Two kinds of discrete integrable hierarchies of evolution equations and some algebraic-geometric solutions

Abstract

With the help of a loop algebra we first present a $(1+1)$-dimensional discrete integrable hierarchy with a Hamiltonian structure and generate a $(2+1)$-dimensional discrete integrable hierarchy, respectively. Then we obtain a new differential-difference integrable system with three-potential functions, whose algebraic-geometric solution is derived from the theory of algebraic curves, where we construct the new elliptic coordinates to straighten out the continuous and discrete flows by introducing the Abel maps as well as the Riemann-Jacobi inversion theorem.

1 Introduction

Integrable nonlinear lattice systems have important applications and rich mathematical structures in mathematical physics, statistical physics, and quantum physics. For example, the Toda lattice equation governs a system of unit masses connected by nonlinear springs whose restoring force is exponential. The equation has been proved to have integrability properties, such as a Lax pair, the Hamiltonian structure, infinite many conservation laws, and so on [1, 2]. The Toda lattice was also solved by using the Casoratian technique systematically on rational or soliton or complex solutions [3, 4]. Therefore, it is interesting how to generate integrable nonlinear lattice systems associated with mathematics and physics by various methods. Suris [5] once derived a new lattice equation related to the relativistic Toda lattice hierarchy via a highly non-trivial Bäcklund transformation. Tu Guizhang [6] applied a compatibility condition of spectral problems and some Lie algebras to propose a powerful method for generating integrable differential-difference hierarchies and the corresponding Hamiltonian structures. Based on the scheme, some related integrable nonlinear lattice hierarchies were obtained; e.g. see [7–11]. In the case where lattice equations including the positive and negative lattices by using the semi-direct sums of Lie algebras have been present [12, 13], their mathematical structures such as Hamiltonian structures usually investigated by the variational identity [14]. Ablowitz et al. [15] considered some exact linearization of difference equations; Nijhoff and Papageorgiou [16] studied similarity reductions; Levi et al. [17] investigated some symmetries of differential and difference equations; Ablowitz and Ladik [18] obtained some differential-difference equations and applied Fourier analysis to review their some integrable properties; Cao Cewen et al. [19] applied the nonlinearization method to importantly pave the way for generating differential-difference equations and algebraic-geometric solutions of $(1+1)$-dimensional and $(2+1)$-dimensional difference equations. Next Geng and Dai [20] proposed some new $(2+1)$-dimensional discrete models and obtained some algebraic-geometric solutions by applying the nonlinearization method. Based on this, Geng et al. [21–27] further improved the method so as to conveniently investigate algebraic-geometric solutions of differential and difference equations by introducing a new matrix consisting of fundamental solutions of spectral problems which satisfy discrete zero-curvature equations. With the help of the nonlinearization method, some interesting work on algebraic-geometric solutions was performed; e.g. see [28, 29].

As for as non-isospectral integrable lattice hierarchies are concerned, as is well known, less work has been done. Gordoa, Pickering and Zhu [30] made great progress in the aspect of constructing new non-isospectral lattice hierarchies in $2+1$ dimensions. Based on this, Pickering, Zhu [31] constructed two $(2+1)$-dimensional discrete linear spectral problems and generalized some known lattice equations. In the paper, we make use of a loop algebra of the Lie algebra $A_{1}$ to deduce a $(1+1)$-dimensional discrete integrable hierarchy and a $(2+1)$-dimensional discrete hierarchy, respectively. Furthermore, we investigate their Hamiltonian structures by the trace identity. The $(1+1)$-dimensional discrete integrable hierarchy obtained in the paper can be reduced to a new $(1+1)$-dimensional integrable nonlinear difference system with three-potential functions, and the $(2+1)$-dimensional discrete integrable hierarchy presented in the paper is obtained by a non-isospectral Lax pair based on the loop algebra and a zero-curvature equation. Finally, we generate the algebraic-geometric solution of the reduced discrete integrable system by introducing Abel coordinates and the Riemann-Jacobi inversion theorem. The latter was once used to investigate binary constrained flows and separation of variables in [32].

2 Two integrable differential-difference hierarchies of evolution equations

We presented a loop algebra of the Lie algebra $A_{1}$ as follows in [32]:

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

where

$$\begin{aligned} &h_{1}(n)=h_{1}\lambda^{2n},\qquad h_{2}(n)=h_{2}\lambda^{2n},\qquad e(n)=e\lambda ^{2n-1},\qquad f(n)=f\lambda^{2n-1}, \\ & h_{1}= \begin{pmatrix}1&0\\0&0 \end{pmatrix} ,\qquad h_{2}= \begin{pmatrix}0&0\\0&1 \end{pmatrix},\qquad e= \begin{pmatrix}0&1\\0&0 \end{pmatrix},\qquad f= \begin{pmatrix}0&0\\1&0 \end{pmatrix}, \end{aligned}$$

which has the commutative operations

$$\begin{aligned} \begin{aligned} &[h_{1},h_{2}]=0,\qquad [h_{1},e]=e,\qquad [h_{1},f]=-f,\qquad [h_{2},e]=-e,\\ &[h_{2},f]=f,\qquad[e,f]=h\equiv h_{1}-h_{2}, \qquad [h,e]=2e,\qquad [h,f]=-2f. \end{aligned} \end{aligned}$$

(1)

In general, we usually apply multiplication operations among elements of the Lie algebra $g=\operatorname{span}\{h_{1},h_{2},e,f\}$. It is easy to see that

$$\begin{aligned} \begin{aligned} &h_{1}h_{1}=h_{1}, \qquad h_{2}h_{2}=h_{2},\qquad h_{1}h_{2}=h_{2}h_{1}=ee=ff=0, \qquad h_{1}e=e,\qquad eh_{1}=0, \\ &h_{1}f=0,\qquad fh_{1}=f,\qquad h_{2}f=f,\qquad fh_{2}=0,\qquad h_{2}e=0,\qquad eh_{2}=e,\\ & ef=h_{1},\qquad fe=h_{2}. \end{aligned} \end{aligned}$$

(2)

In [27], we changed the form of the discrete zero-curvature equation as follows:

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

(3)

where $\Delta=E-1, Ef(n)=f(n+1),U_{n}$ and V are Lax matrices which appear in the spectral problems

$$ \varphi_{n+1}=U_{n}\varphi_{n},\qquad \frac{d\varphi_{n}}{dt}=V \varphi_{n},\qquad \varphi _{n}= \varphi(n,t). $$

(4)

Equation (3) is similar to the stationary zero-curvature equation in continuous spectral problems,

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

The reason why we adopt equation (3) to investigate discrete integrable hierarchies aims at applying the Tu scheme [33] to generate lattice integrable hierarchies, which has been a current way for generating integrable hierarchies of evolution equations. Based on the above version, we had obtained the well-known Toda lattice hierarchy and a differential-difference hierarchy; and further their expanding integrable models were produced, respectively. In the following, we choose $U_{n}$ and V to be of the form [32]

$$\begin{aligned} &U_{n}=h_{1}(1)+q_{n}h_{2}(0)+r_{n}e(1)+s_{n}f(1), \\ &V=\sum_{n\geq0}\bigl[a_{n} \bigl(h_{1}(-n)-h_{2}(-n)\bigr)+b_{n}e(-n)+c_{n}f(-n) \bigr], \end{aligned}$$

and apply equation (3) and the discrete zero-curvature equation,

$$ \frac{dU_{n}}{dt_{m}}=(\Delta V_{(m)})U_{n}-[U_{n},V_{(m)}], $$

(5)

to obtain the following integrable discrete hierarchy:

$$ \textstyle\begin{cases}q_{n,t_{m}}=-r_{n}c_{m}^{(1)}+s_{n}b_{m},\\ r_{n,t_{m}}=b_{m},\\ s_{n,t_{m}}=-c_{m}^{(1)}, \end{cases} $$

(6)

where

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

Assume $a_{0}=\frac{1}{2},b_{0}=r_{n}, c_{0}=s_{n-1}$, then when $m=0$, equation (6) can be reduced to

$$ q_{n,t_{0}}=s_{n}r_{n}-r_{n}s_{n-1},\qquad r_{n,t_{0}}=r_{n},\qquad s_{n,t_{0}}=-s_{n}. $$

(7)

When $m=1$, equation (6) gives rise to ($t_{1}=t$):

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

which can be written as

$$ \textstyle\begin{cases}\partial_{t}\ln q_{n}=s_{n}r_{n+1}-r_{n}s_{n-1},\\ \partial_{t}\ln r_{n}=-r_{n+1}s_{n}-r_{n}s_{n-1}+q_{n}\frac{r_{n+1}}{r_{n}},\\ \partial_{t}\ln s_{n}=s_{n-1}r_{n}+s_{n}r_{n+1}-q_{n}\frac{s_{n-1}}{s_{n}}. \end{cases} $$

(8)

In the following, we still make use of the loop algebra g̃ to generate $(2+1)$-dimensional non-isospectral differential-difference hierarchy by adopting the method presented in [34–36].

Consider the non-isospectral Lax problem

$$ \textstyle\begin{cases}\psi_{n+1}(\lambda)=U_{n}(q_{n},r_{n},s_{n},\lambda)\psi_{n}(\lambda ),\\ \frac{d\psi_{n}(\lambda)}{dt}=\omega(\lambda)\frac{d\psi_{n}(\lambda )}{dy}+V^{(m)}_{n}(q_{n},r_{n},s_{n},\lambda)\psi_{n}(\lambda), \end{cases} $$

(9)

where

$$\lambda=\lambda(t,y),\qquad \frac{d \lambda}{dt}=\lambda_{t}=\omega(\lambda ) \lambda_{y}+\beta(\lambda). $$

The compatibility condition of (9) yields

$$ \frac{\partial U_{n}}{\partial t}-\omega(\lambda)\frac{\partial U_{n}}{\partial y}+\beta( \lambda) \frac{\partial U_{n}}{\partial\lambda} + \bigl(\Delta V_{n}^{(m)} \bigr)U_{n}- \bigl[U_{n},V^{(m)}_{n} \bigr]=0. $$

(10)

Now we take

$$V^{(m)}_{n}= \begin{pmatrix}A&B\\C&D \end{pmatrix}, $$

where

$$ \textstyle\begin{cases}A=\sum_{j=0}^{m}a_{j}(n,t,y)\lambda^{2(m-j)},\qquad B=\sum_{j=0}^{m}b_{j}(n,t,y)\lambda^{2(m-j)+1},\\ C=\sum_{j=0}^{m}c_{i}(n,t,y)\lambda^{2(m-j)+1},\qquad D=\sum_{j=0}^{m}d_{j}(n,t,y)\lambda^{2(m-j)}. \end{cases} $$

(11)

Then equation (3) admits

$$\textstyle\begin{cases}\lambda^{2} EA+\lambda s_{n}EB-\lambda^{2} A-\lambda r_{n}C=2\lambda\beta(\lambda),\\ \lambda\dot{r_{n}}-\lambda\omega(\lambda)r_{n,y}+\beta(\lambda )r_{n}=\lambda r_{n}EA+q_{n}EB-\lambda^{2}B-\lambda r_{n}D,\\ \lambda\dot{s_{n}}-\lambda\omega(\lambda)s_{n,y}+\beta(\lambda )s_{n}=\lambda^{2}EC+\lambda s_{n}ED-\lambda s_{n}A-q_{n}C,\\ \dot{q_{n}}-\omega(\lambda)q_{n,y}=\lambda r_{n}EC+q_{n}ED-\lambda s_{n}B-q_{n}D. \end{cases} $$

Set

$$ \beta(\lambda)=\sum_{j=0}^{m} \beta_{j}\lambda^{2(m-j)+1},\qquad \omega (\lambda)= \lambda^{2m}. $$

(12)

Substituting (11) and (12) into equation (5) yields

$$ \textstyle\begin{cases}-r_{n,y}+\beta_{0}r_{n}=r_{n}Ea_{0}-b_{1}+q_{n}Eb_{0}-r_{n}d_{0},\\ r_{n}\beta_{j}=r_{n}Ea_{j}+q_{n}Eb_{j}-r_{n}d_{j}-b_{j+1},\\ -s_{n,y}+\beta_{0}s_{n}=Ec_{1}+s_{n}Ed_{0}-s_{n}a_{0}-q_{n}c_{0},\\ s_{n}\beta_{j}=Ec_{j+1}+s_{n}Ed_{j}-s_{n}a_{j}-q_{n}c_{j},\\ -q_{n,y}=r_{n}Ec_{1}+q_{n}Ed_{0}-s_{n}b_{1}-q_{n}d_{0},\\ r_{n}Ec_{j+1}+q_{n}Ed_{j}-s_{n}b_{j+1}-q_{n}d_{j}=0,\quad j=1,\ldots,m, \end{cases} $$

(13)

and

$$ \textstyle\begin{cases}r_{n,t_{m}}=-r_{n}\beta_{m}+r_{n}Ea_{m}+q_{n}Eb_{m}-r_{n}d_{m},\\ s_{n,t_{m}}=s_{n}Ed_{m}-s_{n}a_{m}-q_{n}c_{m},\\ \Delta q_{n,t_{m}}=q_{n}\Delta d_{m},\quad j=1,2,\ldots, m. \end{cases} $$

(14)

From equation (13), we find that

$$ (q_{n}-s_{n}r_{n})\Delta d_{j}=r_{n}s_{n} \Delta a_{j}+q_{n}s_{n}Eb_{j}-q_{n}r_{n}c_{j}-2s_{n}r_{n} \beta_{j},\quad j=1,\ldots, m. $$

(15)

For equation (15) to be solvable locally, we let $a_{j}=-d_{j}$, then equations (13)-(15) can be simplified, respectively,

$$ \textstyle\begin{cases}-r_{n,y}+\beta_{0}r_{n}=r_{n}Ea_{0}-b_{1}+q_{n}Eb_{0}+r_{n}a_{0},\\ r_{n}\beta_{j}=r_{n}Ea_{j}+q_{n}Eb_{j}+r_{n}a_{j}-b_{j+1},\\ -s_{n,y}+\beta_{0}s_{n}=Ec_{1}-s_{n}Ea_{0}-s_{n}a_{0}-q_{n}c_{0},\\ s_{n}\beta_{j}=Ec_{j+1}-s_{n}Ea_{j}-s_{n}a_{j}-q_{n}c_{j},\\ -q_{n,y}=r_{n}Ec_{1}-q_{n}Ea_{0}-s_{n}b_{1}-q_{n}d_{0},\\ r_{n}Ec_{j+1}-q_{n}Ea_{j+1}-s_{n}b_{j+1}+q_{n}a_{j}=0,\quad j=1,\ldots,m, \end{cases} $$

(16)

and

$$ \textstyle\begin{cases}r_{n,t_{m}}=-r_{n}\beta_{m}+r_{n}Ea_{m}+r_{n}a_{m}+q_{n}Eb_{m},\\ s_{n,t_{m}}=-s_{n}Ea_{m}-s_{n}a_{m}-q_{n}c_{m},\\ \Delta q_{n,t_{m}}=-q_{n}\Delta a_{m},\quad j=1,2,\ldots, m, \end{cases} $$

(17)

$$ q_{n}\Delta a_{j}=-q_{n}s_{n}Eb_{j}+q_{n}r_{n}c_{j}+2s_{n}r_{n} \beta_{j},\quad j=1,2,\ldots,m. $$

(18)

Assume $b_{0}=\frac{1}{2}s_{n-1}^{-1}, c_{0}=-\frac{1}{2}r_{n}^{-1}$, then one infers from (18) that

$$a_{0}=-n+2\beta_{0}\Delta^{-1} \frac{s_{n}r_{n}}{q_{n}}. $$

In terms of (16), we have

$$\begin{aligned} \begin{aligned} &b_{1}=-r_{n}+r_{n,y}- \beta_{0}r_{n}+2 \beta_{0}\frac{s_{n}r_{n}^{2}}{q_{n}}- \frac {1}{2}\frac{q_{n}}{s_{n}}, \\ &c_{1}=(\beta_{0}-1)s_{n-1}-s_{n-1,y}+2 \beta_{0}\frac {r_{n-1}s_{n-1}^{2}}{q_{n-1}}+\frac{q_{n-1}}{2r_{n-1}}, \\ &\Delta a_{1}=-\Delta r_{n}s_{n-1}+\beta _{0}(E+1)r_{n}s_{n-1}-s_{n}r_{n+1,y}-r_{n}s_{n-1,y}+ \frac{1}{2}q_{n+1}s_{n}s_{n+1}^{-1} + \frac{1}{2}q_{n-1}r_{n}r_{n-1}^{-1} \\ &\phantom{\Delta a_{1}=}{}+2\beta_{0}r_{n}r_{n-1}s_{n-1}^{2}q_{n-1}^{-1}-2 \beta _{0}s_{n}s_{n+1}q_{n+1}^{-1}r_{n+1}^{2}. \end{aligned} \end{aligned}$$

(19)

Substituting the above results into (17) yields a reduction of the $(2+1)$-dimensional non-isospectral discrete hierarchy (17),

$$\textstyle\begin{cases}r_{n,t_{1}}=-\beta_{1}r_{n}-q_{n}r_{n+1}+q_{n}r_{n+1,y}-\beta _{0}q_{n}r_{n+1}+2\beta_{0}\frac{q_{n}s_{n+1}r_{n+1}^{2}}{q_{n+1}} -\frac{1}{2}\frac{q_{n}q_{n+1}}{s_{n+1}}+r_{n}(E+1)a_{1},\\ s_{n,t_{1}}=(1-\beta_{0})q_{n}s_{n-1}+q_{n}s_{n-1,y}-2\beta_{0}\frac {q_{n}r_{n-1}s_{n-1}^{2}}{q_{n-1}}-\frac {q_{n-1}a_{n}}{2r_{n-1}}-s_{n}(E+1)a_{1},\\ \Delta q_{n,t_{1}}=-q_{n}\Delta a_{1}, \end{cases} $$

where $a_{1}$ is given by (19).

Remark 1

Via applying the trace identity proposed by Tu [6], we could deduce the Hamiltonian structure of the $(1+1)$-dimensional discrete integrable hierarchy (6). However, how do we search for the Hamiltonian structure of the $(2+1)$-dimensional non-isospectral discrete integrable hierarchy (14)? This is a problem worth of discussing in the future.

3 Algebraic-geometric solution of the $(1+1)$-dimensional nonlinear discrete integrable system (8)

The nonlinear discrete system (8) possesses the following Lax pair:

$$ \textstyle\begin{cases}E\varphi(n)=U_{n}\varphi (n),\qquad U_{n}=h_{1}(1)+q_{n}h_{2}(0)+r_{n}e(1)+s_{n}f(1),\\ \varphi_{t}(n)=V_{(1)}\varphi(n), \end{cases} $$

(20)

where

$$\begin{aligned} &V_{(1)}= \begin{pmatrix}\frac{1}{2}\lambda^{2}-r_{n}s_{n-1}\lambda&V_{12}\\ V_{21}&-\frac{1}{2}\lambda^{2}+r_{n}s_{n-1}\lambda \end{pmatrix}, \\ &V_{12}=r_{n}\lambda^{2}+\biggl(\lambda- \frac{1}{\lambda }\biggr) \bigl(q_{n}r_{n+1}-r_{n}r_{n+1}s_{n}-r_{n}^{2}s_{n-1} \bigr), \\ &V_{21}=s_{n-1}\lambda^{2}+\biggl(\lambda- \frac{1}{\lambda }\biggr) \bigl(q_{n-1}s_{n-2}-s_{n-1}^{2}r_{n}-s_{n-1}s_{n-2}r_{n-1} \bigr). \end{aligned}$$

With the help of the approaches presented in [35, 36], we could generate Darboux-Bäcklund transformations and exact soliton solutions of equation (8). Of course, the key problem focuses on how to construct suitable Darboux matrices. The problem will be dealt in another paper.

In the following, we want to seek algebraic-geometric solutions based on theories in [19–23, 37]. We first introduce the Lenard gradient sequence $\bar{S}_{j},0\leq j\in\mathbf{Z}$ by the recursion equation

$$ K_{n}\bar{S}_{j}(n)=J_{n}\bar{S}_{j+1},\qquad J_{n} \bar{S}_{0}(n)=0,\quad j\geq0, $$

(21)

with the two operators

$$K_{n}= \begin{pmatrix}0&q_{n}E&0\\-q_{n}&0&0\\r_{n}E&-s_{n}&-q_{n}\Delta \end{pmatrix},\qquad J_{n}= \begin{pmatrix}0&-1&r_{n}E+r_{n}\\E&0&-s_{n}E-s_{n}\\r_{n}E&-s_{n}&-q_{n}\Delta \end{pmatrix}, $$

$\bar{S}_{j}(n)=(S^{(1)}_{j},S^{(2)}_{j},S^{(3)}_{j})^{T}$.

Equation $J_{n}\bar{S}_{0}(n)=0$ possesses a special solution as follows:

$$ \bar{S}_{0}(n)= \begin{pmatrix}s_{n-1}\\r_{n}\\\frac{1}{2} \end{pmatrix}, $$

(22)

and we find that

$$\operatorname{ker} J_{n}= \bigl\{ c \bar{S}_{0}(n) \bigr\} , $$

where c is an arbitrary constant. From equation (21), we easily have

$$ \bar{S}_{1}(n)= \begin{pmatrix}-s_{n-1}^{2}r_{n}-s_{n-1}s_{n-2}r_{n-1}+q_{n-1}s_{n-2}\\ -r_{n}r_{n+1}s_{n}-r_{n}^{2}s_{n-1}+q_{n}r_{n+1}\\-r_{n}s_{n-1} \end{pmatrix}, \ldots. $$

(23)

It is easy to see from (21) that

$$ \textstyle\begin{cases}r_{n}Es_{j+1}^{(3)}+q_{n}Es_{j}^{(2)}-s_{j+1}^{(2)}+r_{n}s_{j+1}^{(3)}=0,\\ Es_{j+1}^{(1)}-s_{n}Es_{j+1}^{(3)}-q_{n}s_{j}^{(1)}-s_{n}s_{j+1}^{(3)}=0,\\ r_{n}Es_{j}^{(1)}-q_{n}Es_{j}^{(3)}-s_{n}s_{j}^{(2)}+q_{n}s_{j}^{(3)}=0. \end{cases} $$

(24)

The $(1+1)$-dimensional integrable discrete hierarchy can be viewed as a generation of the following isospectral problems:

$$ \textstyle\begin{cases}\psi(n+1)=U_{n}\psi (n),\quad U_{n}=h_{1}(1)+q_{n}h_{2}(0)+r_{n}e(1)+s_{n}f(1),\\ \psi(n)_{t_{m}}=V^{(m)}_{n}\psi (n),\quad V^{(m)}_{n}=A^{(m)}_{n}h_{1}(1)+B^{(m)}_{n}e(1)+C_{n}^{(m)}f(1)-A_{n}^{(m)}h_{2}(1), \end{cases} $$

(25)

where

$$A_{n}^{(m)}=\sum_{j=0}^{m}s_{j}^{(3)}(n) \lambda ^{2(m-j)},\qquad B_{n}^{(m)}=\sum _{j=0}^{m}s_{j}^{(2)}(n) \lambda^{2(m-j)},\qquad C_{n}^{(m)}= \sum _{j=0}^{m}s_{j}^{(1)}(n) \lambda^{2(m-j)}. $$

The compatibility condition of (25) admits equation (6), which can be expressed as

$$ \begin{pmatrix}q_{n}\\r_{n}\\s_{n} \end{pmatrix}_{t_{m}}=X_{m}(n)= \begin{pmatrix} -r_{n}c_{m}^{(1)}+s_{n}b_{m}\\ b_{m}\\-c_{m}^{(1)} \end{pmatrix}. $$

3.1 Decomposition of the differential-difference equations

In the subsection, we shall decompose the $(1+1)$-dimensional lattice system (8) into solvable ordinary differential equations. Suppose (25) has two basic solutions $\psi(n)=(\psi^{(1)}(n),\psi^{(2)}(n))^{T}$ and $\varphi(n)=(\varphi^{(1)}(n),\varphi^{(2)}(n))^{T}$. We define a Lax matrix $W_{n}$ in terms of $\psi(n)$ and $\varphi(n)$, which has some generalizations in [38], by

$$ W_{n}= \begin{pmatrix}f(n)&g(n)\\h(n)&-f(n) \end{pmatrix}= \frac{1}{2} \bigl(\varphi(n)\psi(n)^{T}+\psi(n) \varphi(n)^{T} \bigr) \begin{pmatrix}0&-1\\1&0 \end{pmatrix}. $$

(26)

From equation (25) we can verify that

$$ W_{n+1}U_{n}-U_{n}W_{n}=0, \qquad W_{n,t_{m}}= \bigl[V^{(m)}_{n},W_{n} \bigr], $$

(27)

which means that the function $\operatorname{det} W_{n}$ is a constant independent of n and $t_{m}$. It is easy to see that equation (27) can be written as

$$ \textstyle\begin{cases}\lambda^{2} \Delta f(n)+\lambda s_{n}Eg(n)-\lambda r_{n}h(n)=0,\\ \lambda r_{n}Ef(n)+q_{n}Eg(n)-\lambda^{2} g(n)+\lambda r_{n}f(n)=0,\\ \lambda^{2} Eh(n)-\lambda s_{n} Ef(n)-q_{n}h(n)-\lambda s_{n}f(n)=0,\\ \lambda r_{n} Eh(n)-q_{n}Ef(n)-\lambda s_{n} g(n)+q_{n}f(n)=0, \end{cases} $$

(28)

and

$$ \textstyle\begin{cases}f(n)_{t_{m}}=B_{n}^{(m)}h(n)-C_{n}^{(m)}g(n),\\ g(n)_{t_{m}}=2g(n)A_{n}^{(m)}-2B_{n}^{(m)}f(n),\\ h(n)_{t_{m}}=2C_{n}^{(m)}f(n)-2A_{n}^{(m)}h(n), \end{cases} $$

(29)

where

$$\begin{aligned} \begin{aligned} &f(n)=\sum_{j=0}^{N}f_{j}(n) \lambda^{2(N-j)+2},\qquad g(n)=\sum_{j=0}^{N}g_{j}(n) \lambda^{2(N-j)+1},\\ & h(n)=\sum_{j=0}^{N}h_{j}(n) \lambda ^{2(N-j)+1}. \end{aligned} \end{aligned}$$

(30)

Substituting (30) into (28) and comparing the coefficients of the same powers of λ give rise to

$$ K_{n}G_{j}(n)=J_{n}G_{j+1}(n), \qquad J_{n}G_{0}(n)=0,\qquad K_{n}G_{N}(n)=0, $$

(31)

where $G_{j}(n)=(h_{j}(n),g_{j}(n),f_{j}(n))^{T}$. It is easy to see that equation $J_{n}G_{0}(n)=0$ has the general solution

$$ G_{0}(n)=\alpha_{0}\bar{S}_{0}(n), $$

(32)

here $\alpha_{0}$ is a constant. Acting with $(J_{n}^{-1}K_{n})^{k}$ on equation (32), we obtain

$$ G_{k}(n)=\alpha_{0}\bar{S}_{k}(n)+ \alpha_{1}\bar{S}_{k-1}(n)+\cdots+\alpha_{k}\bar{S}_{0}(n), $$

(33)

where $\alpha_{0},\alpha_{1},\ldots, \alpha_{k}$ are constants. Inserting (33) into equation $K_{n}G_{N}(n)=0$ gives a discrete stationary equation

$$ \alpha_{0}X_{N}(n)+\alpha_{1}X_{N-1}(n)+ \cdots+\alpha_{N}X_{0}(n)=0, $$

(34)

which implies $(q_{n},r_{n},s_{n})$ is the finite-band solution. Assume $\alpha_{0}=1$, we can obtain from (32) and (33) that

$$ \textstyle\begin{cases}f_{0}(n)=\frac{1}{2}, \qquad g_{0}(n)=r_{n},\qquad h_{0}(n)=s_{n-1},\\ f_{1}(n)=-r_{n}s_{n-1},\\ g_{1}(n)=-r_{n}r_{n+1}s_{n}-r_{n}^{2}s_{n-1}+q_{n}r_{n+1},\\ h_{1}(n)=-s_{n-1}^{2}r_{n}-s_{n-1}s_{n-2}r_{n-1}+q_{n-1}s_{n-2}. \end{cases} $$

(35)

We apply $g(n)$ and $h(n)$ as polynomials of λ to define the elliptic coordinates $\{\mu_{j}(n)\}$ and $\{\nu_{j}(n)\}$:

$$ \textstyle\begin{cases}g(n)=r_{n}\mathop{\pi}\limits _{j=1}^{N}(\lambda^{2}-\mu _{j}(n)^{2})\equiv r_{n}\mathop{\pi}\limits_{j=1}^{N}(\tilde{\lambda}-\tilde{\mu}_{j}(n)),\\ h(n)=s_{n-1}(\mathop{\pi}\limits_{j=1}^{N}(\lambda^{2}-\nu_{j}(n)^{2}))\equiv s_{n-1}\mathop{\pi}\limits _{j=1}^{N}(\tilde{\lambda}-\tilde{\nu}_{j}(n)), \end{cases} $$

(36)

where we denote $\lambda^{2},\mu_{j}(n)^{2},\nu_{j}(n)^{2}$ by $\tilde{\lambda},\tilde{\mu}_{j}(n)$ and $\tilde{\nu}_{j}(n)$, respectively. By comparing coefficients of the same power for λ, we have

$$ \textstyle\begin{cases}g_{1}(n)=-r_{n}\sum_{j=1}^{N}\tilde{\mu}_{j}(n),\qquad h_{1}(n)=-s_{n-1}\sum_{j=1}^{N}\tilde{\nu}_{j}(n),\\ g_{2}(n)=r_{n}\sum_{i< j}\tilde{\mu}_{i}(n)\tilde{\mu}_{j}(n),\qquad h_{2}(n)=s_{n-1}\sum_{i< j}\tilde{\nu}_{i}(n)\tilde{n}u_{j}(n). \end{cases} $$

(37)

Combined with (33), equation (37) can be written as

$$ \textstyle\begin{cases}r_{n+1}s_{n}+r_{n}s_{n-1}-\frac{r_{n+1}}{r_{n}}q_{n}=\sum_{j=1}^{N}\tilde{\mu}_{j}(n)+\alpha_{1},\\ s_{n-1}r_{n}+s_{n-2}r_{n-1}-\frac{s_{n-2}}{s_{n-1}}q_{n-1}=\sum_{j=1}^{N}\tilde{\nu}_{j}(n)+\alpha_{1}. \end{cases} $$

(38)

Thus, equation (8) can be written as

$$ \textstyle\begin{cases}\partial_{t}\ln q_{n}=s_{n}r_{n+1}-r_{n}s_{n-1},\\ \partial_{t}\ln r_{n}=-\sum_{j=1}^{N}\tilde{\mu}_{j}(n)-\alpha_{1},\\ \partial_{t} \ln s _{n}=E\sum_{j=1}^{N}\tilde{\nu}_{j}(n)+\alpha_{1}. \end{cases} $$

(39)

Consider the function $\operatorname{det} W_{n}$, which is a $(4N+4)$th-order polynomial in λ:

$$ -\operatorname{det}W_{n}=f^{2}(n)+g(n)h(n)= \frac{1}{4} \lambda^{2}\mathop{\pi}\limits _{j=1}^{2N+1} \bigl( \lambda^{2}-\lambda_{j}^{2} \bigr)= \frac{1}{4}\tilde{\lambda}\mathop{\pi}\limits _{j=1}^{2N+1}(\tilde{\lambda}-\tilde{\lambda}_{j})=\frac {1}{4}R(\tilde{\lambda}). $$

(40)

Substituting (30) into (40) yields

$$\alpha_{1}=-\frac{1}{2}\sum_{j=1}^{2N+1} \tilde{\lambda}_{j}. $$

One infers that

$$ f(n)|_{\tilde{\lambda}=\tilde{\mu}_{k}(n)}=\frac{1}{2}\sqrt{R \bigl(\tilde{\mu}_{k}(n) \bigr)}, \qquad f(n)|_{{\tilde{\lambda}=\tilde{\nu}_{j}(n)}}=\frac{1}{2}\sqrt {R \bigl(\tilde{\nu}_{j}(n) \bigr)}, $$

(41)

and

$$\textstyle\begin{cases}g(n)_{t_{0}}|_{\tilde{\lambda}=\tilde{\mu}_{k}(n)}=(2s_{0}^{(3)}(n)g(n)-2f(n)s_{0}^{(2)}(n))|_{\tilde{\lambda}=\tilde{\nu}_{j}(n)}\\ \phantom{g(n)_{t_{0}}|_{\tilde{\lambda}=\tilde{\mu}_{k}(n)}}=g(n)_{t_{0}}|_{\tilde{\lambda}=\tilde{\mu}_{k}(n)}=r_{n}(\partial _{t_{0}}\tilde{\mu}_{k}(n))\mathop{\pi}\limits _{i\neq j,i=1}^{N}(\tilde{\mu}_{k}(n)-\tilde{\mu}_{i}(n)),\\ h(n)_{t_{0}}|_{\tilde{\lambda}=\tilde{\nu}_{k}(n)}=(2f(n)s_{0}^{(1)}-2h(n)s_{0}^{(3)})|_{\tilde{\lambda}=\tilde{\nu}_{k}(n)}=s_{n-1}(\partial_{t_{0}}\tilde{\nu}_{k}(n)) \mathop{\pi}\limits _{i\neq j,i=1}^{N}(\tilde{\nu}_{k}(n)-\tilde{\nu}_{i}(n)), \end{cases} $$

from which we have

$$ \textstyle\begin{cases}\frac{\partial_{t_{0}}\tilde{\mu}_{k}(n)}{\sqrt{R(\tilde{\mu}_{k}(n))}}=-\frac{1}{\mathop{\pi}\limits _{i\neq k,i=1}^{N}(\tilde{\mu}_{k}(n)-\tilde{\mu}_{i}(n))},\\ \frac{\partial_{t_{0}}\tilde{\nu}_{k}(n)}{\sqrt{R(\tilde{\nu}_{k}(n))}}=\frac {1}{\mathop{\pi}\limits _{i\neq k,i=1}^{N}(\tilde{\nu}_{k}(n)-\tilde{\nu}_{i}(n))}. \end{cases} $$

(42)

Taking $t=t_{1}$, in terms of (29), we get

$$\begin{aligned} &g(n)_{t}|_{\tilde{\lambda}=\tilde{\mu}_{k}(n)}=2g(n) \biggl[ \frac{1}{2}\tilde{\lambda}^{2}-r_{n}s_{n-1} \tilde{\lambda}\biggr] \\ &\phantom{g(n)_{t}|_{\tilde{\lambda}=\tilde{\mu}_{k}(n)}=}{}-2f(n) \bigl[r_{n}\tilde{\lambda}\sqrt {\tilde{\lambda}} \bigl(-r_{n}r_{n+1}s_{n}-r_{n}^{2}s_{n-1}+q_{n}r_{n+1} \bigr)|_{\tilde{\lambda}=\tilde{\mu}_{k}(n)} \bigr], \end{aligned}$$

(43)

$$\begin{aligned} &h(n)_{t}|_{\tilde{\lambda}=\tilde{\nu}_{k}(n)}=2f \bigl(\tilde{\nu}_{k}(n) \bigr)\bigl[s_{n-1}\tilde{\nu}_{k}(n)\sqrt{\tilde{\nu}_{k}(n)}\bigr] \\ &\phantom{h(n)_{t}|_{\tilde{\lambda}=\tilde{\nu}_{k}(n)}=}{}+ \bigl(-s_{n-1}^{2}r_{n}-s_{n-1}s_{n-2}r_{n-1} +q_{n-1}s_{n-2} \bigr)\sqrt{\tilde{\nu}_{k}(n)}. \end{aligned}$$

(44)

Again from (36) and (43), (44), we have the following ODEs:

$$ \textstyle\begin{cases}\frac{\partial_{t}\tilde{\mu}_{k}(n)}{\sqrt{R(\tilde{\mu}_{k}(n))}}=-\frac{\tilde{\mu}_{k}(n)-\sum_{j=1}^{N}\tilde{\mu}_{j}(n)-\alpha_{1}}{\mathop{\pi}\limits _{i\neq j,i=1}^{N}(\tilde{\mu}_{k}(n)-\tilde{\mu}_{i}(n))},\\ \frac{\partial_{t}\tilde{\nu}_{k}(n)}{\sqrt{R(\tilde{\nu}_{k}(n))}}=\frac {\tilde{\nu}_{k}(n)-\sum_{j=1}^{N}\tilde{\nu}_{j}(n)-\alpha_{1}}{\mathop{\pi}\limits_{i\neq j,i=1}^{N}(\tilde{\nu}_{k}(n)-\tilde{\nu}_{i}(n))}. \end{cases} $$

(45)

Therefore, if $\tilde{\lambda}_{1},\ldots, \tilde{\lambda}_{2N+1}$ are $2N+1$ distinct parameters, and $\tilde{\mu}_{k}(n),\tilde{\nu}_{k}(n)$ are compatible solutions of (42) and (45), then $q_{n},r_{n},s_{n}$ determined by (38), (39) solve the $(1+1)$-dimensional lattice system (8).

3.2 Straightening out of the continuous flow

We introduce the Riemann surface Γ of the hyper-elliptic curve with genus N:

$$\xi^{2}=R(\tilde{\lambda}), \qquad R(\tilde{\lambda})=\tilde{\lambda}\mathop{\pi}\limits _{j=1}^{2N+1}(\tilde{\lambda}-\tilde{\lambda}_{j}), $$

which has two infinite points $\infty_{1}$ and $\infty_{2}$, not branch points of Γ. We fix a set of regular cycle paths: $a_{1},\ldots, a_{N};b_{1},\ldots, b_{N}$ which are independent and have the intersection numbers

$$a_{k}\circ a_{j}=b_{k}\circ b_{j}=0, \qquad a_{k} \circ b_{j}=\delta_{kj},\quad 1\leq k,j\leq N. $$

On Γ, we choose the holomorphic differentials:

$$\tilde{\omega}_{l}=\frac{\tilde{\lambda}^{l-1}\,d\tilde{\lambda}}{\sqrt {R(\tilde{\lambda})}},\quad l=1,\ldots,N, $$

and we denote

$$A_{kj}= \int_{a_{j}}\tilde{\omega}_{k},\qquad B_{kj}= \int_{b_{j}}\tilde{\omega}_{k}. $$

It can be verified that the matrices $A=(A_{kj})$ and $B=(B_{ij})$ are all $N\times N$ invertible. If we denote matrices C and τ by $C=(c_{kj})=A^{-1}, \tau=(\tau_{kj})=CB$, then the matrix τ can be proved to be symmetric and have positive defined imaginary part. Now we normalize $\tilde{\omega}_{j}$ into the new basis $\omega_{j}$:

$$\omega_{j}=\sum_{i=1}^{N}c_{jl} \tilde{\omega}_{l},\quad l=1,\ldots, N, $$

so that they satisfy

$$ \int_{a_{k}}\omega_{j}=\sum _{l=1}^{N}c_{jl} \int_{a_{k}}\tilde{\omega}_{l}=\sum _{l=1}^{N}c_{jl}A_{lk}= \delta_{jk},\qquad \int_{b_{k}}\omega_{j}=\tau_{jk}. $$

(46)

We again introduce the Abel map $\mathcal {A}(P)$:

$$\mathcal {A}(P)= \int_{P_{0}}^{P}\omega, $$

which can be extended to the whole divisor group of $\Gamma: \mathcal {A}:\operatorname{Div}(\Gamma)\rightarrow\bar{\mathcal {J}}(\Gamma)=C^{N}/\mathcal {J}$, where the lattice $\mathcal {J}$ is spanned by the periodic vectors $\{\delta_{k},\tau_{k}\}$ given by (46). The Abel-Jacobi coordinates are defined as

$$\textstyle\begin{cases}\rho^{(1)}(n)=\mathcal {A}(\sum_{k=1}^{N}P(\tilde{\mu}_{k}(n)))=\sum_{k=1}^{N}\int_{P_{0}}^{P(\tilde{\mu}_{k}(n))}\omega,\\ \rho^{(2)}(n)=\mathcal {A}(\sum_{k=1}^{N}P(\tilde{\nu}_{k}(n)))=\sum_{k=1}^{N}P(\tilde{\mu}_{k}(n))=\sum_{k=1}^{N}\int _{P_{0}}^{P(\tilde{\nu}_{k}(n))}\omega, \end{cases} $$

explicitly,

$$ \textstyle\begin{cases}\rho^{(1)}(n)=\sum_{k=1}^{N}\int_{\tilde{\lambda}(P_{0})}^{\tilde{\mu}_{k}(n)}\omega_{j}=\sum_{k=1}^{N}\sum_{l=1}^{N}c_{jl} \int_{\tilde{\lambda}(P_{0})}^{\tilde{\mu}_{k}(n)}\frac{\tilde{\lambda}^{l-1}\,d\tilde{\lambda}}{\sqrt{R(\tilde{\lambda})}},\\ \rho^{(2)}(n)=\sum_{k=1}^{N}\int_{\tilde{\lambda}(P_{0})}^{\tilde{\nu}_{k}(n)}\omega_{j}=\sum_{k=1}^{N}\sum_{l=1}^{N}c_{jl} \int_{\tilde{\lambda}(P_{0})}^{\tilde{\nu}_{k}(n)}\frac{\tilde{\lambda}^{l-1}\,d\tilde{\lambda}}{\sqrt{R(\tilde{\lambda})}}, \end{cases} $$

(47)

where $\tilde{\lambda}(P_{0})$ is the local coordinate of $P_{0}, P(\tilde{\mu}_{k}(n))=(\tilde{\lambda}=\tilde{\mu}_{k}(n),\xi=\sqrt{R(\tilde{\mu}_{k}(n))}), P(\tilde{\nu}_{k}(n))=(\tilde{\lambda}=\tilde{\nu}_{k}(n),\xi=\sqrt{R(\tilde{\nu}_{k}(n))})\in\Gamma$. We obtain

$$\begin{aligned} &\partial_{t_{0}}\rho^{(1)}(n)=\sum _{l=1}^{N} \sum_{k=1}^{N}c_{jl} \frac{\tilde{\mu}_{k}(n)^{l-1}\partial_{t_{0}}\tilde{\mu}_{k}(n)}{\sqrt{R(\tilde{\mu}_{k}(n))}} \\ &\phantom{\partial_{t_{0}}\rho^{(1)}(n)} =-\sum_{l=1}^{N}\sum _{k=1}^{N}c_{jl}\frac{\tilde{\mu}_{k}(n)^{l-1}}{\mathop{\pi}\limits _{i\neq k,i=1}^{N}(\tilde{\mu}_{k}(n)-\tilde{\mu}_{i}(n))} =-c_{jN}\equiv\Omega^{(1)}_{j}, \end{aligned}$$

(48)

$$\begin{aligned} &\partial_{t}\rho^{(1)}(n)=\partial_{t}\sum _{l=1}^{N}\sum_{k=1}^{N}c_{jl} \int_{\tilde{\lambda}(P_{0})}^{\tilde{\mu}_{k}(n)}\frac{\tilde{\lambda}^{l-1}\,d\tilde{\lambda}}{\sqrt{R(\tilde{\lambda})}} \\ &\phantom{\partial_{t}\rho^{(1)}(n)}=-\sum_{k=1}^{N}c_{jl} \frac{\tilde{\mu}_{k}(n)^{l-1}(\tilde{\mu}_{k}(n)-\sum_{j=1}^{N}\tilde{\mu}_{j}(n)-\alpha_{1})}{\mathop{\pi}\limits _{i\neq k,i=1}^{N}(\tilde{\mu}_{k}(n)-\tilde{\mu}_{i}(n))}\equiv\Omega ^{(2)}_{j},\quad 1\leq j\leq N. \end{aligned}$$

(49)

Similarly, we can obtain

$$\partial_{t_{0}}\rho^{(2)}(n)=-\Omega_{j}^{(1)},\qquad \partial_{t}\rho ^{(2)}(n)=-\Omega^{(2)}_{j},\quad j=1,2, \ldots, N. $$

Remark 2

Equation (49) is a finite sum, but we do not know how to express it by some linear combinations of the elements $c_{ij}$.

3.3 Straightening out of the discrete flow

Suppose the fundamental solution matrix of the first equation in (25) is given by [24]

$$Q_{n}= \bigl(\phi(n),\tilde{\phi}(n) \bigr)= \begin{pmatrix}\phi^{(1)}&\hat{\phi}^{(1)}(n)\\ \phi^{(2)}&\hat{\phi}^{(2)}(n) \end{pmatrix},\qquad Q_{0}=I, $$

which satisfies

$$ Q_{n+1}=U_{n}U_{n-1}\ldots U_{0}. $$

(50)

We can compute that

$$\begin{aligned} &\phi^{(1)}(1)=\lambda^{2},\qquad \phi^{(2)}(1)= \lambda s_{0},\qquad \hat{\phi}^{(1)}(1)=\lambda r_{0},\qquad \hat{\phi}^{(2)}(1)=q_{0}, \\ &\phi^{(1)}(2)=\lambda^{4}+\lambda^{2}r_{1}s_{0}, \qquad \phi^{(2)}(2)=\lambda ^{3}s_{1}+\lambda q_{1}s_{0}, \\ &\hat{\phi}^{(1)}(2)=\lambda^{3}r_{0}+\lambda r_{1}q_{0},\qquad \hat{\phi}^{(2)}=\lambda ^{2}s_{1}r_{0}+q_{1}q_{0}, \ldots. \end{aligned}$$

Assume δ is the eigenvalue of the Lax matrix $W_{n}$ in the solution space of equation $\psi(n+1)=U_{n}\psi(n)$, which is invariant under the action of $W_{n}$ due to $(EW_{n})U_{n}=U_{n}W_{n}$. The corresponding eigenfunction is $\psi(n)$ called the Baker function which obeys

$$ \psi(n+1)=U_{n}\psi(n),\qquad W_{n}\psi(n)= \delta\psi(n). $$

(51)

It is easy to see that

$$\operatorname{det}|\delta-W_{n}|=\delta^{2}-f^{2}(n)-g(n)h(n)=0 $$

has two eigenvalues $\delta^{\pm}=\pm\delta$, where

$$ \delta=\sqrt{f^{2}(n)+g(n)h(n)}=\frac{1}{2} \sqrt{R(\tilde{\lambda})}. $$

(52)

The corresponding Baker function can be taken as

$$\phi^{\pm}(n)=\phi(n)+b^{\pm}\hat{\phi}(n),\qquad b^{\pm}= \frac{\pm\delta-f(0)}{g(0)}, $$

or

$$\begin{aligned} &\phi^{\pm}(n)=\phi(n)+\bar{b}^{\pm}\hat{\phi}(n),\qquad \bar{b}^{\pm}=\frac{h(0)}{\pm \delta+f(0)}; \\ &\hat{\phi}^{\pm}(n)=c^{\pm}\phi(n)+\hat{\phi}(n),\qquad c^{\pm}= \frac{\pm\delta+f(0)}{h(0)}, \end{aligned}$$

or

$$\hat{\phi}^{\pm}(n)=\bar{c}^{\pm}\phi(n)+\hat{\phi}(n),\qquad \bar{c}^{\pm}=\frac {g(0)}{\pm\delta-f(0)}. $$

By following [19], we can prove the following formula of Dubrovin-Novikov type:

$$ \textstyle\begin{cases}p^{+}(n)p^{-}(n)=\frac{r_{n}}{r_{0}}\mathop{\pi}\limits _{j=1}^{N}\frac{\tilde{\lambda}-\tilde{\mu}_{j}(n)}{\tilde{\lambda}-\tilde{\mu}_{j}(0)},\\ q^{+}(n)q^{-}(n)=\frac{s_{n-1}}{s_{-1}}\mathop{\pi}\limits _{j=1}^{N}\frac {\tilde{\lambda}-\tilde{\nu}_{j}(n)}{\tilde{\lambda}-\tilde{\nu}_{j}(0)}, \end{cases} $$

(53)

where

$$ \textstyle\begin{cases}p^{+}(n)=\phi^{(1)}(n)+b^{+}\hat{\phi}^{(2)}(n),\qquad p^{-}{(n)}=\phi^{(1)}(n)+b^{-}{\hat{\phi}^{(2)}(n)},\\ q^{+}{(n)}=c^{+}{\phi^{(1)}(n)}+\hat{\phi}^{(2)}(n),\qquad q^{-}{(n)}=c^{-}{\phi ^{(1)}(n)}+\hat{\phi}^{(2)}(n). \end{cases} $$

(54)

Now we consider the approximation of $b^{\pm}$ and $c^{\pm}$, then we discuss the approximations of the functions (54) so that we have some properties of the Baker function as follows. A direct calculation gives rise to

$$\begin{aligned} &b^{+}=\frac{h(0)}{\delta+f(0)}=2s_{-1}\tilde{\lambda}\bigl(1+O \bigl(\tilde{\lambda}^{-1} \bigr) \bigr), \end{aligned}$$

(55)

$$\begin{aligned} &b^{-}=\frac{-\delta-f(0)}{g(0)}=-\frac{1}{r_{0}}\tilde{\lambda}\bigl\{ 1+O \bigl(\tilde{\lambda}^{-1} \bigr) \bigr\} , \end{aligned}$$

(56)

$$\begin{aligned} &c^{+}=\frac{\delta+f(0)}{h(0)}=\frac{\tilde{\lambda}}{s_{-1}} \bigl\{ 1+O \bigl(\tilde{\lambda}^{-1} \bigr) \bigr\} , \end{aligned}$$

(57)

$$\begin{aligned} &c^{-}=-\frac{g(0)}{\delta+f(0)}=-r_{0}\tilde{\lambda}^{-1} \bigl\{ 1+O \bigl(\tilde{\lambda}^{-1} \bigr) \bigr\} . \end{aligned}$$

(58)

From (55)-(58) and (53), one infers that

$$\begin{aligned} p^{+}(n)p^{-}(n)&=\frac{r_{n}}{r_{0}} \bigl\{ 1+O \bigl(\tilde{\lambda}^{-1} \bigr) \bigr\} ,\\ q^{+}(n)q^{-}(n)&=\frac{s_{n-1}}{s_{-1}} \bigl\{ 1+O \bigl(\tilde{\lambda}^{-1} \bigr) \bigr\} . \end{aligned}$$

The functions $p^{+}(n),p^{-}(n)$ and $q^{+}(n),q^{-}(n)$ can be regarded as values of the singly valued functions $p(n,P)$ and $q(n,P)$ on the upper and lower sheets of Γ, respectively. Hence, we have the following assertion:

$$ \textstyle\begin{cases}p^{+}(n)=(1+2s_{-1}s_{n}r_{0})\tilde{\lambda}^{n}+O(\tilde{\lambda}^{n-1}),\\ p^{-}(n)=(1-s_{n})\tilde{\lambda}^{n}+O(\tilde{\lambda}^{n-1}),\\ q^{+}(n)=\frac{1}{s_{-1}}\tilde{\lambda}^{n+1}+(s_{-1}^{-1}r_{n}s_{n-1}+s_{n}r_{0})\tilde{\lambda}^{n}+O(\tilde{\lambda}^{n-1}),\\ q^{-}(n)=(1-s_{n})r_{0}\tilde{\lambda}^{n-1}+O(\tilde{\lambda}^{n-2}). \end{cases} $$

(59)

As stated by Cao and Geng [19, 20], we can prove the following assertions based on (53)-(59).

Proposition 1

The Baker function $p(n,P)$ has

• (i) N simple zeros at $\tilde{\mu}_{1}(n),\ldots, \tilde{\mu}_{N}(n)$ and a pole of the nth order at $\infty_{2}=(z=0,1),z=\tilde{\lambda}^{-1}$ on the upper sheet of Γ;

• (ii) N simple zeros at $\tilde{\nu}_{1}(n),\ldots, \tilde{\nu}_{N}(n)$ and a zero of the nth order at $\infty_{1}=(z=0,-1)$ on the lower sheet of Γ.

Proposition 2

The Baker function $q(n,P)$ has

• (i) N simple poles at $\tilde{\nu}_{1}(0),\ldots, \tilde{\nu}_{N}(n)$ and a pole of nth order at $\infty_{2}$ on the upper sheet of Γ;

• (ii) N simple zeros at $\tilde{\nu}_{1}(n),\ldots, \tilde{\nu}_{k}(n)$ and a zero of the nth order at $\infty_{1}$ on the lower sheet of Γ.

Theorem

Straightening out the discrete flow

$$\begin{aligned} &\Delta\rho^{(1)}=\rho^{(1)}(n+1)-\rho^{(1)}(n)= \Omega^{(0)}(\operatorname {mod}\mathcal {J}); \\ &\Delta\rho^{(2)}=\rho^{(2)}(n+1)-\rho^{(2)}(n)= \Omega^{(0)}(\operatorname {mod}\mathcal {J}), \end{aligned}$$

where $\Omega^{(0)}=\int_{\infty_{1}}^{\infty_{2}}\omega$.

3.4 Algebraic-geometric solution of equation (8)

The well-known Riemann theta function of Γ is defined by

$$\theta(\xi |\tau)=\sum_{z\in{\mathbf {Z}}^{N}}\operatorname{exp}\bigl(2 \pi i\langle \tau z,z\rangle +2\pi i\langle \xi,z \rangle \bigr),\quad \xi\in\mathbf{ C}^{N}, $$

in which $\xi=(\xi_{1},\ldots, \xi_{N})^{T},\langle \xi,z \rangle =\sum_{j=1}^{N}\xi _{j}z_{j}$.

According to the Riemann theorem, there exists a constant $M^{(i)}\in \mathbf{ C}^{N}$ so that

• (i) $F_{1}=\theta(\mathcal {A} (P)-\rho^{(1)}(n)-M^{(1)})$ has exactly N zeros at $\tilde{\lambda}=\tilde{\mu}_{1}(n),\ldots, \tilde{\mu}_{N}(n)$;

• (ii) $F_{2}=\theta(\mathcal {A}(P)-\rho^{(2)}(n)-M^{(2)})$ has exactly zeros at $\tilde{\lambda}=\tilde{\nu}_{1}(n),\ldots, \tilde{\nu}_{N}(n)$.

The surface Γ is cut along all $a_{k},b_{k}$ to become a simple connected region so that the function defined on Γ is simple valued. Denote the boundary of Γ by γ, then the integrals

$$\frac{1}{2\pi i} \int_{\gamma}\tilde{\lambda}\, d\ln F_{m}=I_{k}( \Gamma),\quad m=1,2;k=1,2, $$

are constants which are independent of $\rho^{(1)}(n)$ and $\rho ^{(2)}(n)$ with $I_{k}(\Gamma)=\sum_{j=1}^{N}\int_{a_{j}}\tilde{\lambda}^{k}\omega_{j}$.

According to the inversion theorem, we have

$$ \textstyle\begin{cases}\sum_{j=1}^{N}\tilde{\mu}_{j}(n)^{k}=I_{k}(\Gamma)-\sum_{s=1}^{2}\mathop{\operatorname{Res}} _{\tilde{\lambda}=\infty _{s}}\tilde{\lambda}^{k} \,d\ln F_{1}(\tilde{\lambda}),\\ \sum_{j=1}^{N}\tilde{\nu}_{j}(n)^{k}=I_{k}(\Gamma)-\sum_{s=1}^{2}\mathop{\operatorname{Res}} _{\tilde{\lambda}=\infty_{s}}\tilde{\lambda}^{k} \,d\ln F_{2}(\tilde{\lambda}). \end{cases} $$

(60)

In the following, we calculate the residues in (60). We introduce local coordinate $z=\tilde{\lambda}^{-1}$ at $\infty_{s}$. Then the hyper-elliptic curve $\xi^{2}=R(\tilde{\lambda})$ in the neighborhood of infinity is given by $\tilde{\xi}^{2}=\tilde{R}(z)$ along with $\tilde{\xi}=z^{2N+2},\tilde{R}(z)=z^{2N} \mathop{\pi}\limits_{j=1}^{2N+1}(1-\tilde{\lambda}_{j} z)$, and $\infty _{s}=(z=0,(-1)^{s-1}\sqrt{\tilde{R}(\tilde{\lambda})}|_{z=0})=(0,(-1)^{s-1}),s=1,2$. We can infer that

$$\begin{aligned} \mathcal {A}\bigl(P\bigl(z^{-1}\bigr)\bigr)&=\biggl(- \int_{\infty_{s}}^{P_{0}}+ \int_{\infty_{s}}^{P}\biggr)\omega \\ &=-\eta_{s}-(-1)^{s-1} \sum_{l=1}^{N}c_{jl} \int_{0}^{z}\frac {z^{N-1}\,dz}{\sqrt{\tilde{R}(z)}} \\ &=- \eta_{s}-(-1)^{s-1}\bigl[c_{jN}z+O \bigl(z^{2}\bigr)\bigr],\quad \eta_{s}= \int_{\infty_{s}}^{P_{0}}\omega . \end{aligned}$$

Since the theta function is an even function, $F_{m}(\tilde{\lambda})$ can be written as

$$\begin{aligned} F_{m}\bigl(z^{-1}\bigr)&=\theta\bigl(\ldots, \rho^{(m)}_{j}+M^{(m)}_{j}+\eta _{s}^{(j)}+(-1)^{s-1}c_{jN}z+O \bigl(z^{2}\bigr),\ldots\bigr)\\ &=\theta_{s}^{(m)}+z(-1)^{s-1} \sum_{j=1}^{N}c_{jN}D_{j} \theta_{s}^{(m)}+O\bigl(z^{2}\bigr), \end{aligned}$$

where $\theta_{s}^{(m)}=\theta(\rho^{(m)}(n)+M^{(m)}+\eta_{s}^{(m)}), \eta _{s}^{(m)}=\int_{\infty_{m}}^{P_{0}}\omega, m=1,2$. $D_{j}$ stands for the derivative with respect to the jth argument of $\theta_{s}^{(m)}$. It is easy to compute that

$$\frac{\partial}{\partial_{t_{0}}}\theta_{s}^{(m)}=\sum _{j=1}^{N}c_{jN}D_{j} \theta_{s}^{(m)}. $$

Thus, we have

$$\begin{aligned} \begin{aligned} &F_{m} \bigl(z^{-1} \bigr)= \theta_{s}^{(m)}-z(-1)^{s-1} \partial_{t_{0}} \partial_{t}\theta _{s}^{(m)}+O \bigl(z^{2} \bigr). \\ &\operatorname{Res}_{\tilde{\lambda}=\infty_{s}}\tilde{\lambda}\, d\ln F_{m}(\tilde{\lambda})=-(-1)^{s-1}\partial_{t_{0}}\ln\theta_{s}^{(m)}+O(z), \quad 1 \leq s,m\leq2, \end{aligned} \end{aligned}$$

(61)

where

$$\begin{aligned} &\theta_{s}^{(1)}=\theta\bigl(\Omega^{(0)}n+t_{0} \Omega^{(1)}+t\Omega^{(2)}+\rho ^{(1)}_{0} \bigr), \\ &\theta_{s}^{(2)}=\theta\bigl(\Omega^{(0)}n-t_{0} \Omega^{(1)}-t\Omega^{(2)}+\rho ^{(2)}\bigr). \end{aligned}$$

Hence, equations (60) and (61) lead to

$$\begin{aligned} \textstyle\begin{cases}\sum_{j=1}^{N}\tilde{\mu}_{j}(n)=I_{1}(\Gamma)-\partial _{t_{0}}\ln\frac{\theta_{2}^{(1)}}{\theta_{1}^{(1)}},\\ \sum_{j=1}^{N}\tilde{\nu}_{j}(n)=I_{1}(\Gamma)-\partial_{t_{0}}\ln\frac {\theta_{1}^{(2)}}{\theta_{2}^{(2)}}. \end{cases}\displaystyle \end{aligned}$$

(62)

Substituting (62) into (39) yields

$$\begin{aligned} &r_{n}=\operatorname{exp}\Biggl[-\partial_{t^{-1}} \partial_{t_{0}}\ln\frac{\theta _{2}^{(1)}}{\theta_{1}^{(1)}}-\frac{1}{2}t\sum _{j=1}^{2N+1}\tilde{\lambda}_{j}\Biggr], \\ &s_{n}=\operatorname{exp}\Biggl[-\partial_{t^{-1}} \partial_{t_{0}}\ln\frac{E\theta _{2}^{(1)}}{E\theta_{1}^{(1)}}-\frac{1}{2}t\sum _{j=1}^{2N+1}\tilde{\lambda}_{j}\Biggr], \\ & q_{n}=\operatorname{exp}\Biggl\{ \partial_{t^{-1}}\Delta \Biggl[\operatorname{exp}\Biggl(-\partial _{t}^{-1} \partial_{t_{0}}\ln\frac{\theta_{2}^{(1)}}{\theta_{1}^{(1)}}-\frac {1}{2}t\sum _{j=1}^{2N+1}\Biggr) \operatorname{exp}\biggl(- \partial_{t^{-1}}\partial_{t_{0}}\ln\frac{E\theta _{2}^{(1)}}{E\theta_{1}^{(1)}}- \frac{1}{2}t\sum_{j=1}^{2N+1}\tilde{\lambda}_{j}\biggr)\Biggr]\Biggr\} , \end{aligned}$$

which is the algebro-geometric solution to equation (8).

Remark 3

We have obtained the algebraic-geometric solutions of the $(1+1)$-dimensional nonlinear discrete system (8). It is also an interesting and challenging work to address how to directly generate algebraic-geometric solutions of some $(2+1)$-dimensional reduced discrete integrable systems of the $(2+1)$-dimensional differential-difference hierarchy (17) just like the model for generating algebraic-geometric solutions in $1+1$ dimensions. In addition, it is important for investigating numerical solutions of the discrete integrable system (8) like the way presented in [39]. It is also interesting to discuss some properties presented in [40–43]. These problems will be discussed in the future.