N-Dimensional Lattice Integrable Systems and Their bi-Hamiltonian Structure on the Time Scale Using the R-Matrix Approach

Abstract

A time scale is a special measure chain that can unify continuous and discrete spaces, enabling the construction of integrable equations. In this paper, with the Lax operator generated by the displacement operator, N-dimensional lattice integrable systems on the time scale are given by the R-matrix approach. The recursion operators of the lattice systems are derived on the time scale. Finally, two integrable hierarchies of the discrete chain with a bi-Hamiltonian structure are obtained. In particular, we give the structure of two-field and four-field systems.

1. Introduction

In the realm of nonlinear evolution systems, integrable systems in (1 + 1) dimensions have been extensively investigated [1,2]. The time evolutionary variable is represented by one of the dimensions, while the other dimension represents the space variable. To incorporate research on the integrable system into a more comprehensive frame, one possible approach put forward by Stefan Hilger in 1988 [3] is to establish these systems on time scales. The method of studying integrable systems on the time scale is the unity of the lattice soliton system [4,5,6], field-like soliton system [7,8], q-discrete soliton system [9,10,11] and other nonlinear evolution systems [12]. Time scale $\mathbb{T}$ is a nonempty and arbitrarily chosen subset of real numbers, introduced to encompass all potential intervals on the real line $\mathbb{R}$, including discrete $\mathbb{Z}$, continuous $\mathbb{R}$, and so on [13,14]. To build the integrable system on the time scale, one approach involves an extension of the Gelfand–Dickey method [4], while another approach focuses on formulating different kinds of discrete dynamics on $\mathbb{R}$ [15].

The primary objective of this study is to present a theory that systematically constructs integrable discrete systems on a time scale with the R-matrix approach [16,17]. The R-matrix formalism represents a highly efficient and systematic approach for constructing integrable systems, originating from Gelfand and Dickey [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]. The key aspect of the R-matrix approach lies in its utilization of the Lax equations on suitable Lie algebras for constructing integrable systems. The constructions of the simplest R-matrices involve decomposing a Lie algebra into two Lie subalgebras [19]. Błaszak constructed the R-matrix theory of lattice integrable systems [20] and dispersionless systems. Furthermore, integrable systems are obtained by the R-matrix approach on time scales [21,22,23]. On this basis, we study the lattice system on the time scale through R-matrix formalisms [24]. The lattice system on the time scale obtained by the R-matrix method is a more general case, which includes the results of the previous study [20]. When different time scales are taken, the lattice system and q-lattice system can be obtained. In addition to the potential for constructing integrable systems, this formalism offers significant advantages for the bi-Hamiltonian structure and the hierarchy of symmetric and conserved quantities [25].

This paper is structured as follows: In the first part of this paper, we present the basic concept of a regular discrete time scale and the conceptual formalism of the R-matrix in a systematic way. In the second part, the simplest R-matrix is then built on the Lie algebra, which is introduced as the algebra of the displacement operator and is divided into two subalgebras. We give the family of equations of the N-dimensional lattice field systems on the time scale and discuss the infinite and finite fields. For the infinite field, we rewrite the first two flows of the equation to obtain a (2 + 1)-dimensional one-field system. For the finite field, the equations of the two-field systems to four-field systems are given. This part concludes with the establishment of the lattice system’s recursion operators. In the last section, with the recursive operators obtained in the previous part, we study bi-Hamiltonian structures of lattice hierarchies on the regular discrete obtained by means of the R-matrix. In the $\mathbb{T}=\mathbb{Z}$ case, the bi-Hamilton for lattice integrable systems can be derived. We can consider the content of this paper as the discretization of the field soliton system. In conclusion, we discuss the main findings of this study as well as some outstanding issues.

2. Basic Concepts about the Regular Time Scale and R-Matrix Theory

In this section, we review several concepts related to the time scale and R-matrix theory, especially regular discrete time scales [26,27].

Definition 1.

Let $\mathbb{T}$ be the time scale. For s ∈ $\mathbb{T}$, the forward jump operator σ: $\mathbb{T}$→$\mathbb{T}$ and backward jump operator ρ: $\mathbb{T}$→$\mathbb{T}$ are defined by

$$\sigma \left(s\right)=inf\left\{y\in \mathbb{T}:y>s\right\},\rho \left(s\right)=inf\left\{y\in \mathbb{T}:y<s\right\}.$$

(1)

Definition 2.

The forward and backward jump distance functions μ, v: $\mathbb{T}$→$R^{+}$ are given as follows

$$\mu \left(s\right)=\sigma \left(s\right)-s,v\left(s\right)=s-\rho \left(s\right),$$

(2)

where $\mu \left(s\right)$ and $v\left(s\right)$ are called graininess functions as well.

When $\mathbb{T}$= $\mathbb{R}$, $\mathbb{T}$= $\mathbb{Z}$, the forward jump operator, backward jump operator and distance functions become

$$\begin{array}{cc}\hfill & \sigma \left(s\right)=inf(x,\infty )=s,\sigma \left(s\right)=inf\left\{s+1,s+2,\dots \right\}=s+1,\hfill \\ \hfill & \rho \left(s\right)=sup(-\infty ,s)=s,\rho \left(s\right)=sup\left\{s-1,s-2,\dots \right\}=s-1,\hfill \\ \hfill & \mu \left(s\right)=s,\mu \left(s\right)=\sigma \left(s\right)-s=1.\hfill \end{array}$$

(3)

Definition 3.

If $s^{*}$ is left dense and $s_{*}$ is right dense, each point in $\mathbb{T}\setminus \{s_{*},s^{*}\}$ is scattered on both sides, and $\mathbb{T}$ is called the regular discrete time scale, which can be given by

$$\mu \left(s\right)=\sigma \left(s\right)-s\ne 0,s\in \mathbb{T}\setminus \{s_{*},s^{*}\}.$$

(4)

Definition 4.

A linear mapping R: $\mathcal{G}$→$\mathcal{G}$ satisfying the bracket below

$${[a,b]}_R:=[Ra,b]+[a,Rb].$$

(5)

This is known as the classical R-matrix. Obviously, (5) is skew symmetric, and when the Jacobi identity of (5) is checked, the Yang–Baxter equation is simply demonstrated to be a sufficient condition for fulfilling the R-matrix, that is,

$$[Ra,Rb]-R{[a,b]}_R+\alpha [a,b]=0,$$

(6)

there are only two relevant cases of the Yang–Baxter equation, namely, $\alpha =0$ and $\alpha \ne 0$.

Definition 5.

Let $\mathcal{G}$ be an algebra, and its decomposition can be defined as

$$\mathcal{G}={\mathcal{G}}_{\ge k}\oplus {\mathcal{G}}_{<k}.$$

(7)

Only if k = 0, 1, subspace ${\mathcal{G}}_{\ge k}$ and ${\mathcal{G}}_{<k}$ are closed Lie subalgebras of $\mathcal{G}$. The classical R-matrix can be defined as

$$R:=\frac{1}{2}(P_{\ge k}-P_{<k}),k=0,1,$$

(8)

where $P_{\ge k}$ and $P_{<k}$ are projections onto ${\mathcal{G}}_{\ge k}$, ${\mathcal{G}}_{<k}$. Therefore, Yang–Baxter Equation (6) for α = $\frac{1}{4}$ is satisfied by the classical R-matrix (8) since it is defined via projection on Lie subalgebras.

3. R-Matrix Approach to N-Dimensional Lattice Integrable Systems on the Regular Time Scale

In this part, we will use the R-matrix approach to construct lattice soliton systems on the regular time scale. In this kind of time scale, one can define the operator algebra of the displacement operator $\mathcal{E}$ and generate the Lax operator [24]. Let us start with the Lax operators $L\in \mathcal{G}$ below:

$$\begin{array}{cc}\hfill k& =0:L={\mathcal{E}}^{N+\alpha }+u_{N+\alpha -1}{\mathcal{E}}^{N+\alpha -1}+u_{N+\alpha -2}{\mathcal{E}}^{N+\alpha -2}+\dots +u_{\alpha }{\mathcal{E}}^{\alpha },\hfill \\ \hfill k& =1:L={\tilde{u}}_{N+\alpha }{\mathcal{E}}^{N+\alpha }+{\tilde{u}}_{N+\alpha -1}{\mathcal{E}}^{N+\alpha -1}+\dots +{\tilde{u}}_{\alpha +1}{\mathcal{E}}^{\alpha +1}+{\mathcal{E}}^{\alpha },\hfill \end{array}$$

(9)

where $u_i\left(x\right)$ and $\widetilde{u_i}\left(x\right)$ are the function defined on the time scale $x\in \mathbb{T}$ and depend only on the evolution parameters $t_n$; $\alpha$ is an integer that satisfies $-N\le \alpha \le -1$.

To further study the recursion relation of the equation family and simplify the operation, the restriction $N+\alpha =1$ of the Lax operator is increased, so the Lax operators (9) become as follows:

$$k=0:L=\mathcal{E}+\sum _{i=1}^N{u}_{1-i}{\mathcal{E}}^{1-i},$$

(10)

$$k=1:L=\sum _{i=0}^{N-1}{\tilde{u}}_{1-i}{\mathcal{E}}^{1-i}+{\mathcal{E}}^{1-N}.$$

(11)

Then, the algebra of the shift operators can be given by

$$\mathcal{G}={\mathcal{G}}_{\ge k}\oplus {\mathcal{G}}_{<k}=\left\{\sum _{i\ge k}{u}_i\left(x\right){\mathcal{E}}^i\right\}\oplus \left\{\sum _{i<k}{u}_i\left(x\right){\mathcal{E}}^i\right\},$$

(12)

equipped with the Lie bracket $[\mathcal{A},\mathcal{B}]=\mathcal{A}\mathcal{B}-\mathcal{B}\mathcal{A}$, where $\mathcal{A}$, $\mathcal{B}$∈$\mathcal{G}$, and subspaces ${\mathcal{G}}_{\ge k}$, ${\mathcal{G}}_{<k}$ are closed Lie algebras of $\mathcal{G}$ only if $k=1,k=0$. The classical R-matrices can be defined if we decompose $\mathcal{G}$ into the Lie subalgebras

$$R:=\frac{1}{2}(P_{\ge k}-P_{<k})=P_{\ge k}-\frac{1}{2}=\frac{1}{2}-P_{<k},$$

(13)

where $P_{\ge k}$ and $P_{<k}$ are the projections onto the Lie subalgebras ${\mathcal{G}}_{\ge k}$ and ${\mathcal{G}}_{<k}$ for $k=0$, $k=1$, respectively, such that

$$P_{\ge k}\left(A\right)=\sum _{\ge k}{a}_i{\mathcal{E}}^i,P_{<k}\left(A\right)=\sum _{<k}{u}_i\left(x\right){\mathcal{E}}^i,forA=\sum _i{a}_i{\mathcal{E}}^i\in \mathcal{G}.$$

(14)

According to the classical R-matrix (13) and the powers of the finite-field Lax functions (10) and (11), the following hierarchy of evolution equations is derived.

$$\frac{dL}{d{t}_n}=[R\left(L^n\right),L]=[{\left(L^n\right)}_{\ge k},L]=-[{\left(L^n\right)}_{<k},L],(k=0,1,n\in \mathbb{N}).$$

(15)

When $k=0$, for finite-field Lax operator (10), $R\left(L\right)=\mathcal{E}+u_0$. Thus, the first flow is given by (15)

$$\sum _{i=1}^N{\left(u_{1-i}\right)}_{t_1}{\mathcal{E}}^{1-i}=\sum _{i=1}^N{u}_{1-i}(1-E^{1-i})u_0{\mathcal{E}}^{1-i}+\sum _{i=1}^{N-1}(E-1)u_{-i}{\mathcal{E}}^{1-i}.$$

(16)

Here, the displacement operator $\mathcal{E}$ has the following algorithm:

$${\mathcal{E}}^{m}u=\left(E^{m}u\right){\mathcal{E}}^m,Eu\left(x\right)=u\left(\sigma \left(x\right)\right),E^{-1}\left(u\left(x\right)\right)=v\left(x\right),$$

(17)

for $R\left(L^2\right)={\mathcal{E}}^2+u_0^2+u_{-1}+E{u}_{-1}+u_0\mathcal{E}+E{u}_0\mathcal{E}$, one calculates the second flow

$$\begin{array}{cc}\hfill \sum _{i=1}^N{\left(u_{1-i}\right)}_{t_2}{\mathcal{E}}^{1-i}& =\sum _{i=1}^N{u}_{1-i}(1-E^{1-i})u_0^2{\mathcal{E}}^{1-i}+\sum _{i=1}^N{u}_{1-i}(E+1)(1-E^{1-i})u_{-1}\hfill \\ \hfill & +\sum _{i=1}^{N-1}E{u}_{-i}(E+1)u_0{\mathcal{E}}^{1-i}-\sum _{i=1}^{N-1}{u}_{-i}(E^{1-i}+E^{-i})u_0{\mathcal{E}}^{1-i}\hfill \\ \hfill & +\sum _{i=1}^{N-2}(E^2-1)u_{-1-i}{\mathcal{E}}^{1-i}.\hfill \end{array}$$

(18)

Therefore, the N-dimensional field system of the lattice system on the time scale is obtained. N-dimensional lattice integrable systems on the time scale are systems where they are time continuous and the dynamic field function u is defined on the time scale. When taking a special time scale, the lattice system, the q-lattice system, can be obtained. Obviously, Equations (16) and (18) are a finite field. When N tends to infinity, we can obtain the following infinite-field system

$$\begin{array}{cc}\hfill {\left(u_{1-i}\right)}_{t_1}& =u_{1-i}(1-E^{1-i})+(E-1)u_{-i},\hfill \\ \hfill {\left(u_{1-i}\right)}_{t_2}& =u_{1-i}(1-E^{1-i})u_0^2+u_{1-i}(E+1)(1-E^{1-i})u_{-1}+E{u}_{-i}(E+1)u_0\hfill \\ \hfill & -u_{-i}(E^{1-i}+E^{-i})u_0+(E^2-1)u_{-1-i}.\hfill \end{array}$$

(19)

From the above N-dimensional lattice system, we can obtain the following (2 + 1)-dimensional case on the time scale [28] by taking the first two formulas in (16) and the first formula in (18),

$$\begin{array}{cc}\hfill & {\left(u_0\right)}_{t_1}=(E-1)u_{-1},\hfill \\ \hfill & {\left(u_{-1}\right)}_{t_1}=u_{-1}(1-E^{-1})u_0+(E-1)u_{-2},\hfill \\ \hfill & {\left(u_0\right)}_{t_2}=E{u}_{-1}(E+1)u_0-u_{-1}(E^{-1}+1)u_0+(E^2-1)u_{-2}.\hfill \end{array}$$

(20)

Letting $u_0=w,t_1=y$ and $t_2=t$, the above formula becomes

$$w_y=(E-1)u_{-1},$$

(21)

$${\left(u_{-1}\right)}_y=u_{-1}(1-E^{-1})w+(E-1)u_{-2},$$

(22)

$$w_t=E{u}_{-1}(E+1)w-u_{-1}(E^{-1}+1)w+(E^2-1)u_{-2}.$$

(23)

Applying $(E+1)$ to (22) starting from the left, we have

$$(E^2-1)u_{-2}=(E+1)[{\left(u_{-1}\right)}_y-u_{-1}(1-E^{-1})w].$$

(24)

Then, applying the operator $(E-1)$ to Equation (23) and substituting Equations (21) and (24), we ultimately derive the (2 + 1)-dimensional one-field system as follows:

$$(E-1)w_t=(E+1)w_{yy}-2w_{y}w+E{w}_y(E+1)(E^{-1}+E)w.$$

(25)

When $k=1$, according to the Lax operator (11), $R\left(L\right)=\widetilde{u_1}$, and the first flow is obtained as

$$\sum _{i=0}^{N-1}{\left({\tilde{u}}_{1-i}\right)}_{t_1}{\mathcal{E}}^{1-i}=\sum _{i=0}^{N-1}{\tilde{u}}_{1}E{\tilde{u}}_{1-i}{\mathcal{E}}^{1-i}-\sum _{i=0}^{N-1}{\tilde{u}}_{-i}{E}^{-i}{\tilde{u}}_1{\mathcal{E}}^{1-i},$$

(26)

for $R\left(L^2\right)=\widetilde{u_1}E\widetilde{u_1}{\mathcal{E}}^2+\widetilde{u_1}E\widetilde{u_0}\mathcal{E}+\widetilde{u_0}\widetilde{u_1}\mathcal{E}$, the second flow can be obtained as

$$\begin{array}{cc}\hfill & \sum _{i=0}^{N-1}{\left({\tilde{u}}_{1-i}\right)}_{t_2}{\mathcal{E}}^{1-i}=\sum _{i=0}^{N-2}{\tilde{u}}_{1}E{\tilde{u}}_1{E}^2{\tilde{u}}_{-1-i}{\mathcal{E}}^{1-i}-\sum _{i=0}^{N-2}{\tilde{u}}_{-1-i}{E}^{-1-i}{\tilde{u}}_1{E}^{-i}{\tilde{u}}_1{\mathcal{E}}^{1-i}\hfill \\ \hfill & +\sum _{i=0}^{N-1}{\tilde{u}}_1(E+1){\tilde{u}}_{0}E{\tilde{u}}_{-i}{\mathcal{E}}^{1-i}-\sum _{i=0}^{N-1}{\tilde{u}}_{-i}{E}^{-i}{\tilde{u}}_1(E^{1-i}+E^{-i}){\tilde{u}}_0{\mathcal{E}}^{1-i},\hfill \end{array}$$

(27)

where ${\tilde{u}}_{1-N}=1$. Similarly, (27) is a finite-field (N-dimensional) system, and the infinite field is as follows:

$$\begin{array}{cc}\hfill {\left({\tilde{u}}_{1-i}\right)}_{t_1}& ={\tilde{u}}_{1}E{\tilde{u}}_{1-i}-{\tilde{u}}_{-i}{E}^{-i}{\tilde{u}}_1,\hfill \\ \hfill {\left({\tilde{u}}_{1-i}\right)}_{t_2}& ={\tilde{u}}_{1}E{\tilde{u}}_1{E}^2{\tilde{u}}_{-1-i}-{\tilde{u}}_{-1-i}{E}^{-1-i}{\tilde{u}}_1{E}^{-i}{\tilde{u}}_1+{\tilde{u}}_1(E+1){\tilde{u}}_{0}E{\tilde{u}}_{-i}\hfill \\ \hfill & -{\tilde{u}}_{-i}{E}^{-i}{\tilde{u}}_1(E^{1-i}+E^{-i}){\tilde{u}}_0.\hfill \end{array}$$

(28)

The N-dimensional lattice integrable systems on the time scale are obtained (16), (18), (26) and (27). Let us list several examples.

Case 1. Two-field systems

(1) Let $k=0$, $\alpha =-1$. Thus, the Lax operator is given by

$$L=\mathcal{E}+u_{-1}{\mathcal{E}}^{-1}+u_0\equiv p{\mathcal{E}}^{-1}+u+\mathcal{E}.$$

(29)

The Formula (15) results in the evolution of L, and when $n=3$, the hierarchy can be obtained

$$\begin{array}{cc}\hfill n=3:\frac{dp}{d{t}_3}& =u{p}^2+u^{3}p-p{E}^{-1}\left(p\right)E^{-2}\left(u\right)-2p{E}^{-1}\left(u\right)E^{-1}\left(p\right)\hfill \\ \hfill & -p{E}^{-1}\left(u^3\right)+2puE\left(p\right)-p^2{E}^{-1}\left(u\right)+pE\left(u\right)E\left(p\right),\hfill \\ \hfill \frac{du}{d{t}_3}& =E\left(p\right)[u^2+uE\left(u\right)+E\left(u^2\right)+E\left(p\right)+E^2\left(p\right)]\hfill \\ \hfill & -p[E^{-1}\left(p\right)+E^{-1}\left(u^2\right)+u{E}^{-1}\left(u\right)+u^2+p].\hfill \end{array}$$

(30)

The above is the family of Toda equations on the regular time scale; this is a more general situation. Different families of time-continuous and space discrete equations can be obtained when different time scales are taken. When $\mathbb{T}$= $\mathbb{Z}$, $E\left(p\right(x\left)\right)=p(x+1)$, the third family of equations in Equation (30) generate the following famous Toda lattice hierarchy [20].

$$\begin{array}{cc}\hfill \frac{dp}{d{t}_3}& =u\left(x\right)[p^2\left(x\right)+p\left(x\right)u^2\left(x\right)+2p\left(x\right)p(x+1)+u(x+1)p(x+1)]\hfill \\ \hfill & -p\left(x\right)[p(x-1)u(x-2)+2p(x-1)v(x-1)+u^3(x-1)+p\left(x\right)u(x-1)],\hfill \\ \hfill \frac{du}{d{t}_3}& =p(x+1)[u^2\left(x\right)+u\left(x\right)u(x+1)+u^2(x+1)+p(x+1)+p(x+2)]\hfill \\ \hfill & -p\left(x\right)[p(x-1)+p\left(x\right)+u^2(x-1)+u\left(x\right)u(x-1)+u^2\left(x\right)].\hfill \end{array}$$

(31)

A characteristic of integrable systems with infinite families of reciprocal symmetric equations is the existence of recursion operators. Recursion operators acting on appropriate domains can generate local domain equation families [29]. The recursion operator of a given system is an operator of such a property that it produces another symmetry when it acts on one symmetry of the considered system. The next equation of a family of equations can be obtained by acting on the previous equation by a recursion operator. To explore the recursive relationship between the $(n+1)$ equation and the n equation, we introduce the concept of recursive operators, which also prepares for the calculation of bi-Hamiltonian structures in the next section.

The recursion operator has the properties that:

$$\Phi \left(L_{t_n}\right)=L_{t_{n+\alpha +N}},n\in {\mathbb{Z}}_{+}.$$

(32)

When $k=0$, the recursion operator of the corresponding Lax family of equations is constructed by solving the following formula:

$$L_{t_{n+\alpha +N}}=L_{t_n}L+[R,L].$$

(33)

By analyzing the order of the highest terms and lowest terms of R, the remainder in the form of a shift operator can be obtained as

$$R=a_n{\mathcal{E}}^{N+\alpha -1}+\dots +b_n{\mathcal{E}}^{\alpha }.$$

(34)

When $k=1$, the remainder R can be defined as follows:

$$R=a_n{\mathcal{E}}^{N+\alpha }+\dots +b_n{\mathcal{E}}^{\alpha +1}.$$

(35)

Hence, considering infinite Toda lattice (29), we have

$$R=a_n+b_n{\mathcal{E}}^{-1}.$$

(36)

Solving by substituting Equation (36) into Lax hierarchy (33), we find

$$\begin{array}{cc}\hfill a_n& ={(E-1)}^{-1}{u}_{t_n},\hfill \\ \hfill b_n& =\frac{E^{-1}p}{p{E}^{-1}-E^{-1}p}{p}_{t_n},\hfill \end{array}$$

(37)

by substituting Equation (37) into Equation (34) and then into Equation (33), the relationship between $u_{t_n+1}$, $p_{t_n+1}$ and $u_{t_n}$, $p_{t_n}$ can be obtained as

$$\begin{array}{cc}\hfill p_{t_{n+1}}& =p(1+E^{-1})u_{t_n}+\frac{E^{-1}up{E}^{-1}-u{E}^{-1}p}{p{E}^{-1}-E^{-1}p}{p}_{t_n},\hfill \\ \hfill u_{t_{n+1}}& =u{u}_{t_n}+\frac{p{E}^{-1}-p}{p{E}^{-1}-E^{-1}p}{p}_{t_n}.\hfill \end{array}$$

(38)

(2) Let $k=1$, $\alpha =-1$. The Lax operator is given as follows:

$$L={\mathcal{E}}^{-1}+\widetilde{u_0}+\widetilde{u_1}\mathcal{E}\equiv {\mathcal{E}}^{-1}+\tilde{u}+\tilde{p}\mathcal{E},$$

(39)

and the first equation ($n=1$) of the hierarchy can be obtained as

$$\frac{d\tilde{u}}{d{t}_1}=\tilde{p}-E^{-1}\tilde{p},\frac{d\tilde{p}}{d{t}_1}=\tilde{p}(E\tilde{u}-\tilde{u}).$$

(40)

As above, to study the relation between the $(n+1)$ equation and the n equation in the hierarchy, the remainder R has the following form:

$$R=\widetilde{a_n}\mathcal{E}+\widetilde{b_n}.$$

(41)

Solving above into Lax hierarchy (33)

$$\begin{array}{cc}\hfill \widetilde{a_n}& =\frac{E\tilde{p}}{\tilde{p}E-E\tilde{p}}{\tilde{p}}_{t_n},\hfill \\ \hfill \widetilde{b_n}& ={(E^{-1}-1)}^{-1}{\tilde{u}}_{t_n},\hfill \end{array}$$

(42)

then the recurrence relationship can be obtained as

$$\begin{array}{cc}\hfill & {\tilde{u}}_{t_{n+1}}={\tilde{u}}_{t_n}\tilde{u}+\frac{\tilde{p}E-\tilde{p}}{\tilde{p}E-E\tilde{p}}{\tilde{p}}_{tn},\hfill \\ \hfill & {\tilde{p}}_{t_{n+1}}=p(1+E)\widetilde{u_n}+\frac{E\tilde{u}\tilde{p}E-\tilde{u}E\tilde{p}}{\tilde{p}E-E\tilde{p}}{\tilde{p}}_{tn},\hfill \end{array}$$

(43)

Case 2. Four-field systems

(1) Let $k=0$, $\alpha =-3$. The Lax operator and remainder R can be given by

$$\begin{array}{cc}\hfill L& =\mathcal{E}+v+u{\mathcal{E}}^{-1}+p{\mathcal{E}}^{-2}+r{\mathcal{E}}^{-3},\hfill \\ \hfill R& =a_n+c_n{\mathcal{E}}^{-1}+f_n{\mathcal{E}}^{-2}+b_n{\mathcal{E}}^{-3}.\hfill \end{array}$$

(44)

When $n=1$, we can obtain

$$\begin{array}{cc}\hfill \frac{dr}{d{t}_1}& =r[v-E^{-3}\left(v\right)],\hfill \\ \hfill \frac{dp}{d{t}_1}& =p[v-E^{-2}\left(v\right)]+E\left(r\right)-r,\hfill \\ \hfill \frac{du}{d{t}_1}& =u[v-E^{-1}\left(v\right)]+E\left(p\right)-p,\hfill \\ \hfill \frac{dv}{d{t}_1}& =E\left(u\right)-u.\hfill \end{array}$$

(45)

Comparing the degrees of $\mathcal{E}$ on the left and right sides of Equation (33), the following calculation can be obtained

$$\begin{array}{cc}\hfill a_n& ={(E-1)}^{-1}{v}_{t_n},\hfill \\ \hfill b_n& =\frac{E^{-3}r}{r{E}^{-3}-E^{-3}r}{r}_{t_n},\hfill \\ \hfill c_n& =\frac{E^{-1}r}{r{E}^{-3}-E^{-1}r}{u}_{t_n}+\frac{(E^{-2}p-p{E}^{-2})E^{-2}r}{(r{E}^{-3}-E^{-1}r)(r{E}^{-3}-E^{-2}r)}{p}_{t_n}\hfill \\ \hfill & +\left[\frac{E^{-3}u}{r{E}^{-3}-E^{-1}r}+\frac{(E^{-2}p-p{E}^{-2})(E^{-3}pr{E}^{-3}-E^{-3}rp{E}^{-2})}{(r{E}^{-3}-E^{-1}r)(r{E}^{-3}-E^{-3}r)(r{E}^{-3}-E^{-2})r}\right]r_{t_n},\hfill \\ \hfill f_n& =\frac{E^{-2}r}{r{E}^{-3}-E^{-2}r}{p}_{t_n}+\frac{E^{-3}pr{E}^{-3}-E^{-3}rp{E}^{-2}}{(r{E}^{-3}-E^{-3}r)(r{E}^{-3}-E^{-2}r)}{r}_{t_n}.\hfill \end{array}$$

(46)

Therefore, the recurrence relationship and recursion operator are obtained as

$$\begin{array}{cc}\hfill r_{n+1}& =\left[\frac{E^{-3}vr{E}^{-3}-v{E}^{-3}r}{r{E}^{-3}-E^{-3}r}+\frac{(E^{-1}p-p{E}^{-2})E^{-3}u}{r{E}^{-3}-E^{-1}r}\right.\hfill \\ \hfill & +\frac{(E^{-2}-u{E}^{-1})(E^{-3}pr{E}^{-3}-E^{-3}rp{E}^{-2})}{(r{E}^{-3}-E^{-3}r)(r{E}^{-3}-E^{-2}r)}\hfill \\ \hfill & \left.+\frac{(E^{-1}p-p{E}^{-2})(E^{-2}p-p{E}^{-2})(E^{-3}pr{E}^{-3}-E^{-3}rp{E}^{-2})}{(r{E}^{-3}-E^{-1}r)(r{E}^{-3}-E^{-3}r)(r{E}^{-3}-E^{-2}r)}\right]r_{t_n}\hfill \\ \hfill & +\frac{E^{-1}pr{E}^{-3}-p{E}^{-3}r}{r{E}^{-3}-E^{-1}r}{u}_{t_n}+\frac{rE-r{E}^{-3}}{E-1}{v}_{t_n}\hfill \\ \hfill & +\left[\frac{E^{-2}ur{E}^{-3}-u{E}^{-3}r}{r{E}^{-3}-E^{-2}r}+\frac{(E^{-1}p-p{E}^{-2})(E^{-2}p-p{E}^{-2})E^{-2}r}{(r{E}^{-3}-E^{-1}r)(r{E}^{-3}-E^{-2}r)}\right]p_{t_n},\hfill \end{array}$$

$$\begin{array}{cc}\hfill p_{n+1}& =\left[\frac{(E^{-1}u-u{E}^{-1})(E^{-2}p-p{E}^{-2})(E^{-3}pr{E}^{-3}-E^{-3}rp{E}^{-2})}{(r{E}^{-3}-E^{-1}r)(r{E}^{-3}-E^{-3}r)(r{E}^{-3}-E^{-2}r)}\right.\hfill \\ \hfill & +\frac{r{E}^{-3}-E^{-2}r}{r{E}^{-3}-E^{-3}r}+\frac{(E^{-2}v-v)(E^{-3}pr{E}^{-3}-E^{-3}rp{E}^{-2})}{(r{E}^{-3}-E^{-3}r)(r{E}^{-3}-E^{-2}r)}\hfill \\ \hfill & \left.+\frac{(E^{-1}u-u{E}^{-1})E^{-3}u}{r{E}^{-3}-E^{-1}r}\right]r_{t_n}+\frac{E^{-1}ur{E}^{-3}-u{E}^{-2}r}{r{E}^{-3}-E^{-1}}{u}_{t_n}\hfill \\ \hfill & +\left[\frac{E^{-2}vr{E}^{-3}-v{E}^{-2}r}{r{E}^{-3}-E^{-2}r}+\frac{(E^{-1}u-u{E}^{-1})(E^{-2}p-p{E}^{-2})E^{-2}r}{(r{E}^{-3}-E^{-1}r)(r{E}^{-3}-E^{-2}r)}\right]p_{t_n}\hfill \\ \hfill & +\frac{pE-p{E}^{-2}}{E-1}{v}_{t_n},\hfill \\ \hfill u_{n+1}& =\left[\frac{(E^{-1}v-v{E}^{-1})(E^{-2}p-p{E}^{-2})(E^{-3}pr{E}^{-3}-E^{-3}rp{E}^{-2})}{(r{E}^{-3}-E^{-1}r)(r{E}^{-3}-E^{-3}r)(r{E}^{-3}-E^{-2}r)}\right.\hfill \\ \hfill & \left.+\frac{(1-E)(E^{-3}pr{E}^{-3}-E^{-3}rp{E}^{-2})}{(r{E}^{-3}-E^{-3}r)(r{E}^{-3}-E^{-2}r)}+\frac{(E^{-1}v-v{E}^{-1})E^{-3}u}{r{E}^{-3}-E^{-1}r}+\right]r_{t_n}\hfill \\ \hfill & +\frac{E^{-1}vr{E}^{-3}-v{E}^{-2}r}{r{E}^{-3}-E^{-1}}{u}_{t_n}+\frac{uE-u{E}^{-1}}{E-1}{v}_{t_n}\hfill \\ \hfill & +\left[\frac{r{E}^{-3}-E^{-1}r}{r{E}^{-3}-E^{-2}r}+\frac{(E^{-1}v-v{E}^{-1})(E^{-2}p-p{E}^{-2})E^{-2}r}{r{E}^{-3}-E^{-2}r}\right]p_{t_n},\hfill \\ \hfill v_{n+1}& =\left[\frac{(1-E)(E^{-2}p-p{E}^{-2})(E^{-3}pr{E}^{-3}-E^{-3}rp{E}^{-2})}{(r{E}^{-3}-E^{-1}r)(r{E}^{-3}-E^{-3}r)(r{E}^{-3}-E^{-2}r)}+\frac{(1-E)E^{-3}u}{r{E}^{-3}-E^{-1}r}\right]r_{t_n}\hfill \\ \hfill & +\frac{r{E}^{-3}-r}{r{E}^{-3}-E^{-1}r}{u}_{t_n}+v{v}_{t_n}+\frac{(1-E)(E^{-2}p-p{E}^{-2})E^{-2}r}{(r{E}^{-3}-E^{-1}r)(r{E}^{-3}-E^{-2}r)}{p}_{t_n}.\hfill \end{array}$$

(47)

(2) Let $k=1$, $\alpha =-3$, so the Lax operator has the below form:

$$L={\mathcal{E}}^{-3}+\tilde{p}{\mathcal{E}}^{-2}+\tilde{u}{\mathcal{E}}^{-1}+\tilde{v}+\tilde{w}\mathcal{E}.$$

(48)

When $n=1$, we can obtain

$$\begin{array}{cc}\hfill \frac{d\tilde{p}}{d{t}_1}& =\tilde{w}-E^{-3}\tilde{w},\hfill \\ \hfill \frac{d\tilde{u}}{d{t}_1}& =\tilde{w}E\left(\tilde{p}\right)-\tilde{p}{E}^{-2}\left(\tilde{w}\right),\hfill \\ \hfill \frac{d\tilde{v}}{d{t}_1}& =\tilde{w}E\left(\tilde{u}\right)-\tilde{u}{E}^{-1}\tilde{w},\hfill \\ \hfill \frac{d\tilde{w}}{d{t}_1}& =\tilde{w}[E\left(\tilde{v}\right)-\tilde{v}].\hfill \end{array}$$

(49)

The remainder R of the above Lax hierarchy has the following form:

$$R=\widetilde{a_n}\mathcal{E}+\widetilde{c_n}+\widetilde{f_n}{\mathcal{E}}^{-1}+\widetilde{b_n}{\mathcal{E}}^{-2}.$$

(50)

Comparing the degrees of $\mathcal{E}$ on the right and left sides of Equation (33), we can obtain

$$\begin{array}{cc}\hfill \widetilde{a_n}& =\frac{E\tilde{w}}{\tilde{w}E-E\tilde{w}}\widetilde{w_{t_n}},\hfill \\ \hfill \widetilde{b_n}& =\frac{1}{E^{-3}-1}\widetilde{p_{t_n}},\hfill \\ \hfill \widetilde{c_n}& =\left[\frac{E^{-2}\tilde{u}{E}^{-3}-\tilde{u}{E}^{-1}}{{(E^{-3}-1)}^2}+\frac{(E^{-1}\tilde{p}-\tilde{p}{E}^{-2})(E^{-2}\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2})}{{(E^{-3}-1)}^3}\right]\widetilde{p_{t_n}}\hfill \\ \hfill & +\frac{E^{-1}\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2}}{{(E^{-3}-1)}^2}\widetilde{u_{t_n}}+\frac{1}{E^{-3}-1}\widetilde{v_{t_n}},\hfill \\ \hfill \widetilde{f_n}& =\frac{E^{-2}\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2}}{{(E^{-3}-1)}^2}\widetilde{p_{t_n}}+\frac{1}{E^{-3}-1}\widetilde{u_{t_n}}.\hfill \end{array}$$

(51)

Then, the recurrence relationship and recursion operator are obtained as

$$\begin{array}{cc}\hfill \widetilde{p_{n+1}}& =\left[\frac{E^{-2}\tilde{v}{E}^{-3}-\tilde{v}}{E^{-3}-1}+\frac{(\tilde{p}-\tilde{p}{E}^{-2})(E^{-2}\tilde{u}{E}^{-3}-\tilde{u}{E}^{-1})}{{(E^{-3}-1)}^2}\right.\hfill \\ \hfill & +\frac{(E^{-1}\tilde{u}-\tilde{u}{E}^{-1})(E^{-2}\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2})}{{(E^{-3}-1)}^2}\hfill \\ & \left.+\frac{(\tilde{p}-\tilde{p}{E}^{-2})(E^{-1}\tilde{p}-\tilde{p}{E}^{-2})(E^{-2}\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2})}{{(E^{-3}-1)}^3}\right]\widetilde{p_{t_n}}\hfill \\ \hfill & +\left[\frac{E^{-1}\tilde{u}{E}^{-3}-\tilde{u}{E}^{-1}}{E^{-3}-1}+\frac{(\tilde{p}-\tilde{p}{E}^{-2})(E^{-1}\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2})}{{(E^{-3}-1)}^2}\right]\widetilde{u_{t_n}}\hfill \\ \hfill & +\frac{\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2}}{E^{-3}-1}\widetilde{v_{t_n}}+\frac{\tilde{w}E-E^{-2}\tilde{w}}{\tilde{w}E-E^{-2}\tilde{w}}\widetilde{w_{t_n}},\hfill \\ \hfill \widetilde{u_{n+1}}& =\left[\frac{E^{-2}\tilde{w}{E}^{-3}-\tilde{w}E}{E^{-3}-1}+\frac{(\tilde{u}-\tilde{u}{E}^{-1})(E^{-2}\tilde{u}{E}^{-3}-\tilde{u}{E}^{-1})}{{(E^{-3}-1)}^2}\right.\hfill \\ \hfill & +\frac{(\tilde{u}-\tilde{u}{E}^{-1})(E^{-1}\tilde{p}-\tilde{p}{E}^{-2})(E^{-2}\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2})}{{(E^{-3}-1)}^3}\hfill \\ \hfill & \left.+\frac{(E^{-2}\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2})(E^{-1}\tilde{v}-\tilde{v}E)}{{(E^{-3}-1)}^2}\right]\widetilde{p_{t_n}}\hfill \\ \hfill & +\left[\frac{E^{-1}\tilde{v}{E}^{-3}-\tilde{v}}{E^{-3}-1}+\frac{(\tilde{u}-\tilde{u}{E}^{-1})(E^{-1}\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2})}{{(E^{-3}-1)}^2}\right]\widetilde{u_{t_n}}\hfill \\ \hfill & +\frac{\tilde{u}{E}^{-3}-\tilde{u}{E}^{-1}}{E^{-3}-1}\widetilde{v_{t_n}}+\frac{E\tilde{p}\tilde{w}E-\tilde{p}{E}^{-1}\tilde{w}}{\tilde{w}E-E\tilde{w}}\widetilde{w_{t_n}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill \widetilde{v_{n+1}}& =\frac{(E^{-1}\tilde{w}-\tilde{w}E)(E^{-2}\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2})}{{(E^{-3}-1)}^2}\widetilde{p_{t_n}}+\frac{E^{-1}\tilde{w}{E}^{-3}-\tilde{w}E}{E^{-3}-1}\widetilde{u_{t_n}}\hfill \\ \hfill & +\tilde{v}\widetilde{v_{t_n}}+\frac{E\tilde{u}\tilde{w}}{\tilde{w}E-E\tilde{w}}\widetilde{w_{t_n}},\hfill \\ \hfill \widetilde{w_{n+1}}& =\left[\frac{(\tilde{w}-\tilde{w}E)(E^{-1}\tilde{p}-\tilde{p}{E}^{-2})(E^{-2}\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2})}{{(E^{-3}-1)}^3}\right.\hfill \\ \hfill & \left.+\frac{(\tilde{w}-\tilde{w}E)(E^{-2}\tilde{u}{E}^{-3}-\tilde{u}{E}^{-1})}{{(E^{-3}-1)}^2}\right]\widetilde{p_{t_n}}+\frac{\tilde{w}{E}^{-3}-\tilde{w}E}{E^{-3}-1}\widetilde{v_{t_n}}\hfill \\ \hfill & +\frac{(\tilde{w}-\tilde{w}E)(E^{-1}\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2})}{{(E^{-3}-1)}^{-2}}\widetilde{u_{t_n}}+\frac{E\tilde{v}\tilde{w}E-E\tilde{w}\tilde{v}}{\tilde{w}W-E\tilde{w}}\widetilde{w_{t_n}}.\hfill \end{array}$$

(52)

To sum up, starting from Lax operators (10) and (11) and taking $N=2,N=4$, a family of evolution equations is generated according to the classical R-matrix method. The two-field and four-field equations of the lattice system on the regular discrete time scale are obtained.

4. R-Matrix Theory for bi-Hamiltonian Structures on the Regular Time Scale

In this section, the bi-Hamiltonian structures of lattice hierarchies on the regular discrete obtained in the preceding section by means of the R-matrix method [30,31] are investigated. One of the advantages of the classical R-matrix approach is to construct bi-Hamiltonian structures. To find the bi-Hamiltonian structure of the lattice hierarchy, the linear Poisson tensor and related Hamiltonian are constructed, and trace functions are introduced into the displacement operator algebra:

$$\mathrm{Tr}\left(\mathrm{A}\right):={\int }_{\mathbb{T}}{\displaystyle \frac{1}{\mathit{u}}}{\mathrm{a}}_0\Delta \mathrm{x},$$

(53)

where A = ${\displaystyle \sum _i}{a}_i{\mathcal{E}}^i$.

Theorem 1.

Using trace (53), the inner product is given by bilinear mapping on $\mathbf{g}$

$${(.,.)}_{\mathbf{g}}:\mathbf{g}\times \mathbf{g}\to \mathbb{K},{(A,C)}_{\mathbf{g}}:=\mathrm{Tr}\left(\mathrm{AC}\right).$$

(54)

This inner product is non-degenerate, symmetric and adjoint invariant.

In order to define the Hamiltonian structure of (15), we need to define the adjoint of the R-matrix (13) $R^{†}$. We can obtain [30,32]

$$R^{†}=P_{\ge k}^{†}-\frac{1}{2},k=0,1.$$

(55)

The existence of a proper inner product (54) can be identified by its dual ${\mathbf{g}}^{*}$, and $\Theta (\mathbf{g}\cong {\mathbf{g}}^{*})$ is a space of smooth functions on $\mathbf{g}$. The linear Poisson tensor [5,32] can be derived as

$${\pi }_{0}dH=R^{†}[dH,L]+[RdH,L]={\left([dH,L]{\mathcal{E}}^{-1}\right)}_{<k}\mathcal{E}+[L,d{H}_{<k}],k=0,1,$$

(56)

where H ∈$\Theta \left(\mathbf{g}\right)$. We give an explicit form of the differential of dH with respect to the Lax operators (10) and (11) by assuming the following formula:

$${(dH,L_t)}_{\mathbf{g}}={\int }_{\mathbb{T}}\sum _{i=k}^{N+k-2}\frac{\delta H}{\delta u_i}{\left(u_i\right)}_t\Delta x.$$

(57)

Therefore, the differentials have the form $dH={\displaystyle \sum _{i=1-k}^{\infty }}{\mathcal{E}}^{i-N}\frac{\delta H}{\delta u_{N-i}}$.

The quadratic Poisson tensor ${\pi }_1$ is a more subtle instance, and its construction is eliminated from the R-matrix format. In the previous section, recursive computators of the Lax family of equations are constructed such that $\Phi \left(L_{t_n}\right)=L_{t_{n+N+\alpha }}$. Since ${\pi }_0$ is known, ${\pi }_1$ is obtained as

$${\pi }_1=\Phi {\pi }_0.$$

(58)

The recursive operator is heritable in the vector space spanned by the symmetry of the related Lax family of equations, so the Poisson tensors ${\pi }_0$, ${\pi }_1$ are compatible and equation family (15) has a bi-Hamiltonian structure [14,33,34]:

$$L_{t_n}={\pi }_{0}d{H}_n={\pi }_{1}d{H}_{n-N+\alpha }.$$

(59)

Here, the respective Hamiltonians can be defined by

$$H_n\left(L\right)=\frac{N+\alpha }{N+\alpha +n}\mathrm{Tr}\left({\mathrm{L}}^{\frac{\mathrm{n}}{\mathrm{N}+\alpha }+1}\right).$$

(60)

In this paper, we only study the bi-Hamiltonian structures of a two-dimensional field and four-dimensional field.

Case 1. Bi-Hamiltonian structures of two-field systems

(1) In the case of $k=0$, the Lax operator is given by $L=p{\mathcal{E}}^{-1}+u+\mathcal{E}$. Then, we have

$$L_{t_1}={\pi }_{0}d{H}_1={\pi }_{1}d{H}_0.$$

(61)

Calculating the Poisson tensor of the above formula according to (56), we have

$${\pi }_0=\left(\begin{array}{cc}0& p(1-E^{-1})\\ (E-1)p& 0\end{array}\right).$$

According to the recursion relation (38) formula in the previous section, the following recursion derivator is obtained

$$\Phi =\left(\begin{array}{cc}\frac{E^{-1}up{E}^{-1}-u{E}^{-1}p}{p{E}^{-1}-E^{-1}p}& p(1+E^{-1})\\ \frac{p{E}^{-1}-p}{p{E}^{-1}-E^{-1}p}& u\end{array}\right).$$

So, the quadratic Poisson tensor ${\pi }_1$ is given by

$${\pi }_1=\Phi {\pi }_0=\left(\begin{array}{cc}p(1+E^{-1})(E-1)p& \frac{(E^{-1}up{E}^{-1}-u{E}^{-1}p)p(1-E^{-1})}{p{E}^{-1}-E^{-1}p}\\ u(E-1)p& \frac{(p{E}^{-1}-p)p(1-E^{-1})}{p{E}^{-1}-E^{-1}p},\end{array}\right),$$

together with the Hamiltonians:

$$\begin{array}{cc}\hfill H_0& ={\int }_{\mathbb{T}}\frac{1}{\mu }u\Delta x,\hfill \\ \hfill H_1& =\frac{1}{2}{\int }_{\mathbb{T}}\frac{1}{\mu }(u^2+p+Ep)\Delta x.\hfill \end{array}$$

(62)

(2) In the case of $k=1$, the Lax operator is given by (39). Calculating the Poisson tensor of the above formula according to (56),

$$\widetilde{{\pi }_0}=\left(\begin{array}{cc}0& (1-E^{-1})\tilde{p}\\ \tilde{p}(E-1)& 0\end{array}\right).$$

According to the recursion relation Formula (43), the following recursion derivator is obtained:

$$\tilde{\Phi }=\begin{array}{c}\hfill \left(\begin{array}{cc}\tilde{u}& \frac{\tilde{p}E-\tilde{p}}{\tilde{p}E-E\tilde{p}}\\ \tilde{p}(1+E)& \frac{E\tilde{u}\tilde{p}E-\tilde{u}E\tilde{p}}{\tilde{p}E-E\tilde{p}}\end{array}\right).\end{array}$$

(63)

So, the quadratic Poisson tensor is

$$\widetilde{{\pi }_1}=\tilde{\Phi }{\pi }_0=\left(\begin{array}{cc}\frac{\tilde{p}(E-1)(\tilde{p}E-\tilde{p})}{\tilde{p}E-E\tilde{p}}& \tilde{u}(1-E^{-1})\tilde{p}\\ \frac{\tilde{p}(E-1)(E\tilde{u}\tilde{p}E-\tilde{u}E\tilde{p})}{\tilde{p}E-E\tilde{p}}& \tilde{p}(1+E)(1-E^{-1})\tilde{p}\end{array}\right),$$

together with the Hamiltonians:

$$\begin{array}{cc}\hfill H_0& ={\int }_{\mathbb{T}}\frac{1}{\mu }\tilde{u}\Delta x,\hfill \\ \hfill H_1& =\frac{1}{2}{\int }_{\mathbb{T}}\frac{1}{\mu }({\tilde{u}}^2+\tilde{p}+E{\tilde{p}}^{-1})\Delta x.\hfill \end{array}$$

(64)

Case 2. Bi-Hamiltonian structures of four-field systems

(1) In the case of $k=0$, $\alpha =-3$, where $L=r{\mathcal{E}}^{-3}+p{\mathcal{E}}^{-2}+u{\mathcal{E}}^{-1}+v+\mathcal{E}$, we then calculate the Poisson tensor of the above formula according to (56)

$${\pi }_0=\left(\begin{array}{cccc}0& 0& 0& r(E^{-3}-1)\\ 0& 0& r{E}^{-2}-Er& p(E^2-1)\\ 0& r{E}^{-1}-E^{2}r& p{E}^{-1}-Ep& u(E^{-1}-1)\\ (1-E^3)r& (1-E^2)p& (1-E)u& 0\end{array}\right).$$

According to the recursion relation (47), we can obtain the following recursion derivator

$$\Phi ={\left(a_{ij}\right)}_{4\times 4},i=1,\dots ,4;j=1,\dots ,4,$$

where

$$\begin{array}{cc}\hfill a_{11}& =\frac{(E^{-1}p-p{E}^{-2})(E^{-2}p-p{E}^{-2})(E^{-3}pr{E}^{-3}-E^{-3}rp{E}^{-2})}{(r{E}^{-3}-E^{-1}r)(r{E}^{-3}-E^{-3}r)(r{E}^{-3}-E^{-2}r)}\hfill \\ \hfill & +\frac{E^{-3}vr{E}^{-3}-v{E}^{-3}r}{r{E}^{-3}-E^{-3}r}+\frac{(E^{-1}p-p{E}^{-2})E^{-3}u}{r{E}^{-3}-E^{-1}r}\hfill \\ \hfill & +\frac{(E^{-2}-u{E}^{-1})(E^{-3}pr{E}^{-3}-E^{-3}rp{E}^{-2})}{(r{E}^{-3}-E^{-3}r)(r{E}^{-3}-E^{-2}r)},\hfill \\ \hfill a_{12}& =\frac{E^{-2}ur{E}^{-3}-u{E}^{-3}r}{r{E}^{-3}-E^{-2}r}+\frac{(E^{-1}p-p{E}^{-2})(E^{-2}p-p{E}^{-2})E^{-2}r}{(r{E}^{-3}-E^{-1}r)(r{E}^{-3}-E^{-2}r)},\hfill \\ \hfill a_{13}& =\frac{E^{-1}pr{E}^{-3}-p{E}^{-3}r}{r{E}^{-3}-E^{-1}r},a_{14}=\frac{rE-r{E}^{-3}}{E-1},\hfill \end{array}$$

(65)

$$\begin{array}{cc}\hfill a_{21}& =\frac{(E^{-1}u-u{E}^{-1})(E^{-2}p-p{E}^{-2})(E^{-3}pr{E}^{-3}-E^{-3}rp{E}^{-2})}{(r{E}^{-3}-E^{-1}r)(r{E}^{-3}-E^{-3}r)(r{E}^{-3}-E^{-2}r)}\hfill \\ \hfill & +\frac{r{E}^{-3}-E^{-2}r}{r{E}^{-3}-E^{-3}r}+\frac{(E^{-1}u-u{E}^{-1})E^{-3}u}{r{E}^{-3}-E^{-1}r}\hfill \\ \hfill & +\frac{(E^{-2}v-v)(E^{-3}pr{E}^{-3}-E^{-3}rp{E}^{-2})}{(r{E}^{-3}-E^{-3}r)(r{E}^{-3}-E^{-2}r)},\hfill \\ \hfill a_{22}& =\frac{E^{-2}vr{E}^{-3}-v{E}^{-2}r}{r{E}^{-3}-E^{-2}r}+\frac{(E^{-1}u-u{E}^{-1})(E^{-2}p-p{E}^{-2})E^{-2}r}{(r{E}^{-3}-E^{-1}r)(r{E}^{-3}-E^{-2}r)},\hfill \\ \hfill a_{23}& =\frac{E^{-1}ur{E}^{-3}-u{E}^{-2}r}{r{E}^{-3}-E^{-1}},a_{24}=\frac{pE-p{E}^{-2}}{E-1},\hfill \\ \hfill a_{31}& =\frac{(E^{-1}v-v{E}^{-1})E^{-3}u}{r{E}^{-3}-E^{-1}r}+\frac{(1-E)(E^{-3}pr{E}^{-3}-E^{-3}rp{E}^{-2})}{(r{E}^{-3}-E^{-3}r)(r{E}^{-3}-E^{-2}r)}\hfill \\ \hfill & +\frac{(E^{-1}v-v{E}^{-1})(E^{-2}p-p{E}^{-2})(E^{-3}pr{E}^{-3}-E^{-3}rp{E}^{-2})}{(r{E}^{-3}-E^{-1}r)(r{E}^{-3}-E^{-3}r)(r{E}^{-3}-E^{-2}r)},\hfill \\ \hfill a_{32}& =\frac{r{E}^{-3}-E^{-1}r}{r{E}^{-3}-E^{-2}r}+\frac{(E^{-1}v-v{E}^{-1})(E^{-2}p-p{E}^{-2})E^{-2}r}{r{E}^{-3}-E^{-2}r},\hfill \\ \hfill a_{33}& =\frac{E^{-1}vr{E}^{-3}-v{E}^{-2}r}{r{E}^{-3}-E^{-1}},a_{34}=\frac{uE-u{E}^{-1}}{E-1},\hfill \\ \hfill a_{41}& =\frac{(1-E)E^{-3}u}{r{E}^{-3}-E^{-1}r}+\frac{(1-E)(E^{-2}p-p{E}^{-2})(E^{-3}pr{E}^{-3}-E^{-3}rp{E}^{-2})}{(r{E}^{-3}-E^{-1}r)(r{E}^{-3}-E^{-3}r)(r{E}^{-3}-E^{-2}r)}\hfill \\ \hfill a_{42}& =\frac{(1-E)(E^{-2}p-p{E}^{-2})E^{-2}r}{(r{E}^{-3}-E^{-1}r)(r{E}^{-3}-E^{-2}r)},a_{43}=\frac{r{E}^{-3}-r}{r{E}^{-3}-E^{-1}r},a_{44}=v.\hfill \end{array}$$

(66)

So, the quadratic Poisson tensor is ${\pi }_1=\Phi {\pi }_0={\left(b_{ij}\right)}_{4\times 4},i=1,\dots ,4;j=1,\dots ,4$, where

$$\begin{array}{cc}\hfill b_{i1}& =(1-E^3)r{a}_{i4},\hfill \\ \hfill b_{i2}& =(r{E}^{-1}-E^{2}r)a_{i4},\hfill \\ \hfill b_{i3}& =(r{E}^{-2}-Er)a_{i2}+(p{E}^{-1}-Ep)a_{i3}+(1-E)u{a}_{i4},\hfill \\ \hfill b_{i4}& =r(E^{-3}-1)a_{i1}+p(E^2-1)a_{i2}+u(E^{-1}-1)a_{i3},\hfill \end{array}$$

(67)

together with the Hamiltonians

$$\begin{array}{cc}\hfill H_0& ={\int }_{\mathbb{T}}\frac{1}{\mu }v\Delta x,\hfill \\ \hfill H_1& =\frac{1}{2}{\int }_{\mathbb{T}}\frac{1}{\mu }(v^2+u+Eu)\Delta x.\hfill \end{array}$$

(68)

(2) In the case of $k=1$, $\alpha =-3$. Here, $L={\mathcal{E}}^{-3}+\tilde{p}{\mathcal{E}}^{-2}+\tilde{u}{\mathcal{E}}^{-1}+\tilde{v}+\tilde{w}\mathcal{E}$, and calculating the Poisson tensor of the above formula according to (47)

$${\pi }_0=\left(\begin{array}{cccc}0& E-E^{-2}& 0& 0\\ E^2-E& E\tilde{p}-\tilde{p}{E}^{-1}& 0& 0\\ 0& 0& 0& (1-E^{-1})\tilde{w}\\ 0& 0& \tilde{w}(E-1)& 0\end{array}\right),$$

the recursion derivator can be obtained by recursion relation (52),

$$\tilde{\Phi }={\left({\tilde{a}}_{ij}\right)}_{4\times 4},i=1,\dots ,4;j=1,\dots ,4,$$

where

$$\begin{array}{cc}\hfill {\tilde{a}}_{11}& =\frac{E^{-2}\tilde{v}{E}^{-3}-\tilde{v}}{E^{-3}-1}+\frac{(\tilde{p}-\tilde{p}{E}^{-2})(E^{-2}\tilde{u}{E}^{-3}-\tilde{u}{E}^{-1})}{{(E^{-3}-1)}^2}\hfill \\ \hfill & +\frac{(\tilde{p}-\tilde{p}{E}^{-2})(E^{-1}\tilde{p}-\tilde{p}{E}^{-2})(E^{-2}\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2})}{{(E^{-3}-1)}^3}\hfill \\ \hfill & +\frac{(E^{-1}\tilde{u}-\tilde{u}{E}^{-1})(E^{-2}\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2})}{{(E^{-3}-1)}^2},\hfill \\ \hfill {\tilde{a}}_{12}& =\frac{E^{-1}\tilde{u}{E}^{-3}-\tilde{u}{E}^{-1}}{E^{-3}-1}+\frac{(\tilde{p}-\tilde{p}{E}^{-2})(E^{-1}\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2})}{{(E^{-3}-1)}^2},\hfill \\ \hfill {\tilde{a}}_{13}& =\frac{\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2}}{E^{-3}-1},{\tilde{a}}_{14}=\frac{\tilde{w}E-E^{-2}\tilde{w}}{\tilde{w}E-E^{-2}\tilde{w}},\hfill \\ \hfill {\tilde{a}}_{21}& =\frac{E^{-2}\tilde{w}{E}^{-3}-\tilde{w}E}{E^{-3}-1}+\frac{(\tilde{u}-\tilde{u}{E}^{-1})(E^{-2}\tilde{u}{E}^{-3}-\tilde{u}{E}^{-1})}{{(E^{-3}-1)}^2}\hfill \\ \hfill & +\frac{(\tilde{u}-\tilde{u}{E}^{-1})(E^{-1}\tilde{p}-\tilde{p}{E}^{-2})(E^{-2}\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2})}{{(E^{-3}-1)}^3}\hfill \\ & +\frac{(E^{-2}\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2})(E^{-1}\tilde{v}-\tilde{v}E)}{{(E^{-3}-1)}^2},\hfill \end{array}$$

(69)

$$\begin{array}{cc}\hfill & {\tilde{a}}_{22}=\frac{E^{-1}\tilde{v}{E}^{-3}-\tilde{v}}{E^{-3}-1}+\frac{(\tilde{u}-\tilde{u}{E}^{-1})(E^{-1}\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2})}{{(E^{-3}-1)}^2},\hfill \\ \hfill & {\tilde{a}}_{23}=\frac{\tilde{u}{E}^{-3}-\tilde{u}{E}^{-1}}{E^{-3}-1},{\tilde{a}}_{24}=\frac{E\tilde{p}\tilde{w}E-\tilde{p}{E}^{-1}\tilde{w}}{\tilde{w}E-E\tilde{w}},\hfill \\ \hfill & {\tilde{a}}_{31}=\frac{(E^{-1}\tilde{w}-\tilde{w}E)(E^{-2}\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2})}{{(E^{-3}-1)}^2},{\tilde{a}}_{32}=\frac{E^{-1}\tilde{w}{E}^{-3}-\tilde{w}E}{E^{-3}-1},\hfill \\ & {\tilde{a}}_{33}=\tilde{v},{\tilde{a}}_{34}=\frac{E\tilde{u}\tilde{w}}{\tilde{w}E-E\tilde{w}},\hfill \\ \hfill & {\tilde{a}}_{41}=\frac{(\tilde{w}-\tilde{w}E)(E^{-1}\tilde{p}-\tilde{p}{E}^{-2})(E^{-2}\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2})}{{(E^{-3}-1)}^3}\hfill \\ \hfill & +\frac{(\tilde{w}-\tilde{w}E)(E^{-2}\tilde{u}{E}^{-3}-\tilde{u}{E}^{-1})}{{(E^{-3}-1)}^2},\hfill \\ \hfill & {\tilde{a}}_{42}=\frac{(\tilde{w}-\tilde{w}E)(E^{-1}\tilde{p}{E}^{-3}-\tilde{p}{E}^{-2})}{{(E^{-3}-1)}^{-2}},{\tilde{a}}_{43}=\frac{\tilde{w}{E}^{-3}-\tilde{w}E}{E^{-3}-1},{\tilde{a}}_{44}=\frac{E\tilde{v}\tilde{w}E-E\tilde{w}\tilde{v}}{\tilde{w}W-E\tilde{w}}.\hfill \end{array}$$

(70)

Similarly, $\widetilde{{\pi }_1}$ can be obtained from $\widetilde{{\pi }_1}=\tilde{\Phi }\widetilde{{\pi }_0}={\left({\tilde{b}}_{ij}\right)}_{4\times 4},i=1,\dots ,4;j=1,\dots ,4$, where

$$\begin{array}{cc}\hfill {\tilde{b}}_{i1}& ={\tilde{a}}_{i2}(E^2-E),\hfill \\ \hfill {\tilde{b}}_{i2}& ={\tilde{a}}_{i1}(E-E^{-2})+{\tilde{a}}_{i2}(E\tilde{p}-\tilde{p}{E}^{-1}),\hfill \\ \hfill {\tilde{b}}_{i3}& ={\tilde{a}}_{i4}\tilde{w}(E-1),\hfill \\ \hfill {\tilde{b}}_{i4}& ={\tilde{a}}_{i3}(1-E^{-1})\tilde{w}.\hfill \end{array}$$

(71)

The Hamiltonians are shown below:

$$\begin{array}{cc}\hfill H_0& ={\int }_{\mathbb{T}}\frac{1}{\mu }{\tilde{v}}^2\Delta x,\hfill \\ \hfill H_1& =\frac{1}{2}{\int }_{\mathbb{T}}\frac{1}{\mu }({\tilde{v}}^2+\tilde{u}{E}^{-1}\tilde{w}+\tilde{w}E\tilde{u})\Delta x.\hfill \end{array}$$

(72)

When $\mathbb{T}$= $\mathbb{Z}$, the above family of bi-Hamilton equations is bi-Hamilton for lattice integrable systems. Moreover, the Hamiltonian formalism provides a link between classical and quantum mechanics; this is important for understanding integrable systems from a physical perspective. This paper’s content can be regarded as the discretization of field soliton systems. In some particular cases, the quasi-classical limits of discrete soliton systems producing dispersive soliton equations can be considered by introducing appropriate deformation parameters on regular discrete time scales, which is also the future research direction.

5. Conclusions

In this paper, N-dimensional lattice integrable systems on a regular time scale are given by the R-matrix approach. Furthermore, we discuss the infinite and finite fields. For the infinite field, a (2 + 1)-dimensional one-field system is obtained. For the finite field, the equations of the two-field systems and four-field systems are given for $k=0$ and $k=1$. When taking the special time scale $\mathbb{Z}$, the Toda and modified Toda chains are obtained. This further confirms that a time scale can be uniformly continuous and discrete. The recursion operators of the lattice system are constructed to study the recursion relation of the equation family. We derive the recursion operator for each example examined in this paper and propose a method for constructing constants of motion through the introduction of a proper trace form on the time scale. We study bi-Hamiltonian structures of lattice hierarchies on the regular discrete obtained through the R-matrix in the last. The lattice system on the time scale studied in this paper is constructed from Lax operators generated by displacement operators so that different operator forms can be selected to construct continuous or discrete systems. The research method can better unify the soliton system and the lattice system and provides potential values for further research on continuous, discrete and continuous–discrete mixed systems.

In this work, we do not study the bi-Hamiltonian structure of the N-dimensional lattice integrable systems but only consider individual cases. Therefore, we will deeply study the bi-Hamiltonian structure of the N-dimensional field and infinite field in future work.