+\tilde{\eta}_{2}}\biggr), \end{aligned}$$
(99)
$$\begin{aligned}& \begin{aligned}[b] h^{(1)}_{n}(x,n,t)&=\sqrt{\frac{e^{2\dot{a}(t)}}{e^{q_{1}}-e^{Q _{1}}}} \frac{(e^{Q_{2}}-e^{Q_{1}})e^{\epsilon_{1}+\epsilon_{2}}e^{ \tilde{\xi}_{3}-\tilde{\eta}_{2}+a(t)}}{(e^{q_{2}}-e^{Q_{1}})(e^{q _{3}}-e^{Q_{1}})} \\ &\quad{}\times \biggl(1+\frac{(e^{q_{2}}-e^{q_{3}})(e^{Q_{1}}-e^{Q_{2}})}{(e^{Q_{2}}-e^{q_{3}})(e ^{Q_{1}}-e^{q_{2}})}e^{\tilde{\xi}_{2}+\tilde{\eta}_{2}} \biggr). \end{aligned} \end{aligned}$$
(100)

We rederive the two-soliton solution of the nonlinear modified two-dimensional Toda lattice with self-consistent sources (49)-(52) obtained in Section 2, substituting the above functions in equations (97)-(100) into the dependent variable transformation (48).

4 Commutativity of the source generation procedure and Bäcklund transformation

In this section, we show that the commutativity of the source generation procedure and Bäcklund transformation holds for the two-dimensional Toda lattice. For this purpose, we derive another form of the modified two-dimensional Toda lattice with self-consistent sources which is the Bäcklund transformation for the two-dimensional Toda lattice with self-consistent sources given in [25].

We have shown that the Casorati determinants \(f_{n}\), \(f'_{n}\), \(g^{(j)} _{n}\), \(h^{(j)}_{n}\) given in (67)-(70) satisfy the modified two-dimensional Toda lattice with self-consistent sources (8), (33), (36)-(37). Now we take

$$\begin{aligned}& F_{n}=f_{n}=\det \bigl\vert \psi_{i}(n+j-1) \bigr\vert _{1\leq i,j \leq N}=(d _{0},\ldots,d_{N-1},N, \ldots,1), \end{aligned}$$
(101)
$$\begin{aligned}& \begin{aligned}[b] F'_{n}&=f'_{n-1}=\det \bigl\vert \psi_{i}(n+j-1) \bigr\vert _{1\leq i,j \leq N+1} \\ &= (d_{-1}, \ldots,d_{N-1},N+1,\ldots,1), \end{aligned} \end{aligned}$$
(102)
$$\begin{aligned}& \begin{aligned}[b] G^{(j)}_{n}&=\sqrt{2}g^{(j)}_{n-1}=\sqrt{2 \dot{\gamma}_{j}(t)}(d _{-1},\ldots,d_{N-1},N, \ldots,1,\alpha_{j}), \\ &\quad j=1,\ldots,K, \end{aligned} \end{aligned}$$
(103)
$$\begin{aligned}& \begin{aligned}[b] H^{\prime(j)}_{n}&=\sqrt{2}h^{(j)}_{n}=\sqrt{2 \dot{\gamma}_{j}(t)}(d _{0},\ldots,d_{N-1},N+1, \ldots,\hat{j},\ldots,1), \\ &\quad j=1,\ldots,K, \end{aligned} \end{aligned}$$
(104)

and we introduce two new fields

$$\begin{aligned}& G^{\prime(j)}_{n}=\sqrt{2\dot{\gamma}_{j}(t)}(d_{-2}, \ldots,d_{N-1},N+1, \ldots,1,\alpha_{j}),\quad j=1,\ldots,K, \end{aligned}$$
(105)
$$\begin{aligned}& H^{(j)}_{n}=\sqrt{2\dot{\gamma}_{j}(t)}(d_{1}, \ldots,d_{N-1},N, \ldots,\hat{j},\ldots,1),\quad j=1,\ldots,K, \end{aligned}$$
(106)

where the Pfaffian elements are defined in (67)-(74).

In [25], the authors prove that the Casorati determinant \(F_{n}\), \(G^{(j)}_{n}\), \(H^{(j)}_{n}\) solves the following two-dimensional Toda lattice with self-consistent sources [25]:

$$\begin{aligned}& \bigl(D_{x}D_{s}-2e^{D_{n}}+2\bigr)F_{n} \cdot F_{n}=-\sum_{j=1}^{K}e^{D_{n}}G _{n}^{(j)}H_{n}^{(j)}, \end{aligned}$$
(107)
$$\begin{aligned}& \bigl(D_{x}+e^{-D_{n}}\bigr)F_{n}\cdot G_{n}^{(j)}=0,\quad j=1,\ldots,K, \end{aligned}$$
(108)
$$\begin{aligned}& \bigl(D_{x}+e^{-D_{n}}\bigr)H_{n}^{(j)} \cdot F_{n} =0,\quad j=1,\ldots,K. \end{aligned}$$
(109)

It is not difficult to show that the Casorati determinant with \(F'_{n}\), \(G^{\prime(j)}_{n}\), \(H^{\prime(j)}_{n}\) is another solution to the two-dimensional Toda lattice with self-consistent sources (107)-(109).

Furthermore, we can verify that the Casorati determinants \(F_{n}\), \(F'_{n}\), \(G ^{(j)}_{n}\), \(G^{\prime(j)}_{n}\), \(H^{(j)}_{n}\), \(H^{\prime(j)}_{n}\) given in (101)-(106) satisfy the following bilinear equations:

$$\begin{aligned}& 2\bigl(D_{s}e^{-1/2D_{n}}-e^{1/2D_{n}}\bigr)F_{n} \cdot F'_{n}=-\sum_{j=1}^{K}e ^{1/2D_{n}}G_{n}^{(j)}\cdot H^{\prime(j)}_{n}, \end{aligned}$$
(110)
$$\begin{aligned}& \bigl(D_{x}+e^{-D_{n}}\bigr)F_{n}\cdot F'_{n}=0,\quad j=1,\ldots,K, \end{aligned}$$
(111)
$$\begin{aligned}& \bigl(D_{x}+e^{-D_{n}}\bigr)H_{n}^{(j)} \cdot H^{\prime(j)}_{n} =0,\quad j=1,\ldots ,K, \end{aligned}$$
(112)
$$\begin{aligned}& \bigl(D_{x}+e^{-D_{n}}\bigr)G_{n}^{(j)} \cdot G^{\prime(j)}_{n} =0,\quad j=1,\ldots ,K, \end{aligned}$$
(113)
$$\begin{aligned}& \begin{aligned}[b] e^{1/2D_{n}}F_{n}\cdot H^{\prime(j)}_{n}&=e^{-1/2D_{n}}F_{n} \cdot H^{\prime(j)} _{n}-e^{-1/2D_{n}}H_{n}^{(j)} \cdot F'_{n}, \\ &\quad j=1,\ldots,K, \end{aligned} \end{aligned}$$
(114)
$$\begin{aligned}& \begin{aligned}[b] e^{1/2D_{n}} G_{n}^{(j)}\cdot F'_{n}&=e^{-1/2D_{n}}G_{n}^{(j)} \cdot F'_{n}-e^{-1/2D_{n}}F_{n}\cdot G^{\prime(j)}_{n}, \\ &\quad j=1,\ldots,K, \end{aligned} \end{aligned}$$
(115)

which is another form of the modified two-dimensional Toda lattice with self-consistent sources. It is proved in [30] that equations (110)-(115) constitute the Bäcklund transformation for the two-dimensional Toda lattice with self-consistent sources (107)-(109). Therefore, the commutativity of source generation procedure and Bäcklund transformation is valid for the two-dimensional Toda lattice.

5 Conclusion and discussion

In this paper, Grammian solutions to the modified two-dimensional Toda lattice are presented. From the Grammian solutions, the modified two-dimensional Toda lattice with self-consistent sources (8), (33), (36)-(37) are produced via the source generation procedure. We show that the modified two-dimensional Toda lattice with self-consistent sources (8), (33), (36)-(37) are resolved into the determinant identities by presenting its Grammian and Casorati determinant solutions. We also construct another form of the modified discrete KP equation with self-consistent sources (110)-(115) which is the Bäcklund transformation for the two-dimensional Toda lattice with self-consistent sources derived in [25].

Now we show that the modified two-dimensional Toda lattice has a continuum limit into the mKP equation [2, 31], and the modified two-dimensional Toda lattice with self-consistent sources (8, 33, 36)-(37) yields the mKP equation with self-consistent sources derived in [32] through a continuum limit. For this purpose, we take

$$\begin{aligned}& D_{n}=2\epsilon D_{X}-2\epsilon^{2}D_{Y},\quad\quad D_{x}=\epsilon^{2}D_{Y}+ \frac{3}{2} \epsilon D_{X},\quad\quad D_{s}=-\frac{16}{3} \epsilon^{3}D_{T}, \\& f(n,x,s)=F(X,Y,T),\quad\quad f'(n,x,s)=F'(X,Y,T), \end{aligned}$$

in the modified two-dimensional Toda lattice (8)-(9), and compare the \(\epsilon^{2}\) order in (8), and the \(\epsilon^{3}\) order in (9), then we obtain the mKP equation [2, 31]:

$$\begin{aligned}& \bigl(D_{Y}+D^{2}_{X}\bigr)F\cdot F' =0, \\& \bigl(D^{3}_{X}-4D_{T}-3D_{X}D_{Y} \bigr)F\cdot F' =0, \end{aligned}$$

where F, \(F'\) denote \(F(X,Y,T)\), \(F'(X,Y,T)\), respectively.

By taking

$$\begin{aligned}& D_{n}=2\epsilon D_{X}-2\epsilon^{2}D_{Y},\quad\quad D_{x}=\epsilon^{2}D_{Y}+ \frac{3}{2} \epsilon D_{X},\quad\quad D_{s}=\frac{4}{3} \epsilon^{3}D_{T}, \\& f(n,x,s)=F(X,Y,T),\quad\quad g^{(j)}(n,x,s)=\frac{2\sqrt{3}}{3} \epsilon^{\frac{3}{2}}G_{j}(X,Y,T), \\& f'(n,x,s)=F'(X,Y,T),\quad\quad h^{(j)}(n,x,s)= \frac{2\sqrt{3}}{3} \epsilon^{\frac{3}{2}}H_{j}(X,Y,T), \end{aligned}$$

for \(j=1,\ldots,K\) in the modified two-dimensional Toda lattice with self-consistent sources (8, 33, 36)-(37), and comparing the \(\epsilon^{2}\) order in (8), (36)-(37), and the \(\epsilon^{3}\) order in (33), we obtain the mKP equation with self-consistent sources [32]:

$$\begin{aligned}& \bigl(D_{Y}+D^{2}_{X}\bigr)F\cdot F' =0, \\& \bigl(D_{T}-3D_{X}D_{Y}+D^{3}_{X} \bigr)F\cdot F' =-\sum_{j=1}^{K}G_{j}H_{j}, \\& \bigl(D_{Y}+D^{2}_{X}\bigr)F \cdot G_{j} =0, \quad j=1,\ldots,K, \\& \bigl(D_{Y}+D^{2}_{X}\bigr)H_{j} \cdot F' =0, \quad j=1,\ldots,K, \end{aligned}$$

where F, \(F'\), \(G_{j}\), \(H_{j}\) denote \(F(X,Y,T)\), \(F'(X,Y,T)\), \(G_{j}(X,Y,T)\), \(H_{j}(X,Y,T)\) for \(j=1,\ldots,K\), respectively.

Recently, generalized Wronskian (Casorati) determinant solutions are constructed for continuous and discrete soliton equations [33–39]. Besides soliton solutions, a broader class of solutions such as rational solutions, negatons, positons and complexitons solutions are obtained from the generalized Wronskian (Casorati) determinant solutions [33–38]. In [39], a general Casoratian formulation is presented for the two-dimensional Toda lattice equation from which various examples of Casoratian type solutions are derived. It is interesting for us to construct the two-dimensional Toda lattice equation with self-consistent sources having a generalized Casorati determinant solution via the source generation procedure. This will bring us a broader class of solutions such as negatons, positons, and complexiton type solutions of the two-dimensional Toda lattice equation with self-consistent sources.