Theory and Modern Applications

# Periodic solutions to a coupled two-dimensional lattice presented by Blaszak and Szum with Riemann–theta function

## Abstract

A coupled two-dimensional lattice presented by Blaszak and Szum is studied with the aid of Riemann–theta function and the bilinear method. By utilizing a bilinear form of the equation, we have obtained one-periodic and two-periodic solutions. In order to analyze the solution, we study asymptotic behavior and draw the solution plots.

## 1 Introduction

The subject of the discrete system has been attracting interest [13]. Especially, Toda lattice equations have been discussed by many researchers. In [4], Dai was dedicated to the study of integrable variable-coefficient Toda lattice by using the dressing method. Nakamura in [5] discussed the $$3+1$$-dimensional Toda equation and derived the solutions by using the Bessel functions. The authors in [6] obtained the solutions of $$2+1$$-dimensional Toda lattice by using Darboux transformation. Tian and Hu discussed semi-discrete KP and BKP equations by utilizing nonlocal symmetries in [7]. By using the Hirota bilinear method with the help of Riemann–theta function, Nakamura [8, 9] studied some famous equations such as KdV, Boussinesq, Toda, etc., and Dai et al. demonstrated for KP equation and Toda lattice [10, 11]. Recently, a lot of researchers have been concerned with the method [1215]. However, the coupled discrete system and high-dimensional equations have less been studied in the previous literature.

In this paper, we consider the two-dimensional lattice presented by Blaszak and Szum in [16]:

$$\begin{gathered} u_{t}(n)=u(n) \bigl(w(n)-w(n-1)\bigr), \\ v_{t}(n)=u(n+1)-u(n)+w_{y}(n), \\ w_{t}(n+1)+w_{t}(n)=v(n+1)-v(n)-w^{2}(n+1)+w^{2}(n), \end{gathered}$$
(1.1)

which is a coupled discrete system. Tam and Hu in [17] discussed its bilinear forms and its solutions. Yu et al. [18] derived its pfaffianization and molecule solutions. We will obtain one-periodic solution and two-periodic solution by utilizing the bilinear method and the Riemann–theta function presented in [810].

The paper is organized as follows. In Sect. 2, we obtain one-periodic wave solution and study its asymptotic behavior. The solution is also studied graphically. In Sect. 3, we obtain two-periodic wave solutions whose asymptotic behaviors are studied and the plots are given.

## 2 One-periodic solution and its asymptotic behavior

In the section, we study one-periodic solution of (1.1). Through the transformation [17],

$$\begin{gathered} u(n)=\frac{f(n+1)f(n-1)}{f^{2}(n)},\qquad v(n)=\frac{D_{t}^{2}f(n)\cdot f(n+1)}{f(n) f(n+1)},\\ w(n)= \biggl(\ln\frac{f(n+1)}{f(n)}\biggr)_{t}, \end{gathered}$$
(2.1)

then (1.1) can be written as

\begin{aligned}& \bigl(D_{z}e^{\frac{1}{2}D_{n}}-D_{t}^{2}e^{\frac {1}{2}D_{n}}+c_{1} \bigr)f(n)\cdot f(n)=0, \end{aligned}
(2.2)
\begin{aligned}& (D_{t}D_{z}-D_{t}D_{y}-2\cosh D_{n}+2+c_{2})f(n)\cdot f(n)=0, \end{aligned}
(2.3)

where z is an auxiliary variable, $$c_{1}$$ and $$c_{2}$$ are integration constants. The Hirota bilinear differential operator is defined as [19]

$$D^{m}_{x}D^{n}_{y}a(x,y)\cdot b(x,y) \equiv(\partial_{x}-\partial_{x'})^{m}( \partial_{y}-\partial _{y'})^{n}a(x,y)\times b \bigl(x',y'\bigr)\big|x'=x,y'=y$$

and the difference operator is defined as

$$\begin{gathered} e^{D_{n}}a_{n}\cdot b_{n}=a_{n+1}b_{n-1}, \qquad e^{-D_{n}}a_{n}\cdot b_{n}=a_{n-1}b_{n+1}, \\ \cosh D_{n} a_{n}\cdot b_{n}=\frac{1}{2}(a_{n+1}b_{n-1}+a_{n-1}b_{n+1}).\end{gathered}$$

From the definition of Hirota bilinear operator, we have

$$D^{m}_{x}D^{l}_{y} e^{\zeta_{1}} \cdot e^{\zeta_{2}}=(l_{1}-l_{2})^{m}( \rho_{1}-\rho_{2})^{l} e^{\zeta_{1}+\zeta_{2}},$$

where $$\zeta_{j}=l_{j}x+\rho_{j}y+\eta_{j}n+\zeta_{j0}$$ ($$j=1,2$$). Moreover, it is easy to deduce

\begin{aligned}& \cosh D_{n} e^{\zeta_{1}}\cdot e^{\zeta_{2}}=\cosh( \eta_{1}-\eta_{2})e^{\zeta _{1}+\zeta_{2}}, \end{aligned}
(2.4)
\begin{aligned}& G(D_{x},D_{y},\cosh D_{n})e^{\zeta_{1}} \cdot e^{\zeta_{2}}=G(l_{1}-l_{2},\rho_{1}- \rho_{2},\eta_{1}-\eta_{2})e^{\zeta_{1}+\zeta_{2}}. \end{aligned}
(2.5)

### 2.1 One-periodic wave solution

In view of [8, 9], we consider the Riemann–theta function solution of the bilinear form (2.2) and (2.3)

$$f=\sum_{k\in Z^{N}}e^{\pi i\langle\tau k,k\rangle+2\pi i\langle \zeta,k\rangle},$$
(2.6)

where $$\langle\cdot, \cdot\rangle$$ is the inner product, $$k=(k_{1},\ldots,k_{N})^{T}$$, $$\zeta=(\zeta_{1},\ldots,\zeta_{N})^{T}$$ and τ is a symmetric matrix, $$\zeta_{j}=p_{j}t+l_{j}y+\mu_{j}z+\eta_{j}n+\zeta_{0j}$$ ($$j=1,\ldots,N$$). In order to obtain one-periodic wave solution, we consider the case for $$N=1$$, and we denote $$k=k_{1}$$, $$\zeta=\zeta_{1}$$, $$\zeta_{0}=\zeta _{01}$$. The direct calculations show that $$\pi i\langle\tau k,k\rangle=\pi i k^{2}\tau$$, $$2\pi i\langle\zeta,k\rangle=2\pi i k\zeta$$. Thus (2.6) becomes

$$f=\sum_{k=-\infty}^{\infty}e^{2\pi i k\zeta+\pi ik^{2}\tau}.$$
(2.7)

Inserting (2.7) into (2.2) and using the bilinear properties, we have

$$\begin{gathered} F_{1}\bigl(D_{z},D_{t},e^{\frac{1}{2}D_{n}} \bigr)f(n)\cdot f(n) \\ \quad\equiv\bigl(D_{z}e^{\frac{1}{2}D_{n}}-D_{t}^{2}e^{\frac {1}{2}D_{n}}+c_{1} \bigr)f(n)\cdot f(n) \\ \quad=\sum_{k,k'=-\infty}^{\infty}F_{1} \bigl(D_{z},D_{t},e^{\frac {1}{2}D_{n}}\bigr)\exp\bigl(2\pi i k \zeta+\pi ik^{2}\tau\bigr)\cdot\exp\bigl(2\pi i k'\zeta + \pi ik^{\prime2}\tau\bigr) \\ \quad=\sum_{k,m=-\infty}^{\infty}F_{1} \bigl(D_{z},D_{t},e^{\frac{1}{2}D_{n}}\bigr)\exp \bigl(2\pi i k \zeta+\pi ik^{2}\tau\bigr)\cdot\exp\bigl(2\pi i (m-k)\zeta+\pi i(m-k)^{2}\tau\bigr) \\ \quad=\sum_{k,m=-\infty}^{\infty}F_{1}{ \bigl(2\pi i(2k-m)\mu,2\pi i(2k-m)p,e^{\pi i(2k-m)\eta}\bigr)}\\ \qquad{}\times\exp{\bigl(2\pi im \zeta+\pi i\bigl[k^{2}+(k-m)^{2}\bigr]\tau\bigr)} \\ \quad=\sum_{m=-\infty}^{\infty}\tilde{F_{1}}(m)\exp(2\pi im\zeta)=0, \end{gathered}$$

where the new summation index $$m=k+k'$$ has been introduced and $$\tilde{F_{1}}(m)$$ is defined by

$$\tilde{F_{1}}(m)=\sum_{k=-\infty}^{\infty}F_{1}{\bigl(2\pi i(2k-m)\mu,2\pi i(2k-m)p,e^{\pi i(2k-m)\eta}} \bigr)e^{\pi i[k^{2}+(k-m)^{2}]\tau}.$$
(2.8)

Thus,

\begin{aligned}& \tilde{F_{1}}(0) =\sum_{k=-\infty}^{\infty}\bigl(4\pi ik\mu e^{2\pi ik\eta}-16\pi^{2}k^{2}p^{2}e^{2\pi ik\eta} +c_{1}\bigr)e^{2\pi ik^{2}\tau}=0, \end{aligned}
(2.9)
\begin{aligned}& \begin{aligned}[b]\tilde{F_{1}}(1) &=\sum_{k=-\infty}^{\infty}\bigl[2\pi i(2k-1)\mu e^{\pi i(2k-1)\eta}-4\pi^{2} (2k-1)^{2}p^{2} e^{\pi i(2k-1)\eta}+c_{1}\bigr]e^{\pi i[k^{2}+(k-1)^{2}]\tau}\hspace{-20pt}\\ &=0.\end{aligned} \end{aligned}
(2.10)

We denote

$$\begin{gathered} d_{1}=\exp2\pi ik^{2}\tau, \qquad d_{2}=\exp\pi i\bigl[k^{2}+(k-1)^{2}\bigr]\tau ,\\\Delta_{1}=\sum_{k=-\infty}^{\infty}d_{1},\qquad \Delta_{2}=\sum_{k=-\infty}^{\infty}d_{2}, \\ a_{11}=\sum_{k=-\infty}^{\infty}4\pi i k d_{1}\exp2\pi ik\eta, \qquad a_{21}=\sum _{k=-\infty}^{\infty}2\pi i (2k-1) d_{2}\exp\pi i(2k-1)\eta. \end{gathered}$$

Then (2.9) and (2.10) are written as

$$\mu a_{11}+2a_{11,\eta}p^{2}+c_{1} \Delta_{1}=0,\qquad \mu a_{21}+2a_{21,\eta}p^{2}+c_{1} \Delta_{2}=0,$$

from which we have

$$\mu=2p^{2}\frac{a_{21,\eta}\Delta_{1}-a_{11,\eta}\Delta _{2}}{a_{11}\Delta_{2}-a_{21}\Delta_{1}},\qquad c_{1}=2p^{2} \frac {a_{11,\eta}a_{21}-a_{21,\eta}a_{11}}{a_{11}\Delta_{2}-a_{21}\Delta_{1}}.$$
(2.11)

Similarly, substituting (2.7) into (2.3), we derive

$$\begin{gathered} F_{2}(D_{z},D_{t},D_{y}, \cosh D_{n})f(n)\cdot f(n) \\ \quad\equiv(D_{t}D_{z}-D_{t}D_{y}-2 \cosh D_{n}+2+c_{2})f(n)\cdot f(n) \\ \quad=\sum_{k,k'=-\infty}^{\infty}F_{2}(D_{z},D_{t},D_{y},\cosh D_{n})\exp \bigl(2\pi i k\zeta+\pi ik^{2}\tau\bigr)\cdot\exp \bigl(2\pi i k'\zeta+\pi ik^{\prime2}\tau\bigr) \\ \quad=\sum_{k,m=-\infty}^{\infty}F_{2}{ \bigl(2\pi i(2k-m)\mu,2\pi i(2k-m)p,2\pi i(2k-m)l,\cosh2\pi i(2k-m)\eta\bigr)} \\ \qquad{}\times\exp{\bigl(2\pi im\zeta+\pi i\bigl[k^{2}+(k-m)^{2} \bigr]\tau\bigr)} \\ \quad=\sum_{m=-\infty}^{\infty}\tilde{F_{2}}(m)\exp(2\pi im\zeta)=0, \end{gathered}$$

where

\begin{aligned} \tilde{F_{2}}(m)={}&\sum _{k=-\infty}^{\infty}F_{2}{\bigl[2\pi i(2k-m)\mu,2 \pi i(2k-m)p,2\pi i(2k-m)l,\cosh2\pi i(2k-m)\eta\bigr]} \\ &\times\exp{\pi i\bigl[k^{2}+(k-m)^{2}\bigr]\tau}. \end{aligned}

It is easy to know that if $$\tilde{F_{2}}(0)=0$$, $$\tilde{F_{2}}(1)=0$$, then all $$\tilde{F_{2}}(m)=0$$ are proved.

$$\begin{gathered} \tilde{F_{2}}(0)=\sum _{k=-\infty}^{\infty}\bigl(-16\pi^{2}k^{2}p \mu+16\pi ^{2}k^{2}pl-2\cosh4\pi ik\eta+2+c_{2} \bigr)e^{2\pi ik^{2}\tau}=0, \\ \tilde{F_{2}}(1)=\sum_{k=-\infty}^{\infty}\bigl(4\pi^{2}(2k-1)^{2}(-p\mu +pl)-2\cosh2\pi i(2k-1) \eta+2+c_{2}\bigr)\\ \phantom{\tilde{F_{2}}(1)=}{}\times e^{\pi i[k^{2}+(k-1)^{2}]\tau}\\ \phantom{\tilde{F_{2}}(1)}=0. \end{gathered}$$
(2.12)

Letting $$b_{11}=\sum_{k=-\infty}^{\infty}16\pi^{2}k^{2}d_{1}$$, $$b_{12}=\sum_{k=-\infty}^{\infty}\cosh(4\pi i k\eta) d_{1}$$, $$b_{21}=\sum_{k=-\infty}^{\infty}4\pi^{2}(2k-1)^{2}d_{2}$$, $$b_{22}=\sum_{k=-\infty}^{\infty}\cosh2\pi i (2k-1)\eta d_{2}$$, thus, (2.12) can be written as

$$\begin{gathered} p(l-\mu)b_{11}-2b_{12}+(2+c_{2}) \Delta_{1}=0, \\ p(l-\mu)b_{21}-2b_{22}+(2+c_{2}) \Delta_{2}=0.\end{gathered}$$

Solving the above system, we have

$$l=\mu+\frac{2}{p}\frac{b_{22}\Delta_{1}-b_{12}\Delta_{2}}{b_{21}\Delta _{1}-b_{11}\Delta_{2}},\qquad 2+c_{2}=2 \frac{b_{12}b_{21}-b_{22}b_{11}}{b_{21}\Delta_{1}-b_{11}\Delta_{2}},$$
(2.13)

from which we find that parameter l is dependent on μ, η, and p. In view of (2.11), we can see that μ is dependent on η and p.

Then we have derived the Riemann–theta function solution $$f(n)$$ of (2.2) and (2.3). Furthermore, the Riemann–theta function periodic solutions of (1.1) are obtained by using transformation (2.1).

### 2.2 Asymptotic behavior of the one-periodic wave solution

In what follows, we will prove that the soliton solution can be regarded as the limit of the following periodic solution. Therefore, we write $$q=\exp\pi i \tau$$ and take a limit $$q\rightarrow0$$ (or $$\operatorname{Im}\tau\rightarrow\infty$$).

### Theorem 1

Under the condition $$q\rightarrow0$$ (or $$\operatorname{Im}\tau\rightarrow\infty$$), the Riemann–theta function periodic solution (2.7) of (2.2) and (2.3) tends to the one-soliton solutions of (1.1) via (2.1).

$$\begin{gathered} u(n)=\frac{(1+e^{\widetilde{\zeta}+\widetilde{\eta}})(1+e^{\widetilde {\zeta}-\widetilde{\eta}})}{(1+e^{\widetilde{\zeta}})^{2}}, \\ v(n)=-2\pi^{2}p^{2}e^{\frac{\widetilde{\zeta}}{2}} \operatorname{sech} \frac{\widetilde{\zeta}}{2} \frac{1+e^{\widetilde{\eta }}}{1+e^{\widetilde{\zeta}+\widetilde{\eta}}}, \\ w(n)=2\pi ip \frac{e^{\widetilde{\zeta}}(e^{\widetilde{\eta }}-1)}{(1+e^{\widetilde{\zeta}})(1+e^{\widetilde{\zeta}+\widetilde{\eta}})}, \end{gathered}$$
(2.14)

where

$$\begin{gathered} \widetilde{\zeta}=2\pi i(pt+ly+\mu z+\eta n)+ \widetilde{\zeta _{0}},\qquad\widetilde{\eta}=2\pi i \eta,\qquad \widetilde{\zeta _{0}}=\zeta_{0}+\frac{1}{2}\tau, \\ \mu\rightarrow-2\pi p^{2}\cot\pi\eta,\qquad l\rightarrow\mu+ \frac{\cos ^{2} \pi\eta}{p\pi^{2}},\qquad c_{1}\rightarrow0,\qquad c_{2}\rightarrow0. \end{gathered}$$

### Proof

Utilizing $$q=\exp\pi i \tau$$, the quantities defined above are then expanded in powers of q

\begin{aligned}& \Delta_{1}=\sum _{k=-\infty}^{\infty}e^{2\pi ik^{2}\tau }=1+2q^{2}+o \bigl(q^{2}\bigr), \qquad \Delta_{2}=\sum _{k=-\infty}^{\infty}e^{2\pi i[k^{2}+(k-1)^{2}]\tau }=2q^{2}+o \bigl(q^{2}\bigr), \\& a_{11}=\sum_{k=-\infty}^{\infty}4\pi ik e^{2\pi ik\eta} e^{2\pi ik^{2}\tau}=-8\pi\sin(2\pi\eta)q^{2}+o \bigl(q^{2}\bigr), \\& a_{11,\eta}=\sum_{k=-\infty}^{\infty}-8 \pi^{2} k^{2} e^{2\pi ik\eta} e^{2\pi ik^{2}\tau}=-16 \pi^{2} \cos(2\pi\eta)q^{2}+o\bigl(q^{2}\bigr), \\& a_{21}=\sum_{k=-\infty}^{\infty}2\pi i(2k-1) e^{\pi i(2k-1)\eta} e^{\pi i[k^{2}+(k-1)^{2}]\tau}=-4\pi\sin(\pi\eta) q+o \bigl(q^{5}\bigr), \\& a_{21,\eta}=\sum_{k=-\infty}^{\infty}-2 \pi^{2}(2k-1)^{2} e^{\pi i(2k-1)\eta} e^{\pi i[k^{2}+(k-1)^{2}]\tau}=-4 \pi^{2} \cos(\pi\eta) q+o\bigl(q^{5}\bigr), \\& b_{11}=\sum_{k=-\infty}^{\infty}16 \pi^{2}k^{2}e^{2\pi ik^{2}\tau}=32\pi ^{2}q^{2}+o \bigl(q^{8}\bigr), \\& b_{12}=\sum_{k=-\infty}^{\infty}\cosh(4 \pi i k\eta)e^{2\pi ik^{2}\tau }=1+2\cos(4\pi\eta)q^{2}+o \bigl(q^{8}\bigr), \\& b_{21}=\sum_{k=-\infty}^{\infty}4 \pi^{2}(2k-1)^{2}e^{\pi i[k^{2}+(k-1)^{2}]\tau}=8\pi^{2}q+o \bigl(q^{5}\bigr), \\& b_{22}=\sum_{k=-\infty}^{\infty}\cosh2\pi i(2k-1)\eta e^{\pi i[k^{2}+(k-1)^{2}]\tau}=2\cos(2\pi\eta) q+o\bigl(q^{5}\bigr). \end{aligned}

Using (2.11) and (2.13), we have $$\mu\rightarrow-2\pi p^{2}\cot\pi\eta$$, $$l\rightarrow\mu+\frac{\cos2\pi\eta}{2p\pi^{2}}$$, $$c_{1}\rightarrow0$$, $$c_{2}\rightarrow0$$ for $$q\rightarrow0$$.

In order to consider the convergence to the one-periodic wave solution (2.7) in the limit of $$q\rightarrow0$$, under the transformation $$\zeta _{0}=\widetilde{\zeta_{0}}-\frac{1}{2}\tau$$, we can get the following convergent forms:

$$\begin{gathered} f(n)=1+\exp{\widetilde{\zeta}}+o \bigl(q^{2}\bigr), \\ f(n-1)=1+\exp{(\widetilde{\zeta}-\widetilde{\eta})}+o\bigl(q^{2} \bigr), \\ f(n+1)=1+\exp{(\widetilde{\zeta}+\widetilde{\eta})}+o\bigl(q^{2} \bigr), \\ f_{t}(n)=2\pi ip\exp{\widetilde{\zeta}}+o\bigl(q^{2} \bigr), \\ f_{tt}(n)=-4\pi^{2}p^{2}\exp{\widetilde{\zeta}}+o \bigl(q^{2}\bigr), \\ f_{t}(n+1)=2\pi ip\exp(\widetilde{\zeta}+\widetilde{\eta})+o \bigl(q^{2}\bigr), \\ f_{tt}(n+1)=-4\pi^{2}p^{2}\exp(\widetilde{\zeta}+ \widetilde{\eta})+o\bigl(q^{2}\bigr). \end{gathered}$$
(2.15)

After some tedious calculations, we derive (2.14). □

In what follows, Fig. 1, Fig. 2, and Fig. 3 describe the plots of $$u(t,y,n)$$, $$v(t,y,n)$$, and $$w(t,y,n)$$, respectively. From these figures, we find that the plots of $$v(t,y,n)$$ and $$w(t,y,n)$$ have similar forms.

## 3 Two-periodic wave solution and its asymptotic behavior

In what follows, similar to the one-periodic wave solution, we consider a two-periodic wave solution of the coupled two-dimensional lattice (1.1).

### 3.1 Construction of two-periodic wave solution

By letting $$N=2$$ in (2.6), we have $$f(n)=\sum_{k\in Z^{2}} e^{2\pi i \langle\zeta,k\rangle+\pi i\langle\tau k,k\rangle}$$ and substitute it into (2.2). For convenience of calculations, we have introduced different forms of k and $$k'$$. Thus, we obtain

\begin{aligned} F_{1}f_{n} \cdot f_{n}={}&\sum_{k,k'\in Z^{2}} F_{1} \bigl(D_{z},D_{t},e^{\frac{1}{2}D_{n}}\bigr)e^{2\pi i \langle\zeta,k\rangle+\pi i\langle\tau k,k\rangle}\cdot e^{2\pi i \langle\zeta,k'\rangle+\pi i\langle\tau k',k'\rangle} \\ ={}&\sum_{k,k'\in Z^{2}}F_{1}\bigl(2\pi i\bigl\langle k-k',\mu\bigr\rangle ,2\pi i\bigl\langle k-k',p \bigr\rangle ,e^{2\pi i\langle k-k',\eta\rangle }\bigr) \\ &\times\exp\bigl(2\pi i\bigl\langle \zeta,k+k'\bigr\rangle \bigr) \exp\pi i\bigl(\bigl\langle \tau k',k'\bigr\rangle + \langle\tau k,k\rangle\bigr) \\ ={}&\sum_{s'\in Z^{2}}\sum_{k_{1},k_{2}=-\infty}^{\infty}F_{1}{\bigl(2\pi i\bigl\langle 2k-s',\mu\bigr\rangle ,2\pi i\bigl\langle 2k-s',p\bigr\rangle ,e^{2\pi i\langle2k-s',\eta\rangle}\bigr)} \\ &\times\exp{\pi i\bigl(\bigl\langle \tau\bigl(k-s' \bigr),k-s'\bigr\rangle +\langle\tau k,k\rangle\bigr)\exp\bigl(2\pi i \bigl\langle \zeta,s'\bigr\rangle \bigr)} \\ \equiv{}&\sum_{s'\in Z^{2}} \tilde{F_{1}} \bigl(s'_{1},s'_{2}\bigr)\exp{ \bigl(2\pi i\bigl\langle \zeta,s'\bigr\rangle \bigr)}=0. \end{aligned}
(3.1)

By introducing the new summation index $$k+k'=s'$$, $$s'=(s'_{1},s'_{2})^{T}$$, $$k=(k_{1},k_{2})^{T}$$, $$\tilde{F}_{1}(s'_{1},s'_{2})$$ is denoted by

\begin{aligned}[b] \tilde{F}_{1} \bigl(s'_{1},s'_{2}\bigr)={}&\sum _{k_{1},k_{2}=-\infty}^{\infty}F_{1}\bigl[2\pi i \bigl\langle 2k-s',\mu\bigr\rangle ,2\pi i\bigl\langle 2k-s',p \bigr\rangle ,e^{2\pi i\langle2k-s',\eta\rangle}\bigr] \\ &\times\exp\pi i\bigl(\bigl\langle \tau\bigl(k-s' \bigr),k-s'\bigr\rangle +\langle\tau k,k\rangle\bigr) \\ ={}&\sum_{k_{j}=-\infty}^{\infty}F_{1} \Biggl(2\pi i\sum_{j=1}^{2} \bigl(2k_{j}-\bigl(s'_{j}-2 \delta_{jl}\bigr)\bigr)\mu_{j},2\pi i\sum _{j=1}^{2}\bigl(2k_{j}- \bigl(s'_{j}-2\delta_{jl}\bigr) \bigr)p_{j}, \\ &{2\pi i\sum_{j=1}^{2}\bigl(2k_{j}- \bigl(s'_{j}-2\delta_{jl}\bigr)\bigr) \eta_{j}}\Biggr) \\ &\times\exp\pi i\sum_{j,l=1}^{2} \bigl[(k_{j}+\delta_{jl})\tau_{jl}(k_{j}+ \delta _{jl})\\ &+\bigl(\bigl(s_{j}'-2 \delta_{jl}-k_{j}\bigr)+\delta_{jl}\bigr) \tau_{jl}\bigl(\bigl(s_{j}'-2\delta _{jl}-k_{j}\bigr)+\delta_{jl}\bigr)\bigr] \\ ={}&\left \{ \textstyle\begin{array}{l@{\quad}l} \tilde{F}_{1}(s'_{1}-2,s'_{2})e^{2\pi i (s_{1}'-1)\tau_{11}+2\pi i s'_{2}\tau_{12}}, & l \text{ is even},\\ \tilde{F}_{1}(s'_{1},s'_{2}-2)e^{2\pi i (s_{2}'-1)\tau_{22}+2\pi i s'_{1}\tau_{12}}, & l \text{ is odd}. \end{array}\displaystyle \right . \end{aligned}
(3.2)

This relation implies that if $$\tilde{F}_{1}(0,0)=\tilde{F}_{1}(0,1)=\tilde{F}_{1}(1,0)=\tilde{F}_{1}(1,1)=0$$, then $$\tilde{F}_{1}(s'_{1},s'_{2})=0$$ for $$s'_{1}$$, $$s'_{2}\in Z$$.

Denoting

$$\begin{gathered} \delta_{j}(n)=e^{\pi i\langle\tau(k-m^{(j)}),k-m^{(j)}\rangle+\pi i\langle\tau k, k\rangle}, \\ m^{(1)}=(0,0)^{T},\qquad m^{(2)}=(1,0)^{T}, \qquad m^{(3)}=(0,1)^{T},\qquad m^{(4)}=(1,1)^{T}, \end{gathered}$$

we have

$$\begin{gathered} \tilde{F_{1}}(0,0)=\sum _{k_{1}, k_{2}=-\infty}^{\infty}\bigl[2\pi i\bigl\langle 2k-(0,0)^{T},\mu\bigr\rangle e^{\pi i\langle2k-(0,0)^{T},\eta\rangle} \\ \phantom{\tilde{F_{1}}(0,0)=}+4\pi^{2}\bigl\langle 2k-(0,0)^{T},p\bigr\rangle ^{2}e^{\pi i\langle2k-(0,0)^{T},\eta\rangle}+c_{1}\bigr]e^{2\pi i\langle \tau k,k\rangle}, \\ \tilde{F_{1}}(0,1)=\sum_{k_{1},k_{2}=-\infty}^{\infty} \bigl[2\pi i\bigl\langle 2k-(0,1)^{T},\mu\bigr\rangle e^{\pi i\langle2k-(0,1)^{T},\eta\rangle} \\ \phantom{\tilde{F_{1}}(0,1)=}+4\pi^{2}\bigl\langle 2k-(0,1)^{T},p\bigr\rangle ^{2}e^{\pi i\langle2k-(0,1)^{T},\eta\rangle}+c_{1}\bigr]e^{\pi i\langle \tau(k-(0,1)^{T}),k-(0,1)^{T}\rangle+\pi i\langle\tau k,k\rangle}, \\ \tilde{F_{1}}(1,0)=\sum_{k_{1},k_{2}=-\infty}^{\infty} \bigl[2\pi i\bigl\langle 2k-(1,0)^{T},\mu\bigr\rangle e^{\pi i\langle2k-(1,0)^{T},\eta\rangle} \\ \phantom{\tilde{F_{1}}(1,0)=}+4\pi^{2}\bigl\langle 2k-(1,0)^{T},p\bigr\rangle ^{2}e^{\pi i\langle2k-(1,0)^{T},\eta\rangle}+c_{1}\bigr]e^{\pi i\langle \tau(k-(1,0)^{T}),k-(1,0)^{T}\rangle+\pi i\langle\tau k,k\rangle}, \\ \tilde{F_{1}}(1,1)=\sum_{k_{1},k_{2}=-\infty}^{\infty} \bigl[2\pi i\bigl\langle 2k-(1,1)^{T},\mu\bigr\rangle e^{\pi i\langle2k-(1,1)^{T},\eta\rangle} \\ \phantom{\tilde{F_{1}}(1,1)=}+4\pi^{2}\bigl\langle 2k-(1,1)^{T},p\bigr\rangle ^{2}e^{\pi i\langle2k-(1,1)^{T},\eta\rangle}+c_{1}\bigr]e^{\pi i\langle \tau(k-(1,1)^{T}),k-(1,1)^{T}\rangle+\pi i\langle\tau k,k\rangle}. \end{gathered}$$
(3.3)

Denote

$$\begin{gathered} a_{j1}=\sum_{k_{1}, k_{2}=-\infty}^{\infty}2 \pi i\bigl(2k_{1}-m_{1}^{(j)} \bigr)e^{\pi i\langle k-m^{(j)},\eta\rangle}\delta_{1}, \\ a_{j2}=\sum_{k_{1},k_{2} =-\infty}^{\infty}2\pi i \bigl(2k_{2}-m_{2}^{(j)}\bigr)e^{\pi i\langle k-m^{(j)},\eta\rangle} \delta_{2}, \\ a_{j3}=\sum_{k_{1},k_{2}=-\infty}^{\infty}4\pi ^{2}\bigl(2k_{1}-m_{1}^{(j)} \bigr)^{2}e^{\pi i\langle k-m^{(j)},\eta\rangle}\delta_{3}, \\ a_{j4}=\sum_{k_{1},k_{2}=-\infty}^{\infty}4\pi ^{2}\bigl(2k_{2}-m_{2}^{(j)} \bigr)^{2}e^{\pi i\langle k-m^{(j)},\eta\rangle}\delta_{4}, \\ b_{j}=-c_{1}\sum_{k_{1},k_{2}=-\infty}^{\infty} \delta_{j}-8\pi ^{2}\bigl(2k_{1}-m_{1}^{(j)} \bigr) \bigl(2k_{2}-m_{2}^{(j)}\bigr)p_{1}p_{2} e^{\pi i\langle k-m_{1}^{(j)},\eta\rangle}\delta_{j}, \end{gathered}$$

then (3.3) can be written as

$$A\left ( \textstyle\begin{array}{c}\mu_{1}\\ \mu_{2}\\ p^{2}_{1}\\ p^{2}_{2} \end{array}\displaystyle \right )=\vec{b},$$

from which we have $$\mu_{1}=\frac{\triangle_{1}}{\triangle}$$, $$\mu_{2}=\frac{\triangle _{2}}{\triangle}$$, $$p^{2}_{1}=\frac{\triangle_{3}}{\triangle}$$, $$p^{2}_{2}=\frac{\triangle _{4}}{\triangle}$$, where $$\triangle=\det A$$ and $$\triangle_{1}$$, $$\triangle_{2}$$, $$\triangle _{3}$$, $$\triangle_{4}$$ are given by replacing 1st, 2nd, 3rd, 4th columns with b⃗, respectively.

Similarly, by letting $$N=2$$ in (2.6), then we have $$f(n)=\sum_{k\in Z^{2}} e^{2\pi i \langle\zeta,k\rangle+\pi i\langle\tau k,k\rangle}$$ and substitute it into (2.3). For convenience of calculations, we have introduced k and $$k'$$ of different form. We have derived

\begin{aligned}[b] F_{2}(f_{n}\cdot f_{n})={}&\sum_{k,k'\in Z^{2}} F_{2}(D_{z},D_{t},D_{y}, \cosh D_{n})e^{2\pi i \langle\zeta,k\rangle+\pi i\langle\tau k,k\rangle}\cdot e^{2\pi i \langle\zeta,k'\rangle+\pi i\langle\tau k',k'\rangle} \\ ={}&\sum_{k,k'\in Z^{2}}F_{2}\bigl(2\pi i\bigl\langle k-k',\mu\bigr\rangle ,2\pi i\bigl\langle k-k',p \bigr\rangle ,2\pi i\bigl\langle k-k',l\bigr\rangle ,\cosh2\pi i\bigl\langle k-k',\eta\bigr\rangle \bigr) \\ &\times\exp\bigl(2\pi i\bigl\langle \zeta,k+k'\bigr\rangle \bigr) \exp(\pi i\bigl(\bigl\langle \tau k',k'\bigr\rangle + \langle\tau k,k\rangle\bigr) \\ ={}&\sum_{s'\in Z^{2}}\sum_{k_{1},k_{2}=-\infty}^{\infty}F_{2}\bigl(2\pi i\bigl\langle 2k-s',\mu\bigr\rangle ,2\pi i \bigl\langle 2k-s',p\bigr\rangle ,2\pi i\bigl\langle 2k-s',l \bigr\rangle ,\\ &\cosh2\pi i\bigl\langle 2k-s',\eta\bigr\rangle \bigr) \\ &\times\exp{\pi i\bigl(\bigl\langle \eta\bigl(k-s' \bigr),k-s'\bigr\rangle +\langle\tau k,k\rangle\bigr)\exp\bigl(2\pi i \bigl\langle \zeta,s'\bigr\rangle \bigr)} \\ \equiv{}&\sum_{s'\in Z^{2}} \tilde{F_{2}} \bigl(s'_{1},s'_{2}\bigr)\exp{ \bigl(2\pi i\bigl\langle \zeta,s'\bigr\rangle \bigr)}=0. \end{aligned}
(3.4)

By introducing the new summation index $$k+k'=s'$$, $$k=(k_{1},k_{2})^{T}$$, $$\tilde{F}_{2}(s'_{1},s'_{2})$$ is denoted by

\begin{aligned}[b] \tilde{F}_{2} \bigl(s'_{1},s'_{2}\bigr)={}&\sum _{k_{1},k_{2}=-\infty}^{\infty}F_{2}\bigl[2\pi i \bigl\langle 2k-s',\mu\bigr\rangle ,2\pi i\bigl\langle 2k-s',p \bigr\rangle ,2\pi i\bigl\langle 2k-s',l\bigr\rangle ,\cosh2\pi i\bigl\langle 2k-s',\eta\bigr\rangle \bigr] \\ & \times\exp\pi i\bigl(\bigl\langle \tau\bigl(k-s'\bigr),\bigl(k-s' \bigr)\bigr\rangle +\langle\tau k,k\rangle\bigr) \\ ={}&\left \{ \textstyle\begin{array}{l@{\quad}l} \tilde{F}_{2}(s'_{1}-2,s'_{2})e^{2\pi i (s_{1}'-1)\tau_{11}+2\pi i s'_{2}\tau_{12}}, & l \text{ is even},\\ \tilde{F}_{2}(s'_{1},s'_{2}-2)e^{2\pi i (s_{2}'-1)\tau_{22}+2\pi i s'_{1}\tau_{12}}, & l \text{ is odd}, \end{array}\displaystyle \right . \end{aligned}
(3.5)

which means that if $$\tilde{F}_{2}(m^{(j)})=0$$, thus all $$\tilde {F}_{2}(s'_{1},s'_{2})=0$$. Through direct calculations, we have derived

\begin{aligned}[b] \tilde{F}_{2} \bigl(m^{(j)}\bigr)={}&\sum_{k_{1},k_{2}=-\infty}^{\infty}\bigl[-4\pi ^{2}\bigl\langle 2k-m^{(j)},p\bigr\rangle \bigl\langle 2k-m^{(j)},\mu\bigr\rangle +4\pi ^{2}\bigl\langle 2k-m^{(j)},p\bigr\rangle \bigl\langle 2k-m^{(j)},l\bigr\rangle \\ & -2\cosh2\pi i\bigl\langle 2k-m^{(j)},\eta\bigr\rangle +2+c_{2}\bigr]\\ &\times e^{\pi i\langle \tau k,k\rangle+\pi i\langle\tau(k-m^{(j)}),k-m^{(j)}\rangle }. \end{aligned}
(3.6)

Let

\begin{aligned}& c_{j1}=\sum _{k_{1},k_{2}=-\infty}^{\infty}4\pi ^{2}\bigl(2k_{1}-m_{1}^{(j)} \bigr)^{2}\delta_{j}(n), \\ & c_{j2}=\sum_{k_{1},k_{2}=-\infty}^{\infty}4\pi ^{2}\bigl(2k_{2}-m_{2}^{(j)} \bigr)^{2}\delta_{j}(n), \\ & c_{j3}=\sum_{k_{1},k_{2}=-\infty}^{\infty}4\pi ^{2}\bigl(2k_{1}-m_{1}^{(j)}\bigr) \bigl(2k_{2}-m_{2}^{(j)}\bigr)\delta_{j}(n), \\& c_{j4}=\sum_{k_{1},k_{2}=-\infty}^{\infty}\delta_{j}(n), \\& d_{j}=\sum_{k_{1},k_{2}=-\infty}^{\infty}4\cosh2 \pi i\bigl\langle 2k-m^{(j)},\eta\bigr\rangle \delta_{j}(n), \end{aligned}
(3.7)

then $$\tilde{F}_{2}(m^{(j)})=0$$ can be rewritten as

$$C\left ( \textstyle\begin{array}{c}p_{1}(l_{1}-\mu_{1})\\ p_{2}(l_{2}-\mu_{2})\\ p_{1}(l_{2}-\mu_{2})+p_{2}(l_{1}-\mu_{1})\\ 2+c_{2} \end{array}\displaystyle \right )=\vec{d},$$

from which we have $$p_{1}(l_{1}-\mu_{1})=\frac{\triangle_{1}}{\triangle}$$, $$p_{2}(l_{2}-\mu _{2})=\frac{\triangle_{2}}{\triangle}$$, $$p_{1}(l_{2}-\mu_{2})+p_{2}(l_{1}-\mu_{1})=\frac{\triangle_{3}}{\triangle }$$, $$2+c_{2}=\frac{\triangle_{4}}{\triangle}$$, where $$\triangle=\det C$$ and $$\triangle_{1}$$, $$\triangle_{2}$$, $$\triangle _{3}$$, $$\triangle_{4}$$ are given by replacing 1st, 2nd, 3rd, 4th columns with d⃗.

### 3.2 Asymptotic behavior of the two-periodic wave solution

In what follows, we can verify the asymptotic behavior of the two-periodic wave solution to be the well-known two-soliton solution given by the Hirota method.

### Theorem 2

Let $$\lambda_{1}=\exp\tau_{11}\rightarrow0$$, $$\lambda_{2}=\exp\tau _{22}\rightarrow0$$, the periodic solution (2.1) of (1.1) tends to the two-soliton solution

$$\begin{gathered} u(n)=\frac{(1+e^{\tilde{\zeta}_{1}+\tilde{\eta}_{1}}+e^{\tilde{\zeta }_{2}+\tilde{\eta}_{2}} +e^{\tilde{\zeta}_{1}+\tilde{\eta}_{1}+\tilde{\zeta}_{2}+\tilde{\eta }_{2}+2\pi i \tau_{12}})(1+e^{\tilde{\zeta}_{1}-\tilde{\eta }_{1}}+e^{\tilde{\zeta}_{2}-\tilde{\eta}_{2}} +e^{\tilde{\zeta}_{1}-\tilde{\eta}_{1}+\tilde{\zeta}_{2}-\tilde{\eta }_{2}+2\pi i \tau_{12}})}{(1+\exp\tilde{\zeta}_{1}+\exp\tilde{\zeta }_{2}+\exp(\tilde{\zeta}_{1}+\tilde{\zeta}_{2}+2\pi i\tau_{12}))^{2}}, \\ v(n)=\frac {f_{tt}(n)f(n+1)-2f_{t}(n)f_{t}(n+1)+f(n)f_{tt}(n+1)}{f(n)f(n+1)}, \\ w(n)=\frac{f_{t}(n+1)f(n)-f_{t}(n)f(n+1)}{f(n)f(n+1)}, \end{gathered}$$
(3.8)

with

\begin{aligned}& p_{1}=\frac{1-\cosh\tilde{\eta}_{1}}{2\pi^{2}(l_{1}-\mu_{1})},\qquad p_{2}=\frac{1-\cosh\tilde{\eta}_{2}}{2\pi^{2}(l_{2}-\mu_{2})},\\& \tilde{\zeta}_{i}=2\pi i(p_{i}t+l_{i}y+\mu_{i}z+ \eta_{i}n)+\tilde{\zeta _{0i}},\qquad\tilde{ \zeta_{0i}}=\zeta_{0i}+\frac{1}{2} \tau_{ii}, \qquad \tilde{\eta_{i}}=2\pi i \eta_{i},\quad i=1,2, \\& e^{2\pi i\tau_{12}}=-\frac{2\pi^{2}(p_{1}-p_{2})(l_{1}-\mu_{1}+\mu _{2}-l_{2})-\cosh (\tilde{\eta}_{1}-\tilde{\eta}_{2})+1}{2\pi ^{2}(p_{1}+p_{2})(l_{1}-\mu_{1}+l_{2}-\mu_{2})-\cosh(\tilde{\eta }_{1}+\tilde{\eta}_{2})+1}, \\& 2\pi p_{2}^{2}\cos\pi\eta_{2}-\mu_{2} \sin\pi\eta_{2}=0,\qquad 2\pi p_{1}^{2}\cos\pi \eta_{1}-\mu_{1}\sin\pi\eta_{1}=0, \\& \frac{2\pi(p_{1}-p_{2})^{2}\cos\pi(\eta_{1}-\eta_{2})-(\mu_{1}-\mu _{2})\sin\pi(\eta_{1}-\eta_{2})}{ 2\pi(p_{1}+p_{2})^{2}\cos\pi(\eta_{1}+\eta_{2})-(\mu_{1}+\mu_{2})\sin \pi(\eta_{1}+\eta_{2})} \\& \quad=\frac{2\pi^{2}(p_{1}-p_{2})(l_{1}-l_{2}-\mu_{1}+\mu_{2})-\cos2\pi(\eta _{1}-\eta_{2})+1}{ 2\pi^{2}(p_{1}+p_{2})(l_{1}+l_{2}-\mu_{1}-\mu_{2})-\cos2\pi(\eta _{1}+\eta_{2})+1}, \\& f(n)\rightarrow1+\exp\tilde{\zeta}_{1}+ \exp\tilde{\zeta}_{2}+\exp(\tilde {\zeta}_{1}+\tilde{ \zeta}_{2}+2\pi i\tau_{12}), \\& f_{t}(n)\rightarrow2\pi ip_{1}e^{\tilde{\zeta_{1}}}+2\pi ip_{2}e^{\tilde {\zeta_{2}}}+2\pi i(p_{1}+p_{2})e^{\tilde{\zeta_{1}}+\tilde{\zeta _{2}}+2\pi i\tau_{12}}, \\& f_{tt}(n)\rightarrow-4\pi^{2} p^{2}_{1}e^{\tilde{\zeta_{1}}}-4 \pi^{2} p^{2}_{2}e^{\tilde{\zeta_{2}}}-4\pi^{2} (p_{1}+p_{2})^{2}e^{\tilde{\zeta _{1}}+\tilde{\zeta_{2}}+2\pi i\tau_{12}}, \\& f_{t}(n+1)\rightarrow2\pi ip_{1}e^{\tilde{\zeta_{1}}+\tilde{\eta _{1}}}+2\pi ip_{2}e^{\tilde{\zeta_{2}}+\tilde{\eta_{1}}}+2\pi i(p_{1}+p_{2})e^{\tilde{\zeta_{1}}+\tilde{\zeta_{2}}+\tilde{\eta _{1}}+\tilde{\eta_{2}}+2\pi i\tau_{12}}, \\& f_{tt}(n+1)\rightarrow-4\pi^{2} p^{2}_{1}e^{\tilde{\zeta_{1}}+\tilde {\eta_{1}}}-4 \pi^{2} p^{2}_{2}e^{\tilde{\zeta_{2}}+\tilde{\eta _{2}}}-4\pi^{2} (p_{1}+p_{2})^{2}e^{\tilde{\zeta_{1}}+\tilde{\zeta _{2}}+\tilde{\eta_{1}}+\tilde{\eta_{2}}+2\pi i\tau_{12}}. \end{aligned}

### Proof

Let $$\tilde{\zeta_{i}}=2\pi i\zeta_{i}+\pi i \frac{\tau_{ii}}{2}$$, $$\tilde{\eta_{i}}=2\pi i \eta_{i}$$ for $$i=1,2$$. We expand the two-periodic wave solution (2.6) ($$N=2$$) of (2.2) and (2.3):

\begin{aligned}[b] f(n)={}&1+\exp(2\pi i \zeta_{1}+\pi i\tau_{11})+\exp(-2\pi i\zeta_{1}+ \pi i\tau_{11})+\exp(2\pi i\zeta_{2}+\pi i \tau_{22}) \\ &+\exp(-2\pi i\zeta_{2}+\pi i\tau_{22})+\exp\bigl(2\pi i( \zeta_{1}+\zeta_{2})+\pi i(\tau_{11}+2 \tau_{12}+\tau_{22})\bigr) \\ &+\exp\bigl(-2\pi i(\zeta_{1}+\zeta_{2})+\pi i( \tau_{11}+2\tau_{12}+\tau_{22})\bigr)+\cdots \\ \rightarrow{}& 1+\exp\tilde{\zeta}_{1}+\exp\tilde{ \zeta}_{2}+\exp(\tilde {\zeta}_{1}+\tilde{ \zeta}_{2}+2\pi i\tau_{12}), \end{aligned}
(3.9)

then we have

$$\begin{gathered} f_{t}(n)\rightarrow2\pi ip_{1}e^{\tilde{\zeta_{1}}}+2\pi ip_{2}e^{\tilde{\zeta_{2}}}+2\pi i(p_{1}+p_{2})e^{\tilde{\zeta _{1}}+\tilde{\zeta_{2}}+2\pi i\tau_{12}}, \\ f_{tt}(n)\rightarrow-4\pi^{2} p^{2}_{1}e^{\tilde{\zeta_{1}}}-4 \pi^{2} p^{2}_{2}e^{\tilde{\zeta_{2}}}-4\pi^{2} (p_{1}+p_{2})^{2}e^{\tilde{\zeta _{1}}+\tilde{\zeta_{2}}+2\pi i\tau_{12}}, \\ f_{t}(n+1)\rightarrow2\pi ip_{1}e^{\tilde{\zeta_{1}}+\tilde{\eta _{1}}}+2\pi ip_{2}e^{\tilde{\zeta_{2}}+\tilde{\eta_{1}}}+2\pi i(p_{1}+p_{2})e^{\tilde{\zeta_{1}}+\tilde{\zeta_{2}}+\tilde{\eta _{1}}+\tilde{\eta_{2}}+2\pi i\tau_{12}}, \\ f_{tt}(n+1)\rightarrow-4\pi^{2} p^{2}_{1}e^{\tilde{\zeta_{1}}+\tilde {\eta_{1}}}-4 \pi^{2} p^{2}_{2}e^{\tilde{\zeta_{2}}+\tilde{\eta _{2}}}-4\pi^{2} (p_{1}+p_{2})^{2}e^{\tilde{\zeta_{1}}+\tilde{\zeta _{2}}+\tilde{\eta_{1}}+\tilde{\eta_{2}}+2\pi i\tau_{12}}. \end{gathered}$$
(3.10)

For convenience, we denote $$\lambda_{1}=e^{\pi i\tau_{11}}$$, $$\lambda _{2}=e^{\pi i\tau_{22}}$$. In what follows, we expand each function in $$\tilde{F_{1}}(0,0)=\tilde {F_{1}}(0,1)=\tilde{F_{1}}(1,0)=\tilde{F_{1}}(1,1)=0$$, $$\tilde {F_{2}}(0,0)=\tilde{F_{2}}(0,1)=\tilde{F_{2}}(1,0)=\tilde{F_{2}}(1,1)=0$$ into series of $$\lambda_{1}$$, $$\lambda_{2}$$,

\begin{aligned}[b] \tilde{F_{1}}(0,0)={}&c_{1}+ \bigl[4\pi i\mu_{1}\bigl(e^{2\pi i\eta_{1}}-e^{-2\pi i\eta_{1}}\bigr)+16 \pi^{2}p^{2}_{1}\bigl(e^{2\pi i\eta_{1}}+e^{-2\pi i\eta _{1}} \bigr)+c_{1}\bigr]e^{2\pi i\tau_{11}} \\ &+\bigl[4\pi i\mu_{2}\bigl(e^{2\pi i\eta_{2}}-e^{-2\pi i\eta_{2}}\bigr)+16 \pi ^{2}p^{2}_{2}\bigl(e^{2\pi i\eta_{2}}+e^{-2\pi i\eta_{2}} \bigr)+c_{1}\bigr]e^{2\pi i\tau_{22}}\\ &+o\bigl(\lambda_{1}^{r_{1}} \lambda_{2}^{r_{2}}\bigr), \end{aligned}
(3.11)

when $$r_{1}+r_{2}\geq4$$, it is easy to see that $$c_{1}\rightarrow0$$.

\begin{aligned}[b] \tilde{F_{1}}(0,1)={}& \bigl[\bigl(2\pi i\mu_{2}\bigl(e^{\pi i\eta_{2}}-e^{-\pi i\eta _{2}} \bigr)+4\pi^{2}p^{2}_{2}\bigl(e^{\pi i\eta_{2}}+e^{-\pi i\eta_{2}} \bigr)+c_{1}\bigr) \\ &+\bigl(2\pi i(2\mu_{1}-\mu_{2})+4\pi^{2}(2p_{1}-p_{2})^{2}+c_{1} \bigr)e^{2\pi i(\tau_{11}-\tau_{12})} \\ &+\bigl(2\pi i(2\mu_{1}+\mu_{2})+4\pi^{2}(2p_{1}+p_{2})^{2}+c_{1} \bigr)e^{2\pi i(\tau_{11}+\tau_{12})}\bigr]e^{\pi i\tau_{22}}\\&+o\bigl(\lambda_{1}^{r_{1}} \lambda _{2}^{r_{2}}\bigr) \end{aligned}
(3.12)

in view of $$c_{1}\rightarrow0$$, from which we have $$2\pi p^{2}_{2}\cos \pi\eta_{2}-\mu_{2}\sin\pi\eta_{2}=0$$.

\begin{aligned}[b] \tilde{F_{1}}(1,0)={}& \bigl[\bigl(2\pi i\mu_{1}\bigl(e^{\pi i\eta_{1}}-e^{-\pi i\eta _{1}} \bigr)+4\pi^{2}p^{2}_{1}\bigl(e^{\pi i\eta_{1}}+e^{-\pi i\eta_{1}} \bigr)+c_{1}\bigr) \\ &+\bigl(2\pi i(2\mu_{2}-\mu_{1})+4\pi^{2}(2p_{2}-p_{1})^{2}+c_{1} \bigr)e^{2\pi i(\tau_{22}-\tau_{12})} \\ &+\bigl(2\pi i(2\mu_{2}+\mu_{1})+4\pi^{2}(2p_{2}+p_{1})^{2}+c_{1} \bigr)e^{2\pi i(\tau_{22}+\tau_{12})}\bigr]e^{\pi i\tau_{11}}\\ &+o\bigl(\lambda_{1}^{r_{1}} \lambda _{2}^{r_{2}}\bigr). \end{aligned}
(3.13)

In view of $$c_{1}\rightarrow0$$, from (3.13), we have $$2\pi p^{2}_{1}\cos\pi\eta_{1}-\mu_{1}\sin\pi\eta_{1}=0$$.

\begin{aligned}[b] \tilde{F_{1}}(1,1)={}& \bigl[\bigl[2\pi i(\mu_{1}+\mu_{2}) \bigl(e^{\pi i(\eta_{1}+\eta _{2})}-e^{-\pi i(\eta_{1}+\eta_{2})} \bigr)\\ &+4\pi^{2}(p_{1}+p_{2})^{2} \bigl(e^{\pi i(\eta_{1}+\eta_{2})}+e^{-\pi i(\eta_{1}+\eta_{2})}\bigr)\bigr]e^{2\pi i\tau _{12}} \\ &+2\pi i(\mu_{1}-\mu_{2}) \bigl(e^{\pi i(\eta_{1}-\eta_{2})}-e^{\pi i(\eta _{1}-\eta_{2})} \bigr)\\ &+4\pi^{2}(p_{1}-p_{2})^{2} \bigl(e^{\pi i(\eta_{1}-\eta _{2})}+e^{-\pi i(\eta_{1}+\eta_{2})}\bigr)\bigr]e^{\pi i(\tau_{11}+\tau _{22})}+o\bigl( \lambda_{1}^{r_{1}}\lambda_{2}^{r_{2}}\bigr). \end{aligned}
(3.14)

From (3.14), we have

$$e^{2\pi i\tau_{12}}=-\frac{2\pi i(\mu_{1}-\mu_{2})(e^{\pi i(\eta _{1}-\eta_{2})}-e^{-\pi i(\eta_{1}-\eta_{2})})+4\pi ^{2}(p_{1}-p_{2})^{2}(e^{\pi i(\eta_{1}-\eta_{2})}+e^{-\pi i(\eta _{1}-\eta_{2})})}{2\pi i(\mu_{1}+\mu_{2})(e^{\pi i(\eta_{1}+\eta _{2})}-e^{-\pi i(\eta_{1}+\eta_{2})})+4\pi^{2}(p_{1}+p_{2})^{2}(e^{\pi i(\eta_{1}+\eta_{2})}+e^{-\pi i(\eta_{1}+\eta_{2})})}.$$
(3.15)

In the following, we will consider

\begin{aligned}[b] \tilde{F_{2}}(0,0)={}&c_{2}+ \bigl[16\pi^{2}p_{2}(l_{2}-\mu_{2})-2 \cosh4\pi i\eta_{2}+2+c_{2}\bigr]e^{2\pi i\tau_{22}} \\ &+\bigl[16\pi^{2}p_{1}(l_{1}- \mu_{1})-2\cosh4\pi i\eta_{1}+2+c_{2} \bigr]e^{2\pi i\tau_{11}}+o\bigl(\lambda_{1}^{r_{1}} \lambda_{2}^{r_{2}}\bigr). \end{aligned}
(3.16)

From $$\tilde{F_{2}}(0,0)\rightarrow0$$, we have $$c_{2}\rightarrow0$$.

\begin{aligned}[b] \tilde{F_{2}}(0,1)={}& \bigl(8\pi^{2}p_{2}(l_{2}-\mu_{2})-4\cosh2 \pi i\eta _{2}+4+2c_{2}\bigr)e^{\pi i\tau_{22}} \\ &+\bigl[4\pi^{2}(2p_{1}-p_{2}) (2l_{1}-2\mu_{1}+l_{2}-\mu_{2})-2 \cosh2\pi i(2\eta_{1}-\eta_{2})+2+c_{2} \bigr]\\ &\times e^{\pi i(2\tau_{11}-2\tau_{12}+\tau _{22})} \\ &+\bigl[4\pi^{2}(2p_{1}+p_{2}) (2l_{1}+2\mu_{1}-l_{2}-\mu_{2})-2 \cosh2\pi i(2\eta_{1}+\eta_{2})+2+c_{2} \bigr]\\ &\times e^{\pi i(2\tau_{11}+2\tau_{12}+\tau _{22})}+o\bigl(\lambda_{1}^{r_{1}} \lambda_{2}^{r_{2}}\bigr). \end{aligned}
(3.17)

From $$\tilde{F_{2}}(0,1)\rightarrow0$$, in view of $$c_{2}\rightarrow0$$, we have $$2\pi^{2}p_{2}(\mu_{2}-l_{2})+\cosh2\pi i\eta_{2}-1=0$$.

\begin{aligned}[b] \tilde{F_{2}}(1,0)={}&\bigl(8 \pi^{2}p_{1}(l_{1}-\mu_{1})-4\cosh2 \pi i\eta _{1}+4+2c_{2}\bigr)e^{\pi i\tau_{11}} \\ &+\bigl[4\pi^{2}(p_{2}-p_{1}) (2l_{2}-2\mu_{2}-l_{1}+\mu_{1})-2 \cosh2\pi i(2\eta_{2}-\eta_{1})+2+c_{2} \bigr]\\ &\times e^{\pi i(2\tau_{22}-2\tau_{12}+\tau _{11})} \\ &+\bigl[4\pi^{2}(p_{1}+2p_{2}) (2l_{2}-2\mu_{2}+l_{1}-\mu_{1})-2 \cosh2\pi i(2\eta_{2}+\eta_{1})+2+c_{2} \bigr]\\ &\times e^{\pi i(\tau_{11}+2\tau_{12}+2\tau _{22})}+o\bigl(\lambda_{1}^{r_{1}} \lambda_{2}^{r_{2}}\bigr). \end{aligned}
(3.18)

For $$\tilde{F_{2}}(1,0)\rightarrow0$$, we have $$2\pi^{2}p_{1}(\mu _{1}-l_{1})+\cosh2\pi i\eta_{1}-1=0$$.

\begin{aligned}[b] \tilde{F_{2}}(1,1)={}&2 \bigl[4\pi^{2}(p_{1}+p_{2}) (l_{1}+l_{2}- \mu_{1}-\mu_{2}) -2\cosh2\pi i(\eta_{1}+ \eta_{2})+2+c_{2}\bigr]e^{2\pi i\tau_{12}} \\ &+2\bigl[4\pi^{2}(p_{1}-p_{2}) (l_{1}-l_{2}-\mu_{1}+\mu_{2}) -2 \cosh2\pi i(\eta_{1}-\eta_{2})+2+c_{2} \bigr]e^{\pi i(\tau_{11}+\tau _{22})}\hspace{-24pt}\\ &+o\bigl(\lambda_{1}^{r_{1}} \lambda_{2}^{r_{2}}\bigr). \end{aligned}
(3.19)

In view of $$\tilde{F_{2}}(1,1)\rightarrow0$$, we have

$$e^{2\pi i\tau_{12}}=-\frac{2\pi^{2}(p_{1}-p_{2})(l_{1}-\mu_{1}+\mu _{2}-l_{2})-2\cosh2\pi i(\eta_{1}-\eta_{2})+1}{2\pi ^{2}(p_{1}+p_{2})(l_{1}-\mu_{1}-\mu_{2}+l_{2})-2\cosh2\pi i(\eta _{1}+\eta_{2})+1}.$$
(3.20)

This completes the proof of Theorem 2. □

## References

1. Pickering, A., Zhu, Z.-N.: New integrable lattice hierarchies. Phys. Lett. A 349, 439–445 (2006)

2. Wu, Y.Q.: Exact solutions to a Toda like lattice equation in $$2+1$$ dimensions. Chin. Phys. Lett. 29(6), Article ID 060203 (2012)

3. Claire, R.G., Nimmo, J.C.: The relation between a 2D Lotka–Volterra equation and a 2D Toda lattice. J. Nonlinear Math. Phys. 12(2), 169–179 (2001)

4. Dai, H.H., Su, T.: The generalized dressing method with applications to the integration of variable-coefficient Toda equations. Proc. Est. Acad. Sci. 59, 293–298 (2010)

5. Nakamura, A.: Exact cylindrical soliton solutions of the sine-Gordon equation, the sinh-Gordon equation and the periodic Toda equation. J. Phys. Soc. Jpn. 57(10), 3309–3322 (1988)

6. Chen, J.B., Qiao, Z.J.: Darboux transformations of the spectral $$2+1$$-dimensional Toda lattice and its explicit solution. Phys. Scr. 82, Article ID 015003 (2010)

7. Tian, K., Hu, X.B.: Negative semi-discrete Kp and Bkp hierarchies via nonlocal symmetries. J. Phys. A, Math. Theor. 42, Article ID 454022 (2009)

8. Nakamura, A.: A direct method of calculating periodic wave solutions to nonlinear evolution equations. I. Exact two-periodic wave solution. J. Phys. Soc. Jpn. 47(5), 1701–1705 (1979)

9. Nakamura, A.: A direct method of calculating periodic wave solutions to nonlinear evolution equations. II. Exact one- and two-periodic wave solution of the coupled bilinear equations. J. Phys. Soc. Jpn. 48(4), 1365–1370 (1980)

10. Dai, H.H., Fan, E.G., Geng, X.G.: Periodic wave solutions of nonlinear equations by Hirota’s bilinear method (2006). arXiv:nlin/0602015

11. Su, T., Dai, H.H., Ding, G.H.: Periodic-wave solutions of the two-dimensional Toda lattice equation by a direct method. Adv. Differ. Equ. 2016, Article ID 55 (2016)

12. Hon, Y.C., Fan, E.G., Qin, Z.Y.: A kind of explicit quasi-periodic solution and its limit for the Toda lattice equation. Mod. Phys. Lett. B 22(8), 547–553 (2008)

13. Tian, S.F., Zhang, H.Q.: Riemann theta functions periodic wave solutions and rational characteristics for nonlinear equations. J. Math. Anal. Appl. 371, 585–608 (2010)

14. Ma, W.X., Gao, R.G.: Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in $$(2+1)$$-dimensions. Mod. Phys. Lett. A 24(21), 1677–1688 (2009)

15. Wu, Y.Q.: Asymptotic behavior of the periodic wave solution for the $$3+1$$-dimensional KP equation. Appl. Math. Comput. 216, 3154–3161 (2010)

16. Blaszak, M., Szum, A.: Lie algebraic approach to the construction of $$(2+1)$$-dimensional lattice-filed and filed integrable Hamiltonian equations. J. Math. Phys. 42, 225–229 (2001)

17. Tam, H.W., Hu, X.B., Qian, X.M.: Remarks on several $$2+1$$ dimensional lattices. J. Math. Phys. 43, 1008–1017 (2002)

18. Yu, G.F., Li, C.X., Zhao, J.X., Hu, X.B.: On a special two-dimensional lattice by Blaszak and Szum: Pfaffian and molecule solutions. J. Nonlinear Math. Phys. 12, 316–332 (2005)

19. Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)

## Acknowledgements

The author is grateful to the scholars who provided the literature sources. The author is appreciative of reviewers for the valuable suggestions. The author thanks Gong Dong for effective advice.

## Funding

The authors acknowledge the support by the National Natural Science Foundation of China for No. 11301149 and No. 11701147.

## Author information

Authors

### Contributions

ST participated in the computing, drawing, and drafting the manuscript. I have read and approved the final manuscript.

### Corresponding author

Correspondence to Ting Su.

## Ethics declarations

### Competing interests

The author declares that they have not any financial and non-financial competing interests.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and permissions

Su, T. Periodic solutions to a coupled two-dimensional lattice presented by Blaszak and Szum with Riemann–theta function. Adv Differ Equ 2018, 331 (2018). https://doi.org/10.1186/s13662-018-1757-x