1 Introduction

It has been an important work to search for new lattice integrable systems, since such lattice systems not only have rich mathematical structures, e.g., Lax pairs, Bäcklund transformations, Hamiltonian structures, soliton solutions, and so on, but also they have many applications in mathematical physics, statistical physics, and quantum physics. Therefore, one tries to seek for various integrable discrete systems via various methods including mathematical and physical methods, such as the Ablowitz-Ladik lattice, the Toda lattice, the Lotka-Volterra lattice, the differential-difference KdV equation, the Suris lattices, and so on [1–10]. Fan and Yang [11] introduced an isospectral problem and derived a lattice hierarchy which reduced to the Ablowitz-Ladik and the Volterra hierarchies, respectively. As far as the \((2+1)\)-dimensional integrable discrete systems and their some properties are concerned, there are few works. For example, the \((2+1)\)-dimensional Toda lattice was presented and it was verified that it has a Lax pair, a Hamiltonian structure, and soliton solutions [12]. Two \((2+1)\)-dimensional integrable discrete hierarchies with three fields were constructed in terms of discrete zero curvature equations in [13]. Again in the case of a \((2+1)\)-dimensional non-isospectral linear problem, a new \((2+1)\)-dimensional integrable lattice hierarchy, which is a generalization of the discrete second Painlevé hierarchy, was investigated in [14]. By introducing fourth-order Lax matrices, two \((2+1)\)-dimensional integrable lattice hierarchies, which reduce to the two Mlaszak-Marciniak integrable lattice hierarchies, were generated [15]. Tu [16] once applied the Lie-algebra method to deduce the Toda lattice hierarchy and its Hamiltonian structure combined with the variational method. By following the way proposed by Zhang et al. [17] one obtained some integrable discrete hierarchies. One advantage for applying the Lie-algebra method to deduce integrable discrete hierarchies lies in adopting the well-known the Tu scheme [18], which conveniently introduces linear spectral problems and manipulates similar steps as the case of generating continuous integrable systems. Based on the scheme, Zhang and Tam [19] obtained two integrable discrete integrable coupled systems of the Toda lattice, including the linear and nonlinear discrete integrable couplings. All the works mentioned above were performed under matrix Lie algebras. In the paper, we would like to employ the first matrix loop algebra to generate \((1+1)\)- and \((2+1)\)-dimensional integrable discrete hierarchies, which generalize some results obtained in [14], furthermore, we also obtain a positive and negative integrable discrete hierarchy which implements the well-known results presented in [10, 13, 16, 17, 19–22]. We again discuss a discrete integrable coupling of the \((1+1)\)-dimensional integrable discrete hierarchy which possesses an arbitrary parameter derived by using an enlarging matrix loop algebra. Finally, we apply the second matrix loop algebra to generate a new integrable discrete hierarchy which can be reduced to a generalized Toda lattice equation, whose quasi-Hamiltonian structure is obtained. Furthermore, a Darboux transformation of a reduced differential-difference equation system of the latter discrete hierarchy is obtained. We introduce a discrete-operator associated algebra whose elements are just like the form

$$L=u_{\alpha+n}E^{\alpha+n}+u_{\alpha+n-1}E^{\alpha+n-1}+ \cdots+u_{\alpha}E^{\alpha},\quad -n< \alpha\leq-1. $$

Blaszak and Marciniak [23] discovered two types of operator Lie algebras based on the above general associative algebra:

$$\begin{aligned}& k=0:\quad L=E^{\alpha+n}+u_{\alpha+n-1}E^{\alpha+n-1}+ \cdots+u_{\alpha}E^{\alpha},\qquad u_{\alpha+n}=1, \\& k=1:\quad\bar{L}=\bar{u}_{\alpha+n}E^{\alpha+n}+\bar{u}_{\alpha+n-1}E^{\alpha +n-1}+ \cdots+ E^{\alpha},\qquad \bar{u}_{\alpha}=1. \end{aligned}$$

According to the operator Lie algebras, we shall introduce different isospectral problems according to deforms of the spectral problem (54) to deduce various lattice integrable systems, including the Toda lattice system, further we derive their Lax pairs by using the r-matrix theory. In the following, we first recall the simplest matrix Lie algebra,

$$A_{1}=\operatorname{span}\{h_{1},h_{2},e,f\}, $$

where

$$h_{1}= \begin{pmatrix} 1&0\\ 0&0 \end{pmatrix},\qquad h_{2}= \begin{pmatrix} 0&0\\ 0&1 \end{pmatrix},\qquad e= \begin{pmatrix} 0&1\\ 0&0 \end{pmatrix},\qquad f= \begin{pmatrix} 0&0\\ 1&0 \end{pmatrix}, $$

equipped with the commutative relations \(h_{1}h_{1}=h_{1},h_{2}h_{2}=h_{2},h_{1}h_{2}=h_{2}h_{1}=ee=ff=0,h_{1}e=e,eh_{1}=0, h_{1}f=0, fh_{1}=f\), \(h_{2}f=f,fh_{2}=0,h_{2}e=0,eh_{2}=e,ef=h_{1},fe=h_{2}\), from which we have \([h_{1},e]=e,[h_{1},f]=-f, [h_{2},e]=-e, [h_{2},f]=f,[e,f]=h\equiv h_{1}-h_{2},[h,e]=2e,[h,f]=-2f\). The first loop algebra corresponding to the Lie algebra \(A_{1}\) can be defined as

$$\tilde{A}_{1}=\operatorname{span}\bigl\{ h_{1}(n),h_{2}(n),e(n),f(n) \bigr\} , $$

where \(h_{1}(n)=h_{1}\lambda^{n}, h_{2}(n)=h_{2}\lambda^{n},e(n)=e\lambda^{n}, f(n)=f\lambda ^{n}, n\in\mathbf{Z}\).

The second loop algebra is given by

$$\bar{A}_{1}=\operatorname{span}\bigl\{ h_{1}(n),h_{2}(n),e(n),f(n) \bigr\} , $$

where \(h_{1}(n)=h_{1}\lambda^{2n},h_{2}(n)=h_{2}\lambda^{2n},e(n)=e\lambda ^{2n+1},f(n)=f\lambda^{2n+1}\).

The purpose for recalling the above two-loop algebras aims at introducing spectral Lax pairs, then with the help of various compatibility conditions, that is, various zero curvature equations, to generate different discrete integrable hierarchies. It is remarkable that the compatibility of some spectral Lax pairs can be transformed into Lax equations. Discussions of the tensorial form of the Lax pair equations were discovered in a compact and geometrically transparent form in the presence of Cartan’s torsion tensor, therefore, three dimensional spacetimes admitting Lax tensors were analyzed in [24]. Besides, Balean et al. in [25] investigated the connection between Killing tensors and Lax operators, and two examples, i.e., the Toda lattice system and the Rindler system, were analyzed in detail. Further developments on discrete equations focus on fractional difference equations and their different properties emerged. Wu et al. [26] showed that the Caputo-like delta derivative is adopted as the difference operator and the master-slave synchronization for the fractional difference equation was studied with a nonlinear coupling method. A lattice fractional diffusion equation was proposed in Ref. [27], and the numerical simulation of the diffusion procession was discussed for various difference orders. In addition, Wu et al. [28] proposed the fractional logistic map and fractional Lorenz maps of Riemann-Liouville type and the feedback control method was extended to discrete fractional equations. In Ref. [29], by the use of the Riemann-Liouville differences on time scales, the Riesz difference was defined in a consideration for discrete fractional modeling. Specially, the Adomian decomposition method was adopted to solve the fractional partial difference equations numerically. All the results presented in [24–29] could motivate us going on investigating the generating discrete equations and discussing their properties applied to physical and mathematical sciences.

2 Two integrable discrete hierarchies with three fields in \(1+1\) and \(2+1\) dimensions

Tu in [16] proposed a method for generating discrete integrable hierarchies by the use of loop algebras whose specific steps are as follows.

First introduce the spectral problem

$$\psi_{n+1}=U_{n}\psi_{n},\qquad\psi_{n,t}=V_{n} \psi_{n}. $$

Then solve the stationary discrete zero curvature equation

$$(E\Gamma)U_{n}-U_{n}\Gamma=0, $$

where

$$\Gamma= \begin{pmatrix} a&b\\ c&-a \end{pmatrix} =\sum_{m\geq0} \begin{pmatrix} a_{m}(n,t)&b_{m}(n,t)\\ c_{m}(n,t)&-a_{m}(n,t) \end{pmatrix} \lambda^{-m}, $$

to obtain some recurrence relations among \(a_{m},b_{m},c_{m}\).

Third, solve the discrete zero curvature equation

$$\frac{dU_{n}}{dt}=\bigl(EV^{(m)}_{n}\bigr)U_{n}-U_{n}V^{(m)}, $$

where

$$V_{n}^{(m)}=\bigl(\lambda^{m}\Gamma\bigr)_{+}+ \Delta_{m}(u,\lambda)=\sum_{i=0}^{m} \Gamma_{i}\lambda^{m-i}+\Delta_{m}(u,\lambda). $$

Finally, apply the trace identity

$$\frac{\delta}{\delta u}\operatorname{tr}\biggl(W\frac{\partial U_{n}}{\partial\lambda }\biggr)= \lambda^{-\gamma}\frac{\partial}{\partial\lambda}\lambda^{\gamma}\operatorname{tr}(WU_{u_{i}}),\quad i=1,2,\ldots, p, $$

to deduce the Hamiltonian structure of the discrete integrable hierarchies obtained by the discrete zero curvature equations. The above procedure for generating discrete integrable systems is called the Tu scheme.

In the following, we shall apply the Tu scheme and the first loop algebra \(\tilde{A}_{1}\) to generate \((1+1)\)- and \((2+1)\)-dimensional integrable discrete hierarchies, then generalize them to a unified model which is a positive and negative integrable discrete system.

2.1 A \((1+1)\)-dimensional integrable discrete hierarchy

Consider an isospectral problem

$$ \psi_{n+1}=U_{n}\psi_{n},\qquad \psi_{n,t}=V_{n}\psi_{n}, $$
(1)

where \(U_{n}=s_{n}h_{1}(1)+h_{2}(-1)+q_{n}e(0)+r_{n}f(0)\), \(V_{n}=A_{n}(h_{1}(0)-h_{2}(0))+B_{n}e(0)+C_{n}f(0)\), where

$$ A_{n}=\sum_{j\geq0}a_{j} \lambda^{-2j},\qquad B_{n}=\sum_{j\geq 0}b_{j} \lambda^{-2j+1},\qquad C_{n}=\sum_{j\geq0}c_{j} \lambda^{-2j+1}. $$
(2)

Denoting \(\Delta=E-1, Ef(n)=f(n+1),E^{-1}f(n)=f(n-1)\), and solving the stationary discrete zero curvature equation

$$ (\Delta V_{n})U_{n}=[U_{n},V_{n}] $$
(3)

yields

$$ \textstyle\begin{cases} \lambda s_{n}\Delta A_{n}+r_{n}\Delta B_{n}=q_{n}C_{n}-r_{n}B_{n},\\ q_{n}\Delta A_{n}+\lambda^{-1}\Delta B_{n}=\lambda s_{n}B_{n}-2q_{n}A_{n}-\lambda ^{-1}B_{n},\\ \lambda s_{n}\Delta C_{n}-r_{n}\Delta A_{n}=2r_{n}A_{n}+\lambda^{-1}C_{n}-\lambda s_{n}C_{n},\\ q_{n}\Delta C_{n}-\lambda^{-1}\Delta A_{n}=r_{n}B_{n}-q_{n}C_{n}. \end{cases} $$
(4)

Substituting (2) into (4) gives rise to

$$ \textstyle\begin{cases} s_{n}\Delta a_{j}+r_{n}\Delta b_{j}=q_{n}c_{j}-r_{n}b_{j},\\ q_{n}\Delta a_{j}+\Delta b_{j}=s_{n}b_{j+1}-2q_{n}a_{j}-b_{j},\\ s_{n}\Delta c_{j+1}-r_{n}\Delta a_{j}=2r_{n}a_{j}+c_{j}-s_{n}c_{j+1},\\ q_{n}\Delta c_{j+1}-\Delta a_{j}=r_{n}b_{j+1}-q_{n}c_{j+1}. \end{cases} $$
(5)

Taking the initial values \(b_{0}=c_{0}=0,a_{0}=1\), then we get from (5)

$$\begin{aligned}& b_{1}=\frac{2q_{n}}{s_{n}},\qquad c_{1}=\frac{2r_{n-1}}{s_{n-1}}, \qquad a_{1}=\frac {-2q_{n}r_{n-1}}{s_{n}s_{n-1}}, \\& b_{2}=\frac{2q_{n+1}}{s_{n}s_{n+1}}-\frac {2q_{n}r_{n}q_{n+1}}{s_{n}^{2}s_{n+1}}-\frac{2q_{n}^{2}r_{n-1}}{s_{n}^{2}s_{n-1}}, \\& c_{2}=\frac{-2q_{n}r_{n-1}^{2}}{s_{n}s_{n-1}^{2}}-\frac {2q_{n-1}r_{n-1}r_{n-2}}{s_{n-1}^{2}s_{n-2}}+\frac{2r_{n-2}}{s_{n-1}s_{n-2}}, \\& a_{2}=-2(E+1)\frac{q_{n}r_{n-2}}{s_{n}s_{n-1}s_{n-2}}+2(E+1)\frac {q_{n}q_{n-1}r_{n-1}r_{n-2}}{s_{n}s_{n-1}^{2}s_{n-2}}+ \frac {2q_{n}^{2}r_{n-1}^{2}}{s_{n}^{2}s_{n-1}^{2}}, \\& \ldots. \end{aligned}$$

Remark 1

Equation (3) is similar to the stationary zero curvature equation of continuous spectral problems

$$V_{x}=[U,V]. $$

Therefore, by the Tu scheme, we decompose equation (3) into the following form:

$$ -\bigl(\Delta V_{n}^{(m)}\bigr)_{+}U_{n}+ \bigl[U_{n},\bigl(V_{n}^{(m)}\bigr)_{+}\bigr]=\bigl( \Delta V_{n}^{(m)}\bigr)_{-}U_{n}-\bigl[U_{n}, \bigl(V_{n}^{(m)}\bigr)_{-}\bigr], $$
(6)

where

$$ \bigl(V_{n}^{(m)}\bigr)_{+}=\sum _{j=0}^{m}V_{n}\lambda^{2m}= \lambda^{2m}V_{n}-\bigl(V_{n}^{(m)} \bigr)_{-}. $$
(7)

The degree of the elements of the left-hand side of equation (6) is higher than −1, while the right-hand side is smaller than 0. Thus, the degree of both sides of equation (6) is \(-1,0\). Therefore, we obtain

$$-\bigl(\Delta V_{n}^{(m)}\bigr)_{+}U_{n}+ \bigl[U_{n},\bigl(V_{n}^{(m)}\bigr)_{+}\bigr]=(\Delta a_{m})h_{2}(-1)-s_{n}b_{m+1}e(0)+s_{n}Ec_{m+1}f(0). $$

Assuming \(V^{(m)}_{(n)}=(V_{n}^{(m)})_{+}-a_{m}h_{1}(0)+a_{m}h_{2}(0)\), a direct calculation yields

$$-\bigl(\Delta V^{(m)}_{(n)}\bigr)U_{n}+ \bigl[U_{n},V^{(m)}_{(n)}\bigr]=s_{n}\Delta a_{m}h_{1}(1)-Eb_{m}e(0)+c_{m}f(0). $$

The compatibility condition of the following Lax pair:

$$\psi_{n+1}=U_{n}\psi_{n},\qquad\psi_{n,t_{m}}=V^{(m)}_{(n)} \psi_{n} $$

admits an integrable discrete hierarchy

$$ \begin{pmatrix} s_{n}\\ q_{n}\\ r_{n} \end{pmatrix} _{t_{m}}= \begin{pmatrix} -s_{n}\Delta a_{m}\\ Eb_{m}\\ -c_{m} \end{pmatrix}. $$
(8)

Taking \(m=2\), equation (8) reduces to an integrable discrete system with three fields

$$ \textstyle\begin{cases} s_{n,t}=2s_{n}(E^{2}-1)\frac{q_{n}r_{n-2}}{s_{n}s_{n-1} s_{n-2}}-2s_{n}(E^{2}-1)\frac {q_{n}q_{n-1}r_{n-1}r_{n-2}}{s_{n}s_{n-1}^{2}s_{n-2}}-2s_{n}\Delta \frac{q_{n}^{2}r_{n-1}^{2}}{s_{n}^{2}s_{n-1}^{2}},\\ q_{n,t}=\frac{2q_{n+2}}{s_{n+1}s_{n+2}}-\frac{2q_{n+1}r_{n+1}q_{n+2}}{s_{n+1}^{2}s_{n+2}}-\frac{2q_{n+1}^{2}r_{n}}{s_{n}s_{n+1}^{2}},\\ r_{n,t}=\frac{2q_{n}r_{n-1}^{2}}{s_{n}s_{n-1}^{2}}+\frac{2q_{n-1} r_{n-1}r_{n-2}}{s_{n-1}^{2}s_{n-2}}-\frac{2r_{n-2}}{s_{n-1}s_{n-2}}, \end{cases} $$
(9)

which generalizes the positive part of a result in [14] except for constants.

Assuming \(m=1\), equation (8) reduces to the much simpler integrable discrete system

$$ \textstyle\begin{cases} s_{n,t}=\frac{2q_{n+1}r_{n}}{s_{n+1}}-\frac{2q_{n}r_{n-1}}{s_{n-1}},\\ q_{n,t}=\frac{2q_{n+1}}{s_{n+1}},\\ r_{n,t}=\frac{-2r_{n-1}}{s_{n-1}}. \end{cases} $$
(10)

It is easy to see that there exists an explicit relation among the three fields in (10) as follows:

$$s_{n}=q_{n}r_{n}+f(n), $$

where \(f(n)\) is an arbitrary function with respect to variable n.

Let \(s_{n}=1\), equation (9) becomes

$$ \textstyle\begin{cases} q_{n,t}=2q_{n+2}-2q_{n+1}r_{n+1}q_{n+2}-2q_{n+1}^{2}r_{n},\\ r_{n,t}=2q_{n}r_{n-1}^{2}+2q_{n-1}r_{n-1}r_{n-2}-2r_{n-2}, \end{cases} $$
(11)

and

$$ (E+1)q_{n}r_{n-2}-(E+1)q_{n}q_{n-1}r_{n-1}r_{n-2}-q_{n}^{2}r_{n-1}^{2}=c, $$
(12)

where c is a constant independent of \(n,t\). Equation (11) is a modified integrable discrete KdV system with the constraint (12). In fact, substituting (12) into (11) yields a reduced integrable discrete mKdV system

$$\textstyle\begin{cases} q_{n,t}=2q_{n+2}-2q_{n+1}r_{n+1}q_{n+2}-2q_{n+1}^{2}r_{n},\\ r_{n,t}=2\frac{q_{n+1}}{q_{n}}r_{n-1}-2q_{n+1}r_{n}r_{n-1}-2\frac{c}{q_{n}}. \end{cases} $$

In the following, we discuss the quasi-Hamiltonian form of equation (8). A direct computation gives

$$\begin{aligned}& U_{n}^{-1}=\frac{1}{s_{n}-q_{n}r_{n}} \begin{pmatrix} \lambda^{-1}&-q_{n}\\ -r_{n}&\lambda s_{n} \end{pmatrix} \equiv \frac{1}{\rho_{n} } \begin{pmatrix} \lambda^{-1}&-q_{n}\\ -r_{n}&\lambda s_{n} \end{pmatrix}, \\& W\equiv V_{n}U_{n}^{-1}=\frac{1}{\rho_{n}} \begin{pmatrix} \lambda^{-1}A_{n}-B_{n}r_{n}&-q_{n}A_{n}+\lambda s_{n}B_{n}\\ \lambda^{-1}C_{n}+r_{n}A_{n}&-q_{n}C_{n}-\lambda s_{n}A_{n} \end{pmatrix}. \end{aligned}$$

Denoting \(\langle a,b\rangle=\operatorname{tr}(ab)\), we find that

$$\begin{aligned}& \biggl\langle W,\frac{\partial U_{n}}{\partial\lambda}\biggr\rangle =\frac{\lambda ^{-1}A_{n}-r_{n}B_{n}}{\rho_{n}}s_{n}+ \frac{q_{n}C_{n}+\lambda s_{n}A_{n}}{\rho_{n}\lambda ^{2}}, \\& \biggl\langle W,\frac{\partial U_{n}}{\partial s_{n}}\biggr\rangle =\frac{A_{n}-r_{n}B_{n}\lambda}{\rho _{n}},\qquad\biggl\langle W,\frac{\partial U_{n}}{\partial q_{n}}\biggr\rangle =\frac{\lambda ^{-1}C_{n}+r_{n}A_{n}}{\rho_{n}}, \\& \biggl\langle W,\frac{\partial U_{n}}{\partial r_{n}}\biggr\rangle =\frac{\lambda s_{n}B_{n}-q_{n}A_{n}}{\rho _{n}}. \end{aligned}$$

Substituting these results into the trace identity [16] yields

$$ \frac{\delta}{\delta Q_{n}}\biggl(\frac{\lambda^{-1}A_{n}-r_{n}B_{n}}{\rho_{n}}s_{n}+ \frac {q_{n}C_{n}+\lambda s_{n}A_{n}}{\rho_{n}\lambda^{2}}\biggr)=\lambda^{-\gamma}\frac {\partial}{\partial\lambda} \lambda^{\gamma}\begin{pmatrix} \frac{A_{n}-r_{n}B_{n}\lambda}{\rho_{n}}\\ \frac{\lambda ^{-1}C_{n}+r_{n}A_{n}}{\rho_{n}}\\ \frac{\lambda s_{n}B_{n}-q_{n}A_{n}}{\rho_{n}} \end{pmatrix}, $$
(13)

where \(\frac{\delta}{\delta Q_{n}}=(\frac{\delta}{\delta s_{n}},\frac{\delta }{\delta q_{n}},\frac{\delta}{\delta r_{n}})^{T}\).

Inserting (2) into (13), one infers that

$$\frac{\delta}{\delta Q_{n}}\biggl(\frac{q_{n}c_{m}-r_{n}s_{n}b_{m+1}+2s_{n}a_{m}}{\rho _{n}}\biggr)=(\gamma-2m) \begin{pmatrix} \frac{a_{m}-r_{n}b_{m+1}}{\rho_{n}}\\ \frac{c_{m}+r_{n}a_{m}}{\rho_{n}}\\\frac{s_{n}b_{m+1}-q_{n}a_{m}}{\rho_{n}} \end{pmatrix} \equiv(\gamma-2m)P_{m}. $$

It is easy to verify from the initial values in (5) that \(\gamma=0\). Thus, we have

$$ P_{m}=\frac{\delta H_{m+1}}{\delta Q_{n}},\qquad H_{m+1}= \frac {r_{n}s_{n}b_{m+1}-q_{n}c_{m}-2s_{n}a_{m}}{2m\rho_{n}}. $$
(14)

Therefore, equation (8) can be written in Hamiltonian form

$$ \begin{pmatrix} s_{n}\\ q_{n}\\ r_{n} \end{pmatrix} _{t_{m}}= \begin{pmatrix} -s_{n}\Delta a_{m}\\ s_{n}b_{m+1}-q_{n}(E+1)a_{m}\\ -c_{m} \end{pmatrix} =JP_{m}=J \frac{\delta H_{m+1}}{\delta Q_{n}}, $$
(15)

where

$$J= \begin{pmatrix} -s_{n}r_{n}q_{n}\Delta-s_{n}\rho_{n}\Delta&0&-s_{n}r_{n}\Delta\\ -q_{n}^{2}r_{n}E-q_{n}\rho_{n}E&0&s_{n}-q_{n}r_{n}(E+1)\\ r_{n}(\rho_{n}+q_{n}r_{n})&-\rho_{n}&r_{n}^{2} \end{pmatrix} $$

is not obviously a Hamiltonian operator.

Remark 2

Equation (15) is only a form of Hamiltonian structure. Perhaps it becomes a Hamiltonian structure by introducing various modified terms \(\Delta_{n}\) in generating the integrable hierarchy (8); of course, if changing the modified terms \(\Delta_{n}\), the discrete hierarchy is also changed. As to this question, we shall discuss it as presented in [10] in the forthcoming time.

2.2 A \((2+1)\)-dimensional integrable discrete hierarchy

Consider the following \((2+1)\)-dimensional discrete non-isospectral linear problem [13–15]:

$$ \textstyle\begin{cases} E\psi_{n}(\lambda)=U_{n}\psi_{n}(\lambda),\\ \frac{d\psi_{n}(\lambda)}{dt}=\omega(\lambda)\frac{d\psi(\lambda)}{dy} +V_{n}^{(m)}\psi_{n}(\lambda), \end{cases} $$
(16)

where the spectral parameter \(\lambda=\lambda(y,t)\) satisfies a non-isospectral condition

$$ \lambda_{t}=\omega(\lambda)\lambda_{y}+ \beta(\lambda), $$
(17)

here \(\omega(\lambda)\) and \(\beta(\lambda)\) are two functions to be determined. The compatibility condition of (16) along with (17) reads

$$ \frac{\partial U_{n}}{\partial t}=\omega(\lambda)\frac{\partial U_{n}}{\partial y}-\beta( \lambda)\frac{\partial U_{n}}{\partial\lambda} +\bigl(\Delta V_{n}^{(m)} \bigr)U_{n}-\bigl[U_{n},V_{n}^{(m)}\bigr]. $$
(18)

Assume

$$ \textstyle\begin{cases} \omega(\lambda)=\lambda^{2m},\qquad\beta(\lambda)= \sum_{j=0}^{m}\beta_{2j-1}\lambda^{2j-1},\\ A_{n}=\sum_{j=0}^{m}a_{j}(n,y,t)\lambda^{2m-2j},\qquad B_{n}= \sum_{j=0}^{m}b_{j}(n,y,t)\lambda^{2m-2j+1},\\ C_{n}= \sum_{j=0}^{m}c_{j}(n,y,t)\lambda^{2m-2j+1}. \end{cases} $$
(19)

The discrete stationary equation of (18) admits the following:

$$\begin{aligned}& \textstyle\begin{cases} s_{n,y}+s_{n}\Delta a_{0}+r_{n}\Delta b_{1}-q_{n}c_{1}-r_{n}b_{1}=0,\\ q_{n,y}+q_{n}\Delta a_{0}+\Delta b_{0}+s_{n}b_{1}-2q_{n}a_{0}-b_{0}=0,\\ r_{n,y}+s_{n}\Delta c_{1}-r_{n}\Delta a_{0}+2r_{n}a_{0}+c_{0}-s_{n}c_{1}=0,\\ -\beta_{2m-1}+q_{n}\Delta c_{1}-\Delta a_{0}+r_{n}b_{1}-q_{n}c_{1}=0, \end{cases}\displaystyle \end{aligned}$$
(20)
$$\begin{aligned}& \textstyle\begin{cases} -s_{n}\beta_{2m-2j+3}+s_{n}\Delta a_{j}+r_{n}\Delta b_{j}-q_{n}c_{j}-r_{n}b_{j}=0,\\ q_{n}\Delta a_{j}+\Delta b_{j}+s_{n}b_{j+1}-2q_{n}a_{j}-b_{j}=0,\\ s_{n}\Delta c_{j+1}-r_{n}\Delta a_{j}+2r_{n}a_{j}-c_{j}-s_{n}c_{j+1}=0,\\ -\beta_{2m-2j-1}+q_{n}\Delta c_{j+1}-\Delta a_{j}+r_{n}b_{j+1}-q_{n}c_{j+1}=0,\quad j=1,2,\ldots,m-1, \end{cases}\displaystyle \end{aligned}$$
(21)
$$\begin{aligned}& \textstyle\begin{cases} s_{n}\Delta a_{m}+r_{n}\Delta b_{m}-q_{n}c_{m}-r_{n}b_{m}=0,\\ q_{n}\Delta a_{m}+\Delta b_{m}-2q_{n}a_{m}-b_{m}=0,\\ -r_{n}\Delta a_{m}+2r_{n}a_{m}+c_{m}=0,\\ \Delta a_{m}=0. \end{cases}\displaystyle \end{aligned}$$
(22)

Assume

$$\begin{aligned}& \bigl(V_{n}^{(m)}\bigr)_{+}=\sum_{j=0}^{m} \lambda^{2m}V_{n}=\lambda^{2m}V_{n}- \bigl(V_{n}^{(m)}\bigr)_{-}, \\& \begin{aligned}[b] &{-}\bigl(\Delta V_{n}^{(m)} \bigr)_{+}U_{n}+\bigl[U_{n},\bigl(V_{n}^{(m)} \bigr)_{+}\bigr] \\ &\quad=-\omega(\lambda )U_{n,y}+\beta(\lambda)U_{n,\lambda}=(q_{n} \Delta a_{m}+\Delta b_{m}-2q_{n}a_{m}-b_{m})e(0) \\ &\qquad{}+(2r_{n}a_{m}-r_{n}\Delta a_{m}+c_{m})f(0)-(\Delta a_{m})h_{2}(-1). \end{aligned} \end{aligned}$$
(23)

Suppose

$$ V_{(n)}^{(m)}=\bigl(V_{n}^{(m)} \bigr)_{+}+\Delta_{n}=\bigl(V_{n}^{(m)} \bigr)_{+}-a_{m}h_{1}(0)+a_{m}h_{2}(0). $$
(24)

Substituting (23), (24) into equation (18) replacing \(V_{n}^{(m)}\) by \(V_{(n)}^{(m)}\) gives

$$ \textstyle\begin{cases} s_{n,t_{m}}=-s_{n}\Delta a_{m},\\ q_{n,t_{m}}=Eb_{m}-2q_{n}a_{m},\\ r_{n,t_{m}}=c_{m}+4r_{n}a_{m}, \end{cases} $$
(25)

which is a \((2+1)\)-dimensional integrable discrete hierarchy. In the following, we consider some of its reductions. Taking \(b_{0}=c_{0}=0,a_{0}=1\) in (20), we can deduce from (21) and (22) that

$$\begin{aligned}& b_{1}=\frac{1}{s_{n}}(2q_{n}-q_{n,y}),\qquad c_{1}=- \frac {2r_{n-1}+r_{n-1,y}}{s_{n-1}}, \\& a_{1}=n\beta_{2m-1}-\frac{2q_{n}r_{n-1}}{s_{n}s_{n-1}}-2 \Delta^{-1}\biggl(\frac {r_{n}q_{n+1,y}}{s_{n}s_{n+1}}+\frac{q_{n}r_{n-1,y}}{s_{n}s_{n-1}}\biggr), \end{aligned}$$

Let \(m=2\), equation (25) reduces to a new \((2+1)\)-dimensional integrable discrete coupled system

$$\begin{aligned} s_{n,t_{2}} =&-s_{n}\Delta a_{2}=- \beta_{3}s_{n}+(2n+1)\beta_{3}\frac {r_{n}q_{n+1}}{s_{n+1}}+ \frac {2q_{n+1}r_{n+1}r_{n}q_{n+2}+q_{n+1}r_{n+1}r_{n}q_{n+2,y}}{s_{n+1}^{2}s_{n+2}} \\ &{}-\frac{6q_{n+1}^{2}r_{n}^{2}-q_{n+1}^{2}r_{n}q_{n,y}}{s_{n}s_{n+1}^{2}}-\frac {2q_{n+1}q_{n+2}r_{n}-q_{n}r_{n}q_{n+2,y}}{s_{n+1}^{2}s_{n+2}} \\ &{}+(2n-3)\beta_{3} \frac{q_{n}r_{n-1}}{s_{n-1}}+\frac {2r_{n-1}^{2}q_{n}^{2}+q_{n}r_{n-1}^{2}q_{n,y}}{s_{n}s_{n-1}^{2}} \\ &{}-\frac{6q_{n}q_{n-1}r_{n-1}r_{n-2}-q_{n}r_{n-1}q_{n-1}r_{n-2,y}}{s_{n-1}^{2}s_{n-2}} +\frac{2q_{n}r_{n-2}+q_{n}r_{n-2,y}}{s_{n-1}s_{n-2}} \\ &{}-\frac{4q_{n}r_{n-1}}{s_{n-1}}E^{-1} \Delta^{-1}\biggl(\frac {r_{n}q_{n+1,y}}{s_{n}s_{n+1}} +\frac{q_{n}r_{n-1,y}}{s_{n}s_{n-1}}\biggr), \end{aligned}$$
(26)
$$\begin{aligned} q_{n,t_{2}} =&Eb_{2}-2q_{n}a_{2}=(2n+1) \beta_{3}\frac{q_{n+1}}{r_{n+1}}-2\beta _{3}nq_{n}+ \frac{2q_{n+1}r_{n+1}q_{n+2}+q_{n+1}r_{n+1}q_{n+2,y}}{s_{n+1}^{2}s_{n+2}} \\ &{}-\frac{6q_{n+1}^{2}r_{n}-q_{n+1}r_{n,y}}{s_{n}s_{n+1}^{2}}-\frac {2q_{n+1}q_{n+2}-q_{n+1}q_{n+2,y}}{s_{n+1}^{2}s_{n+2}}+2\beta _{3}q_{n}(E+1) \frac{q_{n}r_{n-1}}{s_{n}s_{n-1}} \\ &{}-\frac{4q_{n}^{3}r_{n-1}^{2}}{s_{n}^{2}s_{n-1}^{2}}+4q_{n}(E+1)\frac {q_{n}q_{n-1}r_{n-1}r_{n-2}}{s_{n}s_{n-1}^{2}s_{n-2}} \\ &{} - \frac{4q_{n+1}}{s_{n+1}}E\Delta^{-1}\biggl(\frac{r_{n}q_{n+1,y}}{s_{n} s_{n+1}}+ \frac{q_{n}r_{n-1,y}}{s_{n}s_{n-1}}\biggr) -2q_{n}R(a_{2}), \end{aligned}$$
(27)
$$\begin{aligned} r_{n,t_{2}} =&c_{2}+4r_{n}a_{2}=(3-2n) \beta_{3}\frac{r_{n-1}}{s_{n-1}}-\frac {2r_{n-1}^{2}q_{n}+r_{n-1}^{2}q_{n,y}}{s_{n}s_{n-1}^{2}}+ \frac {6q_{n-1}r_{n-1}r_{n-2}-q_{n-1}r_{n-1}r_{n-2,y}}{s_{n-1}^{2}s_{n-2}} \\ &{}-\frac{2r_{n-2}+r_{n-2,y}}{s_{n-1}s_{n-2}}+4n\beta _{3}r_{n}-4r_{n}(E+1) (3-2n)\frac{q_{n}r_{n-1}}{ s_{n}s_{n-1}} \\ &{}+\frac{8r_{n}q_{n}^{2}r_{n-1}^{2}}{s_{n}^{2}s_{n-1}^{2}}-8r_{n}(E+1)\frac {q_{n}q_{n-1}r_{n-1}r_{n-2}}{s_{n}s_{n-1}^{2}s_{n-2}} \\ &{}+ \frac {16q_{n}r_{n}r_{n-1}}{s_{n}s_{n-1}}E^{-1}\Delta^{-1} \biggl( \frac{r_{n}q_{n+1,y}}{s_{n}s_{n+1}}+\frac {q_{n}r_{n-1,y}}{s_{n}s_{n-1}}\biggr) +4r_{n}R(a_{2}). \end{aligned}$$
(28)

When taking \(\Delta a_{2}=0\), we may take \(a_{2}=\alpha, \beta_{3}=0,s_{n}=1\), equations (26)-(28) reduce to

$$ \textstyle\begin{cases} q_{n,t}=Eb_{2}-2\alpha q_{n},\\ r_{n,t}=c_{2}+4\alpha r_{n}, \end{cases} $$
(29)

which can be written as

$$q_{n}r_{n,t}-r_{n}q_{n,t}=6\alpha q_{n}r_{n}. $$

If \(q_{n}\neq0\), we have

$$ r_{n}=q_{n}g(n,y)e^{6\alpha t}, $$
(30)

where \(g(n,y)\neq0\) is an arbitrary function independent of time t. Hence, equation (29) can be reduced to a \((2+1)\)-dimensional integrable discrete equation

$$g(n,y)q_{n,t}+2\alpha g(n,y)q_{n}=\bar{c}_{2}, $$

here \(\bar{c}_{2}=\frac{c_{2}}{e^{6\alpha t}}\).

2.3 A positive and negative integrable discrete hierarchy

Based on [14], we introduce a \((2+1)\)-dimensional non-isospectral linear problem

$$ \textstyle\begin{cases} \psi_{n+1}=U_{n}\psi_{n},\qquad U_{n}=s_{n}h_{1}(1)+q_{n}e(0)+r_{n}f(0)+h_{2}(-1)+p_{n}h_{2}(1),\\ \frac{d\psi_{n}}{dt}=\omega(\lambda)\frac{d\psi_{n}}{dy}+V_{n}^{(m)}\psi_{n},\qquad \lambda_{t}=\omega(\lambda)\lambda_{y}+\beta(\lambda), \end{cases} $$
(31)

where

$$\begin{aligned}& V_{n}^{(m)}=A_{n}^{(m)}h_{1}(0)+D_{n}^{(m)}h_{2}(0)+B_{n}^{(m)}e(0)+C_{n}^{(m)}f(0), \\& A_{n}^{(m)}=\sum_{j=0}^{m}a_{j} \lambda^{2m-2j}+\sum_{j=0}^{m-1}\bar{a}_{j}\lambda^{-(2m-2j)}, \qquad B_{n}^{(m)}=\sum_{j=1}^{m}b_{j} \lambda^{2(m-j)+1}+\sum_{j=1}^{m}\bar{b}_{j}\lambda^{-(2m-2j+1)}, \\& C_{n}^{(m)}=\sum_{j=1}^{m}c_{j} \lambda^{2(m-j)+1}+\sum_{j=1}^{m}\bar{c}_{j}\lambda^{-(2m-2j+1)}, \qquad D_{n}^{(m)}=\sum_{j=0}^{m}d_{j} \lambda^{2m-2j}+\sum_{j=0}^{m-1}\bar{d}_{j}\lambda^{-(2m-2j)}, \\& \omega(\lambda)=\lambda^{2m}+\lambda^{-2m},\qquad\beta(\lambda)= \sum_{j=2}^{m}\bigl(\alpha_{2j-1} \lambda^{2j-1}+\alpha_{3-2j}\lambda^{3-2j}\bigr). \end{aligned}$$

The compatibility condition of (31) has the same form as equation (18). Substituting the \(U_{n}\) and \(V_{n}^{(m)}\) in (31) into equation (18), combining the operation relations of the loop algebra \(\tilde{A}_{1}\) leads to

$$\begin{aligned}& \textstyle\begin{cases} s_{n}(E-1)a_{0}=-s_{n,y},\qquad q_{n}Ea_{0}+p_{n}b_{1}-q_{n}d_{0}-s_{n}b_{1}=-q_{n,y},\\ s_{n}Ec_{1}+r_{n}Ed_{0}-r_{n}a_{0}-p_{n}c_{1}=-r_{n,y},\\ q_{n}Ec_{1}+Ed_{0}+p_{n}\Delta d_{1}-r_{n}b_{1}-d_{0}=0,\\ s_{n}(E-1)a_{j}+r_{n}Eb_{j}-q_{n}c_{j}=s_{n}\alpha_{2m-2j+1},\\ q_{n}Ec_{j}+p_{n}\Delta d_{j}+Ed_{j-1}-r_{n}b_{j}=2\alpha_{2m-2j+1},\\ s_{n}Ec_{j+1}+r_{n}Ed_{j}-r_{n}a_{j}-p_{n}c_{j+1}-c_{j}=0,\\ q_{n}Ea_{j}+Eb_{j}+p_{n}Eb_{j+1}-s_{n}b_{j+1}-q_{n}d_{j}=0,\quad j=1,2,\ldots, m-1,\\ s_{n}Ec_{m}+r_{n}Ed_{m-1}-p_{n}c_{m}-r_{n}a_{m-1}-c_{m-1}=0,\\ p_{n}Eb_{m}-s_{n}b_{m}+q_{n}Ea_{m-1}-q_{n}d_{m-1}=0,\\ \Delta d_{m}+p_{n}\Delta\bar{d}_{m-1}+q_{n}E\bar{c}_{m}-r_{n}\bar{b}_{m}=\alpha_{-1}, \end{cases}\displaystyle \end{aligned}$$
(32)
$$\begin{aligned}& \textstyle\begin{cases} s_{n}(E-1)\bar{a}_{0}+r_{n}E\bar{b}_{1}-q_{n}\bar{c}_{1}=-s_{n,y},\\ r_{n}E\bar{d}_{0}-r_{n}\bar{a}_{0}-\bar{c}_{1}=-r_{n,y},\\ q_{n}E\bar{c}_{1}+\Delta\bar{d}_{1}+p_{n}\Delta\bar{d}_{0}-r_{n}\bar{b}_{1}=-p_{n,y},\\ \Delta\bar{d}_{0}=-p_{n,y},\\ s_{n}(E-1)\bar{a}_{j}+r_{n}E\bar{b}_{j+1}-q_{n}\bar{c}_{j+1}=s_{n}\alpha_{1-2m+2j},\\ s_{n}E\bar{c}_{j}+r_{n}E\bar{d}_{j}-r_{n}\bar{a}_{j}-\bar{c}_{j+1}-p_{n}\bar{c}_{j}=0,\\ q_{n}E\bar{a}_{j}+E\bar{b}_{j+1}+p_{n}E\bar{b}_{j}-s_{n}\bar{b}_{j}-q_{n}\bar{d}_{j}=0,\\ q_{n}E\bar{c}_{j}+E\bar{d}_{j}-r_{n}\bar{b}_{j}+p_{n}E\bar{d}_{j-1}-\bar{d}_{j}-p_{n} \bar{d}_{j-1}=\alpha_{-2(m-j)-1},\quad j=1,2,\ldots, m-1,\\ r_{n}E\bar{b}_{m}-q_{n}\bar{c}_{m}=s_{n}\alpha_{-1},\\ E\bar{b}_{m}-s_{n}\bar{b}_{m-1}+p_{n}\bar{b}_{m-1}-q_{n}\bar{d}_{m-1}=0. \end{cases}\displaystyle \end{aligned}$$
(33)

The corresponding \((2+1)\)-dimensional positive and negative integrable discrete hierarchy is obtained as follows:

$$\begin{aligned}& \textstyle\begin{cases} s_{n,t_{m}}=s_{n}(E-1)a_{m}+r_{n}Eb_{m}-q_{n}c_{m},\\ r_{n,t_{m}}=r_{n}Ed_{m}-r_{n}a_{m}-c_{m}+(s_{n}E-p_{n})\bar{c}_{m},\\ q_{n,t_{m}}=q_{n}Ea_{m}+Eb_{m}-q_{n}d_{m}+(p_{n}E-s_{n})\bar{b}_{m},\\ p_{n,t_{m}}=q_{n}Ec_{m}+p_{n}(E-1)d_{m}-r_{n}b_{m},\quad m\geq2. \end{cases}\displaystyle \end{aligned}$$
(34)
$$\begin{aligned}& q_{n}Ec_{m}-r_{n}b_{m}=0. \end{aligned}$$
(35)

Given some initial values in terms of (32) and (33), we could obtain some explicit \((2+1)\)-dimensional positive and negative integrable discrete hierarchies as long as \(a_{m},b_{m},c_{m}, d_{m}, \bar{c}_{m}\), and \(\bar{b}_{m}\) are obtained. Here we only discuss the case where \(p_{n}=0\). It is easy to see that (34) reduces to

$$ \textstyle\begin{cases} s_{n,t_{m}}=s_{n}(E-1)a_{m}+r_{n}Eb_{m}-q_{n}c_{m},\\ r_{n,t_{m}}=r_{n}Ed_{m}-r_{n}a_{m}-c_{m}+s_{n}E\bar{c}_{m},\\ q_{n,t_{m}}=q_{n}Ea_{m}+Eb_{m}-q_{n}d_{m}-s_{n}\bar{b}_{m}. \end{cases} $$
(36)

Equation (35) is an obvious generalization of equations (2.18) and (2.19) presented in [14]. Specially, when taking \(m=2\), (35) becomes the following:

$$ \textstyle\begin{cases} s_{n,t_{2}}=s_{n}(E-1)a_{2}+r_{n}Eb_{2}-q_{n}c_{2},\\ r_{n,t_{2}}=r_{n}d_{2}-r_{n}a_{2}-c_{2}+s_{n}E\bar{c}_{2},\\ q_{n,t_{2}}=q_{n}Ea_{2}+Eb_{2}-q_{n}d_{2}-s_{n}\bar{b}_{2}, \end{cases} $$
(37)

and (35) turns to

$$ r_{n}b_{2}=q_{n}Ec_{2}. $$
(38)

From (32) and (33), we can compute that

$$\begin{aligned}& a_{0}=-\Delta^{-1}\frac{s_{n,y}}{s_{n}},\qquad d_{0}= \Delta^{-1}\biggl(\frac {(q_{n}r_{n})_{y}-q_{n}r_{n}s_{n,y}}{\rho_{n}}\biggr), \\& b_{1}=\frac{q_{n,y}}{s_{n}}-\frac{q_{n}}{s_{n}}\Delta^{-1} \frac {s_{n+1,y}}{s_{n+1}}-\frac{q_{n}}{s_{n}}d_{0}, \\& c_{1}=-\frac{r_{n-1,y}}{s_{n-1}}-\frac{r_{n-1}}{s_{n-1}}E^{-1}\Delta ^{-1}\frac{s_{n,y}}{s_{n}}-\frac{r_{n-1}}{s_{n-1}}d_{0}, \\& \begin{aligned}[b] (E-1)a_{1}\equiv{}&\delta_{n}=-\frac{r_{n}q_{n+1,y}}{s_{n}s_{n+1}}- \frac {q_{n}r_{n-1,y}}{s_{n}s_{n-1}}+\frac{r_{n}q_{n+1}}{s_{n}s_{n+1}}E^{2}\Delta ^{-1} \frac{s_{n,y}}{s_{n}} +\frac{q_{n+1}r_{n}}{s_{n}s_{n+1}}Ed_{0}\\ &{}-\frac {q_{n}r_{n-1}}{s_{n}s_{n-1}}E^{-1} \Delta^{-1}\frac{s_{n,y}}{s_{n}}-\frac {q_{n}r_{n-1}}{s_{n}s_{n-1}}d_{0}+ \alpha_{2m-1}, \end{aligned}\\& a_{1}=\Delta^{-1}\delta_{n}, \qquad \bar{b}_{1}=q_{n-1}-q_{n-1,y}+q_{n-1} \Delta^{-1}\frac{(\rho_{n})_{y}}{\rho _{n}},\\& \bar{c}_{1}=r_{n}+r_{n,y}+r_{n} \Delta^{-1}\frac{\rho_{n,y}}{\rho_{n}}, \qquad d_{1}=\Delta^{-1}\frac{2\alpha_{1}s_{n}+\alpha_{3}q_{n}r_{n}}{\rho_{n}}, \\& \begin{aligned}[b] \Delta\bar{a}_{1}={}&\frac{1}{\rho_{n}}\biggl\{ -r_{n}s_{n}q_{n-1}+r_{n}s_{n}q_{n+1,y}-r_{n}s_{n}q_{n-1} \Delta^{-1}\frac{\rho _{n,y}}{\rho_{n}}+s_{n}q_{n}r_{n+1}+s_{n}q_{n}r_{n+1,y}\\ &{}+s_{n}q_{n}r_{n+1} E\Delta^{-1}\frac{\rho_{n,y}}{\rho _{n}}-q_{n}^{2}r_{n}r_{n+1,y}-q_{n}r_{n}^{2}q_{n-1,y}- \alpha_{-3}q_{n}r_{n}-q_{n}r_{n} \Delta (q_{n}r_{n})\\ &{}-q_{n}r_{n}\Delta \biggl(q_{n-1}r_{n}\Delta^{-1}\frac{\rho_{n,y}}{\rho _{n}} \biggr)+\alpha_{-1}s_{n}\biggr\} . \end{aligned} \end{aligned}$$

Substituting the related results obtained above into (37), we can get one \((2+1)\)-dimensional positive and one negative discrete hierarchy with three fields; here we do not write it down again.

When taking \(\partial_{y}=0\), the \((2+1)\)-dimensional integrable discrete system (37) reduces to a \((1+1)\)-dimensional discrete system as follows:

$$\begin{aligned} s_{n,t_{2}} =&\alpha_{1}s_{n}+ \alpha_{3}(n+1)\frac{r_{n}q_{n+1}}{s_{n+1}}-\frac {r_{n}q_{n+1}}{s_{n+1}}E \Delta^{-1}\biggl(\frac{2\alpha_{1}s_{n}+\alpha _{3}q_{n}r_{n}}{\rho_{n}}\biggr) \\ &{}+s_{n}\Delta\biggl[\frac{q_{n}r_{n-1}}{s_{n}s_{n-1}}\Delta^{-1}\biggl( \frac{2\alpha _{1}s_{n}+\alpha_{3}q_{n}r_{n}}{\rho_{n}}\biggr)\biggr] \\ &{}-\alpha_{3}n\frac {q_{n}r_{n-1}}{s_{n-1}}+ \frac{q_{n}r_{n-1}}{s_{n-1}} \Delta^{-1} \biggl(\frac{2\alpha_{1}s_{n}+\alpha_{3}q_{n}r_{n}}{\rho_{n}}\biggr), \end{aligned}$$
(39)
$$\begin{aligned} q_{n,t_{2}} =&\alpha_{1}(n+1)q_{n}+ \frac{q_{n}r_{n}q_{n+1}}{s_{n}s_{n+1}}E\Delta ^{-1} \biggl(\frac{2\alpha_{1}s_{n}+\alpha_{3}q_{n}r_{n}}{\rho_{n}}\biggr)- \alpha_{1}nq_{n}-\alpha _{-3}q_{n}r_{n}q_{n-1} \\ &{}-q_{n}R(d_{2})-s_{n}s_{n-1}q_{n-2}-\alpha _{3}(n-1)s_{n}q_{n-1}+s_{n}r_{n}q_{n-1}^{2}+s_{n}q_{n-1} \bar{a}_{1}, \end{aligned}$$
(40)
$$\begin{aligned} r_{n,t_{2}} =&\alpha_{-1}(n+1)r_{n}+ \alpha _{-3}q_{n}r_{n}r_{n+1}+r_{n}E \bigl(R(d_{2})\bigr)-\alpha_{1}nr_{n} \\ &{}+\biggl(\frac{r_{n-1}}{s_{n-1}}-\frac{q_{n}r_{n}r_{n-1}}{s_{n}s_{n-1}}\biggr) \Delta^{-1} \biggl(\frac{2\alpha_{1}s_{n}+\alpha_{3}q_{n}r_{n}}{\rho_{n}}\biggr)-\alpha_{3}n\frac {r_{n-1}}{s_{n-1}}+s_{n}s_{n+1}r_{n+2} \\ &{}-s_{n}r_{n+1}r_{n+2}q_{n+2} +\alpha_{3}(n+2)s_{n}r_{n+1} -s_{n}E(r_{n}\bar{a}_{1}), \end{aligned}$$
(41)

where

$$\begin{aligned}& \begin{aligned}[b] R(d_{2})={}&\Delta^{-1}\bigl\{ r_{n}s_{n-1}q_{n-2}+r_{n}r_{n-1}q_{n-1}^{2}-(r_{n}q_{n-1}-q_{n}Er_{n}) \bar{a}_{1}\\ &{}-q_{n}s_{n+1}s_{n+2}+q_{n}r_{n+1}r_{n+2}q_{n+2} \bigr\} , \end{aligned}\\& \bar{a}_{1}=\Delta^{-1}\biggl\{ \frac{1}{\rho_{n}} \bigl[s_{n}q_{n}r_{n+1}-q_{n}r_{n} \Delta (q_{n}r_{n})-\alpha_{-1}s_{n}- \alpha_{-3}q_{n}r_{n}\bigr]\biggr\} . \end{aligned}$$

When taking \(s_{n}=1\), equations (39)-(41) can reduce to a new modified integrable discrete system. Specially, if we take various values of the parameters \(\alpha_{1},\alpha_{3}, \alpha_{-1}\), and \(\alpha_{-3}\), we can get different three-field discrete systems. For example, assume \(\alpha _{1}=\alpha_{3}=\alpha_{-1}=\alpha_{-3}=0\), equations (39)-(41) reduce to

$$\begin{aligned} s_{n,t_{2}}={}&{-}\frac{q_{n+1}r_{n}}{s_{n}}+\frac{q_{n}r_{n-1}}{s_{n-1}}+s_{n}\Delta \biggl(\frac{q_{n}r_{n-1}}{s_{n}s_{n-1}}\biggr), \\ r_{n,t_{2}}={}&{-}\frac{q_{n}r_{n}r_{n-1}}{s_{n}s_{n-1}}+\frac {r_{n-1}}{s_{n-1}}+s_{n}s_{n+1}r_{n+2}-s_{n}r_{n+1}r_{n+2}q_{n+2}\\ &{}+r_{n}E\Delta^{-1}\biggl\{ r_{n}s_{n-1}q_{n-2}+r_{n}r_{n-1}q_{n-1}^{2}+(q_{n}Er_{n}-r_{n}q_{n-1}) \Delta^{-1}\frac{q_{n}s_{n}r_{n+1}-q_{n}r_{n}\Delta(q_{n}r_{n})}{\rho _{n}}\\ &{}-q_{n}s_{n+1}r_{n+2}+q_{n}r_{n+1}r_{n+2}q_{n+2} \biggr\} -s_{n}r_{n+2}E\Delta^{-1} \frac{s_{n}q_{n}r_{n+1}-q_{n}r_{n}\Delta(q_{n}r_{n})}{\rho_{n}},\\ q_{n,t_{2}}={}&\frac{q_{n}r_{n}q_{n+1}}{s_{n}s_{n+1}}-\frac {q_{n+1}}{s_{n+1}}-s_{n}s_{n-1}q_{n-2}+s_{n}r_{n}q_{n-1}^{2}\\ &{}-q_{n} \Delta^{-1}\biggl\{ r_{n}s_{n-1}q_{n-2} +r_{n}r_{n-1}q_{n-1}^{2}-q_{n}s_{n+1}r_{n+2}+q_{n}r_{n+1}r_{n+2}q_{n+2}\\ &{}+(q_{n}Er_{n}-r_{n}q_{n-1})\Delta ^{-1}\biggl(\frac{s_{n}q_{n}r_{n+1}-q_{n}r_{n}\Delta(q_{n}r_{n})}{\rho_{n}}\biggr)\biggr\} \\ &{} +q_{n-1}s_{n} \Delta^{-1}\biggl(\frac{s_{n}q_{n}r_{n+1}-q_{n}r_{n}\Delta(q_{n}r_{n})}{\rho _{n}}\biggr). \end{aligned}$$

Remark 3

If we could made use of the constrained condition (39) when deducing the above integrable discrete systems, the local integrable discrete equations could be obtained, here we do not go into that investigation again.

2.4 A discrete integrable coupling system

Obviously, the integrable discrete system (34) is an expanding integrable hierarchy, however, it is not a discrete integrable coupling. Because nonlinear integrable couplings could lead to new integrable systems different from the original ones, it has been an interesting work for us to seek new integrable couplings, specially discrete integrable couplings. In this section we could have discussed the discrete integrable couplings of the \((2+1)\)-dimensional positive and negative integrable discrete hierarchy obtained in the paper; however, for the sake of simpler computations, we only want to investigate discrete nonlinear integrable couplings of the positive part of equation (35). It is remarkable that equation (35) is different from equation (8) - why is that so? Actually, we can verify that if eliminating the constrained condition (36) according to the Tu scheme, equation (35) is just equivalent to equation (8). Therefore, in the following, we apply the Tu scheme to deduce some discrete integrable couplings of the integrable discrete system (8). For this purpose, we must enlarge the Lie algebra \(A_{1}\) as done in [19]: Take

$$Q=\operatorname{span}\{H_{1},H_{2},E,F,T_{1},T_{2},T_{3},T_{4} \}, $$

where

$$\begin{aligned}& H_{1}= \begin{pmatrix} h_{1}&0\\ 0&h_{1} \end{pmatrix},\qquad H_{2}= \begin{pmatrix} h_{2}&0\\ 0&h_{2} \end{pmatrix},\qquad E= \begin{pmatrix} e&0\\ 0&e \end{pmatrix},\qquad F= \begin{pmatrix} f&0\\ 0&f \end{pmatrix},\\& T_{1}= \begin{pmatrix} 0&h_{1}\\ 0&h_{1} \end{pmatrix},\qquad T_{2}= \begin{pmatrix} 0&h_{2}\\ 0&h_{2} \end{pmatrix}, \qquad T_{3}= \begin{pmatrix} 0&e\\ 0&e \end{pmatrix},\qquad T_{4}= \begin{pmatrix} 0&f\\ 0&f \end{pmatrix}. \end{aligned}$$

We denote

$$Q=\operatorname{span}\{H_{1},H_{2},E,F,T_{1},T_{2},T_{3},T_{4} \},\qquad Q=Q_{1}\oplus Q_{2}, $$

here \(Q_{1}=\operatorname{span}\{H_{1},H_{2},E,F\}, Q_{2}=\operatorname{span}\{T_{1},T_{2},T_{3},T_{4}\} \). It is easy to verify that

$$ [Q_{2},Q_{2}]\subset Q_{2},\qquad[Q_{1},Q_{2}] \subset Q_{2}, $$
(42)

which implies the Lie group corresponding to the Lie algebra Q is a symmetric space [20]. Usually, in the case of a symmetric space, the obtained integrable couplings according to the Tu scheme are nonlinear. First of all, we investigate an analog of equation (8) in terms of the Tu scheme which contains an arbitrary parameter. Then we further discuss its discrete integrable coupling system. Based on the above idea, we deduce discrete integrable couplings of equation (46).

Assume

$$ A=\sum_{j\geq0}a_{j} \lambda^{-j},\qquad B=\sum_{j\geq0}b_{j} \lambda^{-j},\qquad C=\sum_{j\geq0}c_{j} \lambda^{-j}, $$
(43)

which is different from (2). Substituting (43) into equation (3) yields

$$ \textstyle\begin{cases} s_{n}\Delta a_{j}+r_{n}\Delta b_{j}=q_{n}c_{j}-r_{n}b_{j},\\ q_{n}\Delta c_{j+2}-\Delta a_{j}=-q_{n}c_{j+2}+r_{n}b_{j+2},\\ q_{n}\Delta a_{j}+\Delta b_{j}=s_{n}b_{j+2}-2q_{n}a_{j}-b_{j},\\ s_{n}\Delta c_{j+2}-r_{n}\Delta a_{j}=-s_{n}c_{j+2}+c_{j}+2q_{n}a_{j}, \end{cases} $$
(44)

which is similar to equation (5), but the terms of odd numbers in (44) are all taken to be zero. Equation (6) has various degrees of elements of loop algebra which are \(-1,0,1\), different from the case where we took (2). Hence, one infers that

$$\begin{aligned} &{-}(\Delta V_{n})_{+}U_{n}+ \bigl[U_{n},(V_{n})_{+}\bigr] \\ &\quad=(s_{n}\Delta a_{m+2}+r_{n}\Delta b_{m+2}+r_{n}b_{m+2})h_{1}(-1) \\ &\qquad{}+(q_{n}\Delta c_{m+2}-r_{n}b_{m+2}+q_{n}c_{m+2})h_{2}(-1)+(s_{n} \Delta c_{m+2}+s_{n}c_{m+2})f(0)-s_{n}b_{m+2}e(0) \\ &\quad\equiv P_{n}. \end{aligned}$$
(45)

Remark 4

Equation (45) could have terms such as \(s_{n}\Delta c_{m+1}f(1), q_{n}\Delta c_{m+1}h_{2}(1), \ldots\) , here we omit them due to equation (44).

Take

$$V_{n}^{(m)}=(V_{n})_{+}+(a_{m}+ \sigma)h_{2}(0)-a_{m}h_{1}(0), $$

where

$$(V_{n})_{+}=\sum_{j=0}^{m} \lambda^{m}V_{n}=\sum_{j=0}^{m} \bigl[a_{j}\bigl(h_{1}(m-j)-h_{2}(m-j) \bigr)+b_{j}e(m-j+1)+c_{j}f(m-j+1)\bigr], $$

σ is an arbitrary constant. A direct calculation reads

$$-\bigl(\Delta V_{n}^{(m)}\bigr)U_{n}+ \bigl[U_{n},V_{n}^{(m)}\bigr]=s_{n}\Delta a_{m}h_{1}(1)+(c_{m}-\sigma r_{n})f(0)+(-Eb_{m}+ \sigma q_{n})e(0)\equiv\Gamma_{n}. $$

Hence, the zero curvature equation

$$U_{n,t_{m}}-\bigl(\Delta V_{n}^{(m)} \bigr)U_{n}+\bigl[U_{n},V_{n}^{(m)}\bigr]=0 $$

admits an integrable discrete hierarchy

$$ \textstyle\begin{cases} s_{n,t_{m}}=-s_{n}\Delta a_{m},\\ q_{n,t_{m}}=Eb_{m}-\sigma q_{n},\\ r_{n,t_{m}}=-c_{m}+\sigma r_{n}. \end{cases} $$
(46)

Comparing equation (46) with equation (8), there is no difference except for the parameter σ as regards the forms. In the following, we only deduce a simple discrete integrable coupling system of equation (46). A loop algebra of the enlarging Lie algebra Q can be given by

$$\tilde{Q}=\operatorname{span}\bigl\{ H_{1}(n),H_{2}(n),E(n),F(n),T_{i}(n),i=1,2,3,4 \bigr\} , $$

where

$$\begin{aligned}& H_{j}(n)=H_{j}\lambda^{n},\qquad E(n)=E \lambda^{n},\qquad F(n)=F\lambda^{n},\\& T_{i}(n)=T_{i}\lambda^{n},\quad j=1,2; i=1,2,3,4. \end{aligned}$$

Applying the loop algebra Q̃ we introduce a Lax pair as follows:

$$ \textstyle\begin{cases} \bar{U}_{n}=s_{n}H_{1}(1)+H_{2}(-1)+q_{n}E(0)+r_{n}F(0)+u_{1}T_{1}(1)+u_{2}T_{3}(0)+u_{3}T_{4}(0),\\ \bar{V}_{n}=A_{n}(H_{1}(0)-H_{2}(0))+B_{n}E(1)+C_{n}F(1)+F_{n}T_{3}(1)+G_{n}T_{4}(1), \end{cases} $$
(47)

where

$$\begin{aligned}& A_{n}=\sum_{j\geq0}a_{j} \lambda^{-j},\qquad B_{n}=\sum_{j\geq 0}b_{j} \lambda^{-j},\qquad C_{n}=\sum_{j\geq0}c_{j} \lambda^{-j},\\& F_{n}=\sum_{j\geq0}f_{j} \lambda^{-j},\qquad G_{n}=\sum_{j\geq0}g_{j} \lambda ^{-j}. \end{aligned}$$

Solving the discrete stationary zero curvature equation

$$ (\Delta\bar{V}_{n})U_{n}=[\bar{U}_{n}, \bar{V}_{n}] $$
(48)

shows that the first part is equation (44), the second part is as follows:

$$ \textstyle\begin{cases} u_{1}\Delta a_{j}+u_{3}\Delta b_{j}+r_{n}\Delta f_{j}+u_{3}\Delta f_{j}=q_{n}g_{j}-r_{n}f_{j} +u_{2}c_{j}+u_{2}g_{j}-u_{3}b_{j}-u_{3}f_{j},\\ u_{2}\Delta c_{j}+q_{n}\Delta g_{j}+u_{2}\Delta g_{j}=-q_{n}g_{j} +r_{n}f_{j}-u_{2}c_{j}-u_{2}g_{j}+u_{3}b_{j}+u_{3}f_{j},\\ u_{2}\Delta a_{j}-\Delta f_{j}=s_{n}f_{j+2}-f_{j}+u_{1}b_{j+2}+u_{1}f_{j+2}-2u_{2}a_{j},\\ -u_{3}\Delta a_{j}+u_{1}\Delta c_{j+2}+s_{n}\Delta g_{j+2}+u_{1}\Delta g_{j+2}\\ \quad=-s_{n}g_{j+2}+g_{j}-u_{1}c_{j+2}-u_{1}g_{j+2}+2u_{3}g_{j}. \end{cases} $$
(49)

Equation (48) decomposes into two parts

$$ -(\Delta\bar{V}_{n})_{+}\bar{U}_{n}+[\bar{U}_{n},\bar{V}_{n}]=(\Delta\bar{V}_{n})_{-}\bar{U}_{n}-[\bar{U}_{n},\bar{V}_{n}]. $$
(50)

Similar to the discussion as above, one infers that

$$\begin{aligned} &{-}(\Delta\bar{V}_{n})_{+}\bar{U}_{n}+[\bar{U}_{n},\bar{V}_{n}] \\ &\quad=P_{n}+[u_{1}\Delta a_{m+2}+u_{3}\Delta b_{m+2}+r_{n}\Delta f_{m+2}+2u_{3}f_{m+2}-q_{n}g_{m+2} \\ &\qquad{}+r_{n}f_{m+2}-u_{2}c_{m+2} -u_{2}g_{m+2}+u_{3}b_{m+2}]T_{1}(-1)+[u_{2} \Delta c_{m+2}+2q_{n}\Delta g_{m+2} \\ &\qquad{}+u_{2}\Delta g_{m+2}-r_{n}f_{m+2}+u_{2}c_{m+2}+u_{2}g_{m+2}-u_{3}b_{m+2}-u_{3}f_{m+2}] T_{2}(-1) \\ &\qquad{}-(s_{n}f_{m+2}+u_{1}b_{m+2}+u_{1}f_{m+2})T_{3}(0) \\ &\qquad{}+(u_{1} \Delta c_{m+2}+s_{n}\Delta g_{m+2}+2u_{1} \Delta g_{m+2}+s_{n}g_{m+2} +u_{1}c_{m+2})T_{4}(0). \end{aligned}$$
(51)

Thus, the discrete zero curvature equation

$$ \bar{U}_{n,t_{m}}-\bigl(\Delta\bar{V}_{n}^{(m)} \bigr)\bar{U}_{n}+\bigl[\bar{U}_{n},\bar{V}_{n}^{(m)} \bigr]=0 $$
(52)

admits a discrete integrable coupling of equation (46):

$$ \textstyle\begin{cases} s_{n,t_{m}}=-s_{n}\Delta a_{m},\\ q_{n,t_{m}}=Eb_{m}-\sigma q_{n},\\ r_{n,t_{m}}=-c_{m}+\sigma r_{n},\\ u_{1,t_{m}}=-u_{1}\Delta a_{m},\\ u_{2,t_{m}}=-Ef_{m}-\sigma u_{2},\\ u_{3,t_{m}}=u_{3}Ea_{m}+\sigma u_{3}. \end{cases} $$
(53)

3 Applications of the second loop algebra

In the section we shall apply the Tu scheme and the second loop algebra \(\bar{A}_{1}\) to deduce a new integrable discrete hierarchy whose quasi-Hamiltonian form will be derived from the trace identity proposed by Tu [16] when \(\alpha=0\). This is a new application of the Tu scheme.

3.1 A new integrable discrete hierarchy and its reductions

Consider the following isospectral problems:

$$\begin{aligned}& \psi_{n+1}=U_{n}\psi_{n},\qquad U_{n}=p_{n}h_{1}(1)+ \alpha h_{1}(0)+s_{n}h_{2}(0)+q_{n}e(0)+r_{n}f(0), \end{aligned}$$
(54)
$$\begin{aligned}& \frac{d}{dt}\psi_{n}=\bigl(Ah_{1}(1)+Dh_{2}(1)+Be(0)+Cf(0) \bigr)\psi_{n}, \end{aligned}$$
(55)

where

$$ \begin{aligned} &A=\sum_{j\geq0}a_{j}(n,t) \lambda^{-2j},\qquad B=\sum_{j\geq 0}b_{j}(n,t) \lambda^{-2j},\\ &C=\sum_{j\geq0}c_{j}(n,t) \lambda ^{-2j},\qquad D=\sum_{j\geq0}d_{j}(n,t) \lambda^{-2j}. \end{aligned} $$
(56)

The stationary discrete zero curvature equation

$$ (\Delta V_{n})U_{n}=[U_{n},V_{n}] $$
(57)

admits

$$ \textstyle\begin{cases} (\lambda^{2} p_{n}+\alpha)\Delta A\lambda^{2}+r_{n}\Delta B\lambda^{2}=q_{n}C\lambda ^{2}-r_{n}B\lambda^{2},\\ q_{n}\Delta A\lambda^{3}+s_{n}\Delta B\lambda=B\lambda(\lambda^{2} p_{n}+\alpha )+q_{n}D\lambda^{3}-q_{n}A\lambda^{3}-s_{n}B\lambda,\\ (\lambda^{3} p_{n}+\lambda\alpha)\Delta C+r_{n}\Delta D\lambda^{3}=r_{n}A\lambda ^{3}+s_{n}C\lambda-C(\lambda^{3}p_{n}+\alpha\lambda)-r_{n}D\lambda^{3},\\ q_{n}\Delta C\lambda^{2}+s_{n}\Delta D\lambda^{2}=r_{n}B\lambda^{2}-q_{n}C\lambda^{2}. \end{cases} $$
(58)

Substituting (56) into (58) yields

$$ \textstyle\begin{cases} p_{n}\Delta a_{j+1}+\alpha\Delta a_{j}+r_{n}\Delta b_{j}=q_{n}c_{j}-r_{n}b_{j},\\ q_{n}\Delta a_{j+1}+s_{n}\Delta b_{j}=p_{n}b_{j+1}+\alpha b_{j}+q_{n}d_{j+1}-q_{n}a_{j+1}-s_{n}b_{j},\\ p_{n}\Delta c_{j+1}+\alpha\Delta c_{j}+r_{n}\Delta d_{j+1}=r_{n}a_{j+1}+s_{n}c_{j}-p_{n}c_{j+1}-\alpha c_{j}-r_{n}d_{j+1},\\ q_{n}\Delta c_{j+1}+s_{n}\Delta d_{j+1}=r_{n}b_{j+1}-q_{n}c_{j+1}. \end{cases} $$
(59)

Taking \(a_{0}=0\), solving the above equations, we find that

$$\begin{aligned}& \Delta d_{0}=0\rightarrow d_{0}=1,\quad b_{0}=- \frac{q_{n}}{p_{n}},\quad c_{0}=-\frac {r_{n-1}}{p_{n-1}},\quad a_{1}= \frac{q_{n}r_{n-1}}{p_{n}p_{n-1}},\quad d_{1}=-\frac {q_{n}r_{n-1}}{p_{n}p_{n-1}}+\delta, \end{aligned}$$

from

$$p_{n}b_{1}=q_{n}Ea_{1}+s_{n}Eb_{0}-q_{n}d_{1}- \alpha b_{0}, $$

we have

$$b_{1}=\frac{q_{n}r_{n}q_{n+1}}{p_{n}^{2}p_{n+1}}+\frac {q_{n}^{2}r_{n-1}}{p_{n}^{2}p_{n-1}}-\frac{q_{n+1}s_{n}}{p_{n}p_{n+1}}+ \frac {q_{n}(\alpha-\delta p_{n})}{p_{n}^{2}}, $$

\(q_{n}Ec_{1}=r_{n}b_{1}-s_{n}\Delta d_{1}\rightarrow c_{1}=\frac {q_{n}r_{n-1}^{2}}{p_{n}p_{n-1}^{2}}+\frac {q_{n-1}r_{n-1}r_{n-2}}{p_{n-2}p_{n-1}^{2}}+\frac{\alpha r_{n-1}-\delta p_{n-1}r_{n-1}}{p_{n-1}^{2}}-\frac{s_{n-1}r_{n-2}}{p_{n-1}p_{n-2}},\ldots \) equation (57) can be decomposed into

$$ -\bigl(\Delta V_{n}^{(m)}\bigr)_{+}U_{n}+ \bigl[U_{n},\bigl(V_{n}^{(m)}\bigr)_{+}\bigr]=\bigl( \Delta V_{n}^{(m)}\bigr)_{-}U_{n}-\bigl[U_{n}, \bigl(V_{n}^{(m)}\bigr)_{-}\bigr], $$
(60)

where

$$\begin{aligned} \bigl(V_{n}^{(m)}\bigr)_{+}&=\sum_{j=0}^{m} \bigl(a_{j}(n,t)h_{1}(m+1-j)+d_{j}(n,t)h_{2}(m+1-j)+b_{j}(n,t)e(m-j)+c_{j}f(m-j) \bigr)\\ &=\lambda ^{2m}V-\bigl(V_{n}^{(m)}\bigr)_{-}. \end{aligned}$$

It is easy to see that the degrees of the left-hand side of (60) are higher than 1, while for the right-hand side they are smaller than 2. Therefore, the degrees of both sides are 1,2. Thus, we have

$$\begin{aligned} &{-}\bigl(\Delta V_{n}^{(m)}\bigr)_{+}U_{n}+ \bigl[U_{n},\bigl(V_{n}^{(m)}\bigr)_{+} \bigr]\\ &\quad=p_{n}\Delta a_{m+1}h_{1}(1)+(q_{n} \Delta a_{m+1}-p_{n}b_{m+1}-q_{n}d_{m+1}+q_{n}a_{m+1})e(0)\\ &\qquad{}+(p_{n} \Delta c_{m+1}+r_{n}\Delta d_{m+1}-r_{n}a_{m+1}+p_{n}c_{m+1}+r_{n}d_{m+1})f(0) \\ &\quad=p_{n}\Delta a_{m+1}h_{1}(1)+(-s_{n} \Delta b_{m}+\alpha b_{m}-s_{n}b_{m})e(0)+(- \alpha\Delta c_{m}+s_{n}c_{m}-\alpha c_{m})f(0). \end{aligned}$$

Letting \(V_{(n)}=(V_{n}^{(m)})_{+}+d_{m+1}h_{2}(0)\), a direct calculation gives

$$\begin{aligned} &{-}(\Delta V_{(n)})U_{n}+[U_{n},V_{(n)}]\\ &\quad=p_{n} \Delta a_{m+1}h_{1}(1)-s_{n}\Delta d_{m+1}h_{2}(0) +(q_{n}d_{m+1}+\alpha b_{m}-s_{n}Eb_{m})e(0)\\ &\qquad{}+(-r_{n}Ed_{m+1}+s_{n}c_{m}- \alpha Ec_{m})f(0). \end{aligned}$$

Hence, the discrete zero curvature equation

$$U_{n,t_{m}}-(\Delta V_{(n)})U_{n}+[U_{n},V_{(n)}]=0 $$

admits the following integrable discrete hierarchy of evolution equations:

$$ \textstyle\begin{cases} p_{n,t_{m}}=-p_{n}\Delta a_{m+1},\\ s_{n,t_{m}}=s_{n}\Delta d_{m+1},\\ q_{n,t_{m}}=p_{n}b_{m+1}-q_{n}Ea_{m+1},\\ r_{n,t_{m}}=r_{n}a_{m+1}-p_{n}Ec_{m+1}. \end{cases} $$
(61)

When \(m=0\), we get a reduction of equation (61) which is a generalized Toda lattice equation

$$ \textstyle\begin{cases} p_{n,t}=\frac{q_{n}r_{n-1}}{p_{n-1}}-\frac{q_{n+1}r_{n}}{p_{n+1}},\\ s_{n,t}=\frac{q_{n}s_{n}r_{n-1}}{p_{n}p_{n-1}}-\frac {s_{n}r_{n}q_{n+1}}{p_{n}p_{n+1}},\\ q_{n,t}=\frac{q_{n}^{2}r_{n-1}}{p_{n}p_{n+1}}-\frac{q_{n+1}s_{n}}{p_{n+1}}+\frac {q_{n}(\alpha-\delta p_{n})}{p_{n}},\\ r_{n,t}=\frac{q_{n}r_{n}r_{n-1}}{p_{n}p_{n-1}}-\frac {q_{n+1}r_{n}^{2}+q_{n}r_{n}r_{n-1}}{p_{n}p_{n+1}}+\frac{\delta p_{n}r_{n}-\alpha r_{n}}{p_{n}}+\frac{s_{n}r_{n-1}}{p_{n-1}}. \end{cases} $$
(62)

When \(\alpha=\delta=s_{n}=0\), equation (62) reduces to a simpler nonlinear integrable discrete system

$$ \textstyle\begin{cases} p_{n,t}=\frac{q_{n}r_{n-1}}{p_{n-1}}-\frac{q_{n+1}r_{n}}{p_{n+1}},\\ q_{n,t}=\frac{q_{n}^{2}r_{n-1}}{p_{n}p_{n+1}},\\ r_{n,t}=\frac{q_{n}r_{n}r_{n-1}}{p_{n}p_{n-1}}-\frac {q_{n+1}r_{n}^{2}+q_{n}r_{n}r_{n-1}}{p_{n}p_{n+1}}. \end{cases} $$
(63)

In the following, we deduce a quasi-Hamiltonian form of the integrable discrete hierarchy (61). It is easy to see that

$$\begin{aligned} W={}&V_{n}U_{n}^{-1}=\frac{1}{M} \bigl[(s_{n}A-r_{n}B)h_{1}(1)+(-q_{n}C+ \alpha D)h_{2}(1)+p_{n}Dh_{2}(2)\\ &{}+(p_{n}B-q_{n}A)e(1)+ \alpha Be(0)+s_{n}Cf(0)-r_{n}Df(1)\bigr], \end{aligned}$$

where

$$M=\alpha s_{n}+(p_{n}s_{n}-q_{n}r_{n}) \lambda^{2}. $$

A direct calculation reads

$$\begin{aligned}& \begin{aligned}[b] \operatorname{tr}\biggl(W\frac{\partial U_{n}}{\partial\lambda}\biggr)={}&M^{-1}\bigl[2 \lambda p_{n}\bigl(As_{n}\lambda^{2}-r_{n}B \lambda^{2}\bigr)+r_{n}\bigl(-q_{n}A \lambda^{3}+\alpha B\lambda +p_{n}B\lambda^{3} \bigr)\\ &{}+q_{n}\bigl(s_{n}C\lambda-r_{n}D \lambda^{3}\bigr)\bigr], \end{aligned} \\& \operatorname{tr}\biggl(W\frac{\partial U_{n}}{\partial p_{n}}\biggr)=M^{-1}\lambda ^{2}(s_{n}A-r_{n}B)\lambda^{2}, \\& \operatorname{tr}\biggl(W\frac{\partial U_{n}}{\partial s_{n}}\biggr)=M^{-1} \bigl(-q_{n}C\lambda ^{2}+\alpha D\lambda^{2}+p_{n}D \lambda^{4}\bigr), \\& \operatorname{tr}\biggl(W\frac{\partial U_{n}}{\partial q_{n}}\biggr)=M^{-1}\lambda \bigl(s_{n}C\lambda -r_{n}D\lambda^{3}\bigr), \\& \operatorname{tr}\biggl(W\frac{\partial U_{n}}{\partial r_{n}}\biggr)=M^{-1}\lambda \bigl(-q_{n}A\lambda^{3}+\alpha B\lambda+p_{n}B \lambda^{3}\bigr). \end{aligned}$$

When \(\alpha=0\), substituting the above results and (56) into the trace identity shows that

$$\begin{aligned} &\frac{\delta}{\delta u}\biggl(\frac {2p_{n}s_{n}a_{m+1}-p_{n}r_{n}b_{m+1}-q_{n}r_{n}a_{m+1} +q_{n}s_{n}c_{m}-q_{n}r_{n}d_{m+1}}{p_{n}s_{n}-q_{n}r_{n}}\biggr)\\ &\quad= (-2m+\gamma) \begin{pmatrix} \frac{s_{n}a_{m+1}-r_{n}b_{m+1}}{p_{n}s_{n}-q_{n}r_{n}}\\ \frac{p_{n}d_{m+1}-q_{n}c_{m}}{p_{n}s_{n}-q_{n}r_{n}}\\ \frac {s_{n}c_{m}-r_{n}d_{m+1}}{p_{n}s_{n}-q_{n}r_{n}}\\ \frac{p_{n}b_{m+1}-q_{n}a_{m+1}}{p_{n}s_{n}-q_{n}r_{n}} \end{pmatrix}. \end{aligned}$$

Therefore, equation (61) can be written when \(\alpha=0\):

$$\begin{aligned} u_{t_{m}} =& \begin{pmatrix} p_{n}\\ s_{n}\\ q_{n}\\ r_{n} \end{pmatrix} _{t_{m}}= \begin{pmatrix} -p_{n}\Delta a_{m+1}\\ s_{n}\Delta d_{m+1}\\ p_{n}b_{m+1}-q_{n}Ea_{m+1}\\ r_{n}a_{m+1}-p_{n}Ec_{m+1} \end{pmatrix} = \begin{pmatrix} -p_{n}\Delta a_{m+1}\\ s_{n}\Delta d_{m+1}\\ p_{n}b_{m+1}-q_{n}Ea_{m+1}\\ r_{n}Ed_{m+1}-s_{n}c_{m} \end{pmatrix} \\ =& \begin{pmatrix} -p_{n}^{2}\Delta&0&0&-p_{n}r_{n}\Delta\\ 0&s_{n}^{2}\Delta&q_{n}s_{n}\Delta&0\\ q_{n}p_{n}-q_{n}p_{n}E&0&0&p_{n}s_{n}-q_{n}r_{n}E\\ 0&-s_{n}r_{n}+s_{n}r_{n}E&-p_{n}s_{n}+q_{n}r_{n}E&0 \end{pmatrix} \begin{pmatrix} s_{n}a_{m+1}-r_{n}b_{m+1}\\ p_{n}d_{m+1}-q_{n}c_{m}\\ s_{n}c_{m}-r_{n}d_{m+1}\\ p_{n}b_{m+1}-q_{n}a_{m+1} \end{pmatrix} \\ =&J \begin{pmatrix} s_{n}a_{m+1}-r_{n}b_{m+1}\\ p_{n}d_{m+1}-q_{n}c_{m}\\ s_{n}c_{m}-r_{n}d_{m+1}\\ p_{n}b_{m+1}-q_{n}a_{m+1} \end{pmatrix}. \end{aligned}$$
(64)

Therefore, equation (64) can be written as

$$ u_{t_{m}}= \begin{pmatrix} p_{n}\\ s_{n}\\ q_{n}\\ r_{n} \end{pmatrix} _{t_{m}} =J \begin{pmatrix} s_{n}a_{m+1}-r_{n}b_{m+1}\\ p_{n}d_{m+1}-q_{n}c_{m}\\ s_{n}c_{m}-r_{n}d_{m+1}\\ p_{n}b_{m+1}-q_{n}a_{m+1} \end{pmatrix} =J \frac{\delta H_{m+1}}{\delta u}, $$
(65)

where

$$H_{m+1}=\frac {2p_{n}s_{n}a_{m+1}-p_{n}r_{n}b_{m+1}-q_{n}r_{n}a_{m+1}+q_{n}s_{n}c_{m}-q_{n}r_{n}d_{m+1}}{(-2m+\gamma )(p_{n}s_{n}-q_{n}r_{n})}, $$

the constant γ can be determined by some initial values of equation (59).

3.2 A Darboux transformation of equation (63)

In order to conveniently deduce the Darboux transformation of equation (63), we first recall the general scheme for Darboux transformations. For spectral problems

$$\psi_{n+1}=U_{n}\psi,\qquad \frac{\psi_{n}}{dt}=V_{n} \psi_{n}, $$

one makes a transformation of the eigenfunction

$$\tilde{\psi}_{n}=T_{n}\psi_{n}, $$

then the above spectral problems are transformed to

$$E\tilde{\psi}_{n}(\lambda)=T_{n+1}U_{n}T_{n}^{-1} \tilde{\psi}_{n}(\lambda),\qquad\frac {d\tilde{\psi}_{n}}{dt}=(T_{n,t}+T_{n}V_{n})T^{-1}_{n} \tilde{\psi}_{n}. $$

Denote

$$\tilde{U}_{n}(\tilde{p}_{n},\tilde{q}_{n})=T_{n+1}U_{n}T^{-1}_{n},\qquad \tilde{V} _{n}(\tilde{p}_{n},\tilde{q}_{n})=(T_{n,t}+T_{n}V_{n})T^{-1}_{n}. $$

We hope to construct the matrix \(T_{n}\) by the use of such the eigenfunctions so that \(T_{n+1}U_{n}T^{-1}_{n}\) and \((T_{n,t}+T_{n}V_{n})T^{-1}_{n}\) have the same structures as \(U_{n}\) and \(V_{n}\). With this purpose, we should take various matrices \(T_{n}\) according to the given different spectral problems.

To obtain the Darboux transformations of equation (63), we rewrite its Lax pair as follows:

$$\begin{aligned}& \psi_{n+1}=U_{n}\psi_{n},\qquad U_{n}=p_{n}h_{1}(1)+q_{n}e(0)+r_{n}f(0), \end{aligned}$$
(66)
$$\begin{aligned}& \frac{d \psi_{n}(\lambda)}{dt}=V_{(n)}\psi_{n}, \qquad V_{(n)}=a_{0}h_{1}{(1)}+d_{0}h_{2}(1)+b_{0}e(0)+c_{0}f(0))+d_{1}h_{2}(0). \end{aligned}$$
(67)

We first make a transformation of the eigenfunction

$$ \tilde{\psi}_{n}=T_{n}\psi_{n}. $$
(68)

By equation (68), equations (66) and (67) can be transformed into

$$\begin{aligned}& \tilde{\psi}_{n+1}=T_{n+1}U_{n}T_{n}^{-1} \tilde{\psi}_{n}\equiv\tilde{U}_{n}\tilde{\psi}_{n}, \end{aligned}$$
(69)
$$\begin{aligned}& \frac{d \tilde{\psi}_{n}}{dt}=(T_{n,t}+T_{n}V_{n})T_{n}^{-1} \tilde{\psi}_{n}\equiv \tilde{V}_{n}\tilde{\psi}_{n}. \end{aligned}$$
(70)

Suppose \(\psi_{n}=(\psi_{1n},\psi_{2n})^{T},\phi_{n}=(\phi_{1n},\phi_{2n})^{T}\) are two linear independent eigenfunctions of the spectral problems (66) and (67) corresponding to the solutions \(p_{n},q_{n},r_{n}\). We want to construct the matrix \(T_{n}\) by using such the two eigenfunctions so that \(\tilde{U}_{n}\) and \(\tilde{V}_{n}\) have the same structures as \(U_{n}\) and \(V_{n}\). For this purpose, we take the matrix \(T_{n}\) as follows:

$$T_{n}= \begin{pmatrix} \lambda^{2}+a_{n}&b_{n}\lambda\\ c_{n}\lambda&\lambda^{2}+d_{n} \end{pmatrix}, $$

where \(a_{n},b_{n},c_{n}\), and \(d_{n}\) will be expressed by \(\psi_{n},\phi_{n}\). Assume that \(\lambda_{1},\lambda_{2}\) are two arbitrary distinct solutions of \(\operatorname{det}T_{n}=0\). Set

$$\Phi_{n}= \begin{pmatrix} \phi_{1n}&\psi_{1n}\\ \phi_{2n}&\psi_{2n} \end{pmatrix},\qquad \tilde{\Phi}_{n}=T_{n} \Phi_{n}, $$

then when λ takes the values \(\lambda_{i}\) (\(i=1,2\)) the two column vectors in \(T_{n}\) and \(\tilde{\Phi}_{n}\) are linear dependent, which means that

$$ \textstyle\begin{cases} a_{n}=\frac{\lambda_{1}\lambda_{2}(\alpha_{2}(n)\lambda_{1}-\alpha_{1}(n)\lambda _{2})}{\alpha_{1}(n)\lambda_{1}-\alpha_{2}(n)\lambda_{2}},\\ b_{n}=\frac{\lambda_{1}^{2}-\lambda_{2}^{2}}{\alpha_{1}(n)\lambda_{1}-\alpha _{2}(n)\lambda_{2}},\\ c_{n}=\frac{(\lambda_{1}^{2}-\lambda_{2}^{2})\alpha_{1}(n)\alpha_{2}(n)}{\alpha _{2}(n)\lambda_{1}-\alpha_{1}(n)\lambda_{2}},\\ d_{n}=\frac{\lambda_{1}\lambda_{2}(\alpha_{1}(n)\lambda_{1}-\alpha_{2}(n)\lambda _{2})}{\alpha_{2}(n)\lambda_{1}-\alpha_{1}(n)\lambda_{2}}, \end{cases} $$
(71)

here

$$\alpha_{i}(n)=-\frac{\gamma_{i}\psi_{2n}(\lambda_{i})-\phi_{2n}(\lambda _{i})}{\gamma_{i}\psi_{1n}(\lambda_{i})-\phi_{1n}(\lambda_{i})},\quad i=1,2, $$

where \(\gamma_{i}\) are suitable constants chosen. From (66), we can easily have

$$ \alpha_{i}(n+1)=\frac{r_{n}\lambda_{i}}{-\lambda_{i}^{2}p_{n}+\alpha_{i}(n)q_{n}\lambda _{i}}\equiv \frac{\nu_{i}(n)}{\mu_{i}(n)},\quad i=1,2. $$
(72)

Thus, one infers that

$$ \textstyle\begin{cases} a_{n+1}=\frac{\lambda_{1}\lambda_{2}(\nu_{2}\mu_{1}\lambda_{1}-\mu_{2}\nu_{1}\lambda _{2})}{\nu_{1}\mu_{2}\lambda_{1}-\mu_{1}\nu_{2}\lambda_{2}},\\ b_{n+1}=\frac{\mu_{1}\mu_{2}(\lambda_{1}^{2}-\lambda_{2}^{2})}{\nu_{1}\mu_{2}\lambda _{1}-\mu_{1}\nu_{2}\lambda_{2}},\\ c_{n+1}=\frac{\nu_{1}\nu_{2}(\lambda_{1}^{2}-\lambda_{2}^{2})}{\nu_{2}\mu_{1}\lambda _{1}-\mu_{2}\nu_{1}\lambda_{2}},\\ d_{n+1}=\frac{\lambda_{1}\lambda_{2}(\mu_{2}\nu_{1}\lambda_{1}-\mu_{1}\nu_{2}\lambda _{2})}{\nu_{2}\mu_{1}\lambda_{1}-\mu_{2}\nu_{1}\lambda_{2}}. \end{cases} $$
(73)

Theorem

Assume \(\tilde{U}_{n}=\tilde{p}_{n}h_{1}(1)+\tilde{q}_{n}e(0)+\tilde{r}_{n}f(0)\), then we have

$$ \tilde{p}_{n}=p_{n},\qquad\tilde{q}_{n}=q_{n}-p_{n}b_{n},\qquad \tilde{r}_{n}=r_{n}+p_{n}c_{n+1} , $$
(74)

which is a set of new solutions of equations (66) and (67). The proof of the theorem is similar to that presented in [15, 30] and [31] by using (71)-(73), here we omit it.

Remark 5

Just like discussions for the applications of the loop algebra \(\tilde{A}_{1}\), we could also investigate the integrable couplings of the integrable discrete hierarchy (61) and the associated \((2+1)\)-dimensional integrable discrete systems for further applications of the second loop algebra \(\bar{A}_{1}\), here we do not again go into details in this paper.

4 Reductions of the isospectral problem (54) and some applications

In this subsection, we shall deform the isospectral problem (54) to obtain the well-known parametrized Toda lattice equation and other new lattice integrable systems including \((2+1)\)-dimensional lattice equations and their Lax pairs by applying the r-matrix method [23, 30]. In the following, we recall the notion on r-matrix. A r-matrix from g to itself is defined by [20]

$$ r_{k}:g\rightarrow g,\quad r_{k}=P_{\geq k}-P_{< k}, $$
(75)

where \(k=0,1\). \(P_{\leq k}\) represents a projection operator from g to a Lie subalgebra

$$g_{\leq k}=\biggl\{ \sum_{i\geq k}u_{i}E^{i} \biggr\} . $$

Similarly,

$$P_{< k}=1-P_{\leq k} $$

stands for a projection operator from g to a Lie subalgebra \(g_{< k}=\{ \sum_{i< k}u_{i}E^{i}\}\). In addition, we have the fact

$$g=g_{\leq k}\oplus g_{< k}. $$

According to the general scheme in [23], we obtain two hierarchies of flows on g:

$$ L_{t_{q}}=\bigl[P_{\leq k}\bigl(L^{q} \bigr),L\bigr],\quad k=0,1. $$
(76)

Equation (54) can be written as

$$ E\phi_{1}=\lambda p_{n}\psi_{1}+ \alpha\psi_{1}+q_{n}\psi_{2},\qquad E\psi_{2}=r_{n} \psi _{1}+s_{n}\psi_{2}. $$
(77)

When \(s_{n}=0\), we have

$$ \psi_{2}=E^{-1}r_{n} \psi_{1}. $$
(78)

Substituting (78) into the first equation of (77) yields

$$Ep_{n}^{-1}\psi_{1}=\lambda\psi_{1}+ \alpha p_{n}^{-1}+\frac {q_{n}}{p_{n}}r_{n-1}E^{-1} \psi_{1}, $$

which can be simplified to

$$ u_{n}E\psi+v_{n}\psi+w_{n}E^{-1} \psi=\lambda\psi, $$
(79)

where \(\psi=\psi_{1},u_{n}=p_{n+1}^{-1},v_{n}=-\alpha p_{n}^{1},w_{n}=-\frac {q_{n}r_{n-1}}{p_{n}}\).

Denote

$$ L=u_{n}E+v_{n}+w_{n}E^{-1}. $$
(80)

It can be verified that all the operators like (80) consist of a Lie algebra g if \(u_{n}=1\) and if equipped with a commutator

$$ [L_{1},L_{2}]=L_{1}L_{2}-L_{2}L_{1}. $$
(81)

Now we take \(k=0, q=1\); equation (76) gives rise to the simplest lattice system

$$\textstyle\begin{cases} w_{n,t_{1}}=(1-E^{-1})v_{n},\\ v_{n,t_{1}}=(E-1)w_{n}. \end{cases} $$

Taking \(s_{n}=-E\), the second equation of (77) gives

$$ \psi_{2}=\frac{1}{2}r_{n-1}E^{-1} \psi_{1}. $$
(82)

Inserting (82) into

$$E^{2}\psi_{1}=\lambda p_{n}\psi_{1}+ \alpha E\psi_{1}+q_{n}r_{n}\psi_{1}+q_{n}s_{n} \psi_{2} $$

leads to the following form:

$$ E^{2}\psi=u_{n}E\psi+\alpha E \psi+v_{n}\psi+w_{n}E^{-1}\psi, $$
(83)

where \(\psi=\psi_{1}\).

Denote

$$L=E^{2}-u_{n}E-v_{n}-w_{n}E^{-1}, $$

then (83) becomes

$$ E^{-1}L\psi=\alpha\psi. $$
(84)

Denote

$$\bar{L}=E^{-1}L=E+u_{n}+v_{n}E^{-1}+w_{n}E^{-2}, $$

then equation (84) becomes

$$ \bar{L} \psi=\alpha\psi. $$
(85)

If we regard the parameter α as a spectral parameter and let \(\alpha=\lambda\), then (85) is just right an isospectral problem of the spatial part

$$ \bar{L}\psi=\lambda\psi. $$
(86)

When \(k=0\), equation (76) reduces to

$$ \bar{L}_{t_{q}}=\bigl[P_{\leq0}\bigl(\bar{L}^{q}\bigr),\bar{L}\bigr]. $$
(87)

Set \(q=1\), it is easy to calculate that

$$ \textstyle\begin{cases} u_{n,t_{1}}=(E-1)v_{n},\\ v_{n,t_{1}}=(E-1)w_{n}+v_{n}(1-E^{-1})u_{n-1},\\ w_{n,t_{1}}=w_{n}(u_{n}-u_{n-2}), \end{cases} $$
(88)

which is a three-field integrable system. When taking \(w_{n}=0\), equation (88) reduces to the well-known reparameterized Toda lattice equation:

$$u_{n,t_{1}}=(E-1)v_{n},\qquad v_{n,t_{1}}=v_{n} \bigl(1-E^{-1}\bigr)u_{n-1}. $$

Taking \(q=2\), one infers that

$$P_{\geq0}\bigl(\bar{L}^{2}\bigr)=E^{2}+(u_{n}+u_{n+1})E+u_{n}^{2}+v_{n}+v_{n+1}. $$

Equation (87) admits the following new three-field lattice system:

$$ \textstyle\begin{cases} u_{n,t_{2}}=w_{n+2}-w_{n}+(E-1)[v_{n}(u_{n}+u_{n+1})],\\ v_{n,t_{2}}=w_{n+1}(u_{n}+u_{n+1})-w_{n}(u_{n-1}+u_{n-2})+v_{n}(E-1)(v_{n-1}+v_{n}+u_{n-1}^{2}),\\ w_{n,t_{2}}=w_{n}[(E^{2}-1)(v_{n-1}+v_{n-2}+u_{n-2}^{2})]. \end{cases} $$
(89)

In the following, we shall deduce the Lax pairs of the lattice systems (88) and (89). Set \(\psi_{1}=E^{-2}\psi, \psi_{2}=E^{-1}\psi, \psi_{3}=\psi\), then the spectral equation (86) gives

$$\textstyle\begin{cases} E\psi_{3}=(\lambda-u_{n})\psi_{3}-v_{n}\psi_{2}-w_{n}\psi_{1},\\ E\psi_{1}=\psi_{2},\\ E\psi_{2}=\psi_{3}, \end{cases} $$

which is equivalent to the following spectral problem:

$$ \Psi_{n+1}=U\Psi, $$
(90)

where \(\Psi=(\psi_{1}\psi_{2},\psi_{3})^{T},U=\left( {\scriptsize\begin{matrix}{} 0&1&0\cr 0&0&1\cr -w_{n}&-v_{n}&\lambda-u_{n} \end{matrix}} \right)\).

When \(q=1\), we represent \(A_{1}=P_{\geq0}(\bar{L})\), one infers that

$$\begin{aligned}& A_{1}\psi_{1}=\psi_{2}+u_{n-2} \psi_{2}, \\& A_{1}\psi_{2}=\psi_{3}+u_{n-1} \psi_{2}, \\& A_{1}\psi_{3}=\lambda\psi_{3}-v_{n} \psi_{2}-w_{n}\psi_{1}, \end{aligned}$$

which conclude that the temporal part of the Lax pair for equation (88) is presented as

$$ V_{1}= \begin{pmatrix} u_{n-2}&1&0\\ 0&u_{n-1}&1\\ -w_{n}&-v_{n}&\lambda \end{pmatrix}. $$
(91)

As for \(q=2\), similarly we can obtain the time part of the Lax pair for equation (89) as follows:

$$ V_{2}= \begin{pmatrix} v_{n-2}+v_{n-1}+u_{n-2}^{2}&u_{n-1}+u_{n-2}&1\\ -w_{n}&v_{n-1}+u_{n-1}^{2}&\lambda+u_{n-1}\\ -(\lambda+u_{n})w_{n}&-\lambda v_{n}-u_{n}v_{n}-w_{n+1}&\lambda^{2}+v_{n} \end{pmatrix}. $$
(92)

4.1 \((2+1)\)-Dimensional lattice systems and Lax pairs

In the following, we want to deduce \((2+1)\)-dimensional integrable lattice equations which correspond to the \((1+1)\)-dimensional lattice systems (88) and (89). Set

$$ \nabla C_{i}=\sum_{i\geq j}a_{j}(n)E^{j},\quad i=1,2, \ldots, $$
(93)

where \(a_{j}(n)\) are to be determined from the following equation via the recurrent procedure [21]:

$$ [\nabla C_{i},\tilde{L}-\partial_{y}]=0, $$
(94)

then we have the following \((2+1)\)-dimensional lattice hierarchy:

$$ \tilde{L}_{t_{i}}=\bigl[P_{\geq0}(\nabla C_{i}),\tilde{L}-\partial_{y}\bigr], $$
(95)

where \(P_{\geq0}(\nabla C_{i})=\sum_{j\geq0}a_{j}(n)E^{j}\).

We take

$$\tilde{L}=\bar{L}-\partial_{y},\qquad\bar{L}=E+u_{n}+v_{n}E^{-1}+w_{n}E^{-2},\qquad \nabla C_{1}=a_{0}(n)+a_{1}(n)E+a_{2}(n)E^{2}, $$

then from (94) we have

$$a_{1}(n)=u_{n}+u_{n+1},\qquad a_{0}=Hu_{n}+v_{n}+v_{n+1}+Hu_{ny},\qquad a_{2}(n)=1, $$

where \(H=(E-1)^{-1}(E+1)\). Therefore, equation (95) admits the following \((2+1)\)-dimensional lattice system:

$$ \textstyle\begin{cases} u_{n,t_{1}}=v_{n+1}(u_{n}+u_{n+1})+w_{n+2}-w_{n}-v_{n}(u_{n}+u_{n-1})\\ \hphantom{u_{n,t_{1}}=}{}+Hu_{ny}+v_{ny}+v_{n+1,y}+Hu_{nyy},\\ v_{n,t_{1}}=v_{n}(E-1)[u_{n-1}+(E+1)v_{n-1}+u_{n-1,y}]+(E-1)(u_{n}w_{n}+w_{n}u_{n-1}),\\ w_{n,t_{1}}=w_{n}(E-1)(E^{-1}+E^{-2})[Hu_{n}+v_{n}v_{n+1}+Hu_{ny}]. \end{cases} $$
(96)

Similar to the previous calculations, we obtain a Lax pair of equation (96) as follows:

$$ \textstyle\begin{cases} U= \left({\scriptsize\begin{matrix}{} 0&0&1\cr 1&0&0\cr -v_{n}&-w_{n}&\lambda-u_{n}+\partial_{y} \end{matrix}}\right),\\ \tilde{V}_{1}= \left({\scriptsize\begin{matrix}{} a_{0}(n-1)-v_{n}&-w_{n}&a_{1}(n-1)+\lambda-u_{n}+\partial_{y}\\ a_{1}(n-2)&a_{0}(n-2)&1\\ a_{1}(n)v_{n}-w_{n+1}&-a_{1}(n)w_{n}&(a_{1}(n)+1)(\lambda-u_{n+1}+\partial_{y})-v_{n+1} \end{matrix}}\right). \end{cases} $$
(97)

According to [23, 30], we can also derive a \((2+1)\)-dimensional lattice system corresponding to the \((1+1)\)-dimensional lattice equation (89) as follows:

$$ \textstyle\begin{cases} u_{n,t_{2}}=(E-1)v_{n}a_{1}(n-1)+a_{2}(n)w_{n+2}-w_{n}a_{2}(n-2)+a_{0y},\\ v_{n,t_{2}}=v_{n}(E-1)a_{0}(n-1)+a_{1}(n)w_{n+1}-w_{n}a_{1}(n-2),\\ w_{n,t_{2}}=w_{n}(a_{0}(n)-a_{0}(n-2)), \end{cases} $$
(98)

where

$$\begin{aligned}& a_{2}(n)=u_{n}+u_{n+1}+u_{n+2}, \\& \begin{aligned}[b] a_{1}(n)={}&v_{n}+v_{n+1}+u_{n+1}(E-1)^{-1}u_{n+2}- \bigl[(E-1)^{-1}u_{n}\bigr](E-1)^{-1}(E+1)u_{n}\\ &{}+(E-1)^{-1}(u_{n}+u_{n+1}+u_{n+2})_{y}, \end{aligned}\\& \begin{aligned}[b] a_{0}(n)={}&u_{n}(E-1)^{-1}a_{1}(n)+w_{n}+w_{n+1}+w_{n+2}+u_{n+1}v_{n+1}+u_{n}v_{n}\\ &{}+(E-1)^{-1}(u_{n+1}+v_{n+2}+u_{n}v_{n+2}-v_{n}u_{n+1}-v_{n}u_{n-1})+(E-1)^{-1} \bigl(a_{1}(n)\bigr)_{y}. \end{aligned} \end{aligned}$$

It is easy to obtain the Lax pair of equations (98)

$$\begin{aligned}& U= \begin{pmatrix} 0&0&1\\ 1&0&0\\ -v_{n}&-w_{n}&\lambda-u_{n}+\partial_{y} \end{pmatrix}, \\& \tilde{V}_{2}= \begin{pmatrix} V_{11}&V_{12}&V_{13}\\ a_{1}(n-2)&a_{0}(n-2)-w_{n}&a_{2}(n)+\lambda-u_{n}+\partial_{y}\\ V_{31}&V_{32}&V_{33} \end{pmatrix}, \end{aligned}$$

where

$$\begin{aligned}& V_{11}=a_{0}(n-1)+a_{2}(n-1)v_{n}-w_{n+1}-( \lambda-u_{n+1}+\partial_{y})v_{n}, \\& V_{12}=a_{2}(n-1)w_{n}-(\lambda-u_{n+1}+ \partial_{y})w_{n}, \\& V_{13}=a_{1}(n-1)+a_{2}(n-2) ( \lambda-u_{n}+\partial_{y})+(\lambda -u_{n+1}+ \partial_{y})^{2}-v_{n+1}, \\& \begin{aligned}[b] V_{31}={}&a_{1}(n)v_{n}-a_{2}(n) ( \lambda-u_{n+1}+\partial _{y})v_{n}-a_{2}(n)w_{n+1}-( \lambda-u_{n+2}+\partial_{y}) (\lambda -u_{n+1}+ \partial_{y})v_{n}\\ &{}+v_{n}v_{n+2} -( \lambda-u_{n+2}+\partial_{y})w_{n+1}, \end{aligned} \\& V_{32}=-a_{1}(n)w_{n}-a_{2}(n) ( \lambda-u_{n+1}+\partial _{y})w_{n}+w_{n}v_{n+2}-( \lambda-u_{n+2}+\partial_{y}) (\lambda-u_{n+1}+ \partial _{y})w_{n}, \\& \begin{aligned}[b] V_{33}={}&a_{0}(n)+a_{1}(n) (\lambda-u_{n}+ \partial_{y})+a_{2}(n) (\lambda -u_{n+1}+ \partial_{y}) (\lambda-u_{n}+\partial_{y})-a_{2}(n)v_{n+1}\\ &{}-w_{n+2}-v_{n+2}(\lambda-u_{n}+\partial_{y})+( \lambda-u_{n+2}+\partial _{y}) (\lambda-u_{n+1}+ \partial_{y}) (\lambda-u_{n}+\partial_{y})\\ &{}-( \lambda -u_{n+2}+\partial_{y})v_{n+1}. \end{aligned} \end{aligned}$$

Remark 6

We have obtained the Lax pairs of equations (88), (89), (96), and (98), from which we could investigate their infinite conservation laws and different Darboux transformations just like in the ways presented before. Hence we do not want to go into a discussion of them again in this paper.