# Integrable discretizations for a generalized sine-Gordon equation and the
reductions to the sine-Gordon equation and the short pulse equation

Han-Han Sheng†, Bao-Feng Feng‡ and Guo-Fu Yu†111Corresponding author. Email
address<EMAIL_ADDRESS>
† School of Mathematical Sciences, CMA-Shanghai, Shanghai Jiao Tong
University,
Shanghai 200240, P.R. China
‡ School of Mathematical and Statistical Sciences,
The University of Texas Rio Grande Valley, Edinburg, Texas 78541, USA

###### Abstract

In this paper, we propose fully discrete analogues of a generalized sine-
Gordon (gsG) equation $u_{tx}=\left(1+\nu\partial_{x}^{2}\right)\sin u$. The
bilinear equations of the discrete KP hierarchy and the proper definition of
discrete hodograph transformations are the keys to the construction. Then we
derive semi-discrete analogues of the gsG equation from the fully discrete gsG
equation by taking the temporal parameter $b\rightarrow 0$. Especially, one
full-discrete gsG equation is reduced to a semi-discrete gsG equation in the
case of $\nu=-1$ (Feng et al. Numer. Algorithms 2023). Furthermore,
$N$-soliton solutions to the semi- and fully discrete analogues of the gsG
equation in the determinant form are constructed. Dynamics of one- and two-
soliton solutions for the discrete gsG equations are discussed with plots. We
also investigate the reductions to the sine-Gordon (sG) equation and the short
pulse (SP) equation. By introducing an important parameter $c$, we demonstrate
that the gsG equation reduces to the sG equation and the SP equation, and the
discrete gsG equation reduces to the discrete sG equation and the discrete SP
equation, respectively, in the appropriate scaling limit. The limiting forms
of the $N$-soliton solutions to the gsG equation also correspond to those of
the sG equation and the SP equation.

Keywords: generalized sine-Gordon equation, short pulse equation, integrable
discretization, Hirota’s bilinear method

##  1 Introduction

In this paper, we are concerned with the integrable discretizations of a
generalized sine-Gordon (gsG) equation

$\displaystyle u_{tx}=\left(1+\nu\partial_{x}^{2}\right)\sin u,$ (1)

where $u=u(x,t)$ is a scalar-valued function, $\nu$ is a real parameter which
can be normalized into $\nu=\pm 1$, $\partial_{x}^{2}$ and the subscripts $t$
and $x$ appended to $u$ denote partial differentiation. The gsG equation (1)
was first derived by Fokas in 1995 using bi-Hamiltonian methods [1]. For
$\nu=-1$, the integrability was established by the Lax pair, and the initial
value problem for decaying initial data was solved by the inverse scattering
method [2]. Eigenfunctions of the Lax pair and traveling-wave solutions were
obtained through the Riemann-Hilbert formalism [2]. A variety of solutions,
including kinks, loop solitons, and breathers, were recognized from the
general soliton solution in parametric form constructed by Hirota’s bilinear
method [3]. The gsG equation (1) with $\nu=1$ was solved by Hirota’s bilinear
method [4]. It should be commented here that the structure of the solutions to
equation (1) with $\nu=1$ is significantly different from that with $\nu=-1$,
as it does not admit multi-valued solutions like loop solitons [3, 4]. Quite
recently, we constructed several semi-discrete analogues of the gsG equation
with $\nu=-1$ and presented the determinant formulae of $N$-soliton solutions
both for the gsG equation and the semi-discrete gsg equation in [5].

The study of discrete integrable systems, which is connected to many other
disciplines such as quantum field theory, numerical algorithms, random
matrices, and orthogonal and biorthogonal polynomials, has recently received a
lot of interest [6]. Compared to continuous integrable systems, there are far
fewer examples of discrete integrable systems and analytical tools available
for studying them. On the other hand, it is widely believed that discrete
integrable systems are more fundamental and universal than their continuous
counterparts. The authors have conducted a substantial amount of research in
finding integrable discretizations of soliton equations, including the short
pulse (SP) equation [7, 8], the (2+1)-dimensional Zakharov equation [9], the
Camassa-Holm equation [10, 11], the Degasperis-Proceli equation [12], and the
modified Camassa-Holm equation [13] via Hirota’s bilinear method. Building
upon the compatibility between an integrable system and its Bäcklund
transformation, a systematic procedure was proposed for obtaining discrete
versions of integrable PDEs using Hirota’s bilinear method [14].

It was demonstrated that the gsG equation with $\nu=\pm 1$ reduces to the SP
equation in the short wave limit and to the sine-Gordon (sG) equation in the
long wave limit [3, 4]. Thus, the gsG equation is an interesting soliton
equation that lies between the SP equation and the sG equation. Since the
semi-discrete and fully discrete sG equation has been well known [15, 16, 17,
18], and the integrable discretization of the SP equation was recently
proposed by one of the authors in [7], it would be quite an interesting
problem to construct the integrable discretizations of the gsG equation (1),
which is indeed the main motivation of our present paper.

Another challenging problem is looking for the reductions from the gsG
equation to the sG equation and the SP equation, both in the continuous case
and in the discrete case. Although the reductions from the gsG equation to the
sG equation and the SP equation were proposed in [3, 4], the reductions in the
discrete case differ substantially from those in the continuous case. The
problem lies in the fact that we not only apply scaling transformations to the
original variables in the equation, i.e., $u$, $x$, and $t$, but also
transform the new variables after hodograph transformation and parameters in
the $\tau$-function such as $y$, $\tau$, and $p$ in [3, 4]. It is difficult to
find the correspondence in the discrete case. To obtain the reductions, we
introduce an important parameter $c$, which can be viewed as the coefficient
of the Bäclund transformation between two sets of bilinear equations of the
two-dimensional Toda-lattice (2DTL) equation.

In this paper, we construct integrable fully-discrete analogues of the gsG
equation (1) with $\nu=\pm 1$ from two sets of discrete bilinear 2DTL
equations, and the corresponding determinant solutions are obtained. In
addition, the semi-discrete gsG equation with $\nu=\pm 1$ is constructed from
the fully discrete gsG, of which the semi-discrete gsG equation with $\nu=-1$
agrees with our previous result in [5]. Moreover, the connections of the
discrete gsG equation to the discrete sG equation and the discrete SP equation
are clarified by appropriate but different scaling limits.

The remainder of the paper is organized as follows. In Section 2, we review
the bilinear equations and determinant solutions of the gsG equation with
$\nu=1$, which can be reduced from the bilinear equations of 2DTL and their
Bäcklund transformation. We demonstrate that equation (1) reduces to the sG
equation and the SP equation with the scaling transformation on the original
variables and the corresponding limits of $c$. The limiting forms of the
$N$-soliton solution also correspond to the known solutions of the sG and SP
equations. In Section 3, starting with two sets of bilinear discrete 2DTL
equations, we derive a fully integrable discrete analog of the gsG equation
(1) with $\nu=\pm 1$ and present its $N$-soliton solutions. Two conserved
quantities of the fully discrete gsG equation are obtained. In Section 4, we
propose the semi-discrete gsG equation from two approaches: the fully discrete
gsG equation and the semi-discrete 2DTL equation, respectively. Reductions to
the corresponding discrete analogues of the sG equation and the SP equation
are also investigated. In Section 5, we present soliton solutions to the semi-
and fully discrete gsG equation and investigate their properties, focusing
mainly on one- and two-soliton solutions. Section 6 is devoted to a brief
summary and discussion. Some detailed proofs are given in Appendices.

##  2 From the 2DTL equation and its Bäcklund transformation to the gsG
equation with $\nu=1$

In this section, we show that the bilinear equations and the multisoliton
solution to the gsG equation (1) with $\nu=1$ given by Matsuno [4] can be
generated from two sets of bilinear equations of the 2DTL equation and its
Bäcklund transformation between them and its determinant solution through a
series of reductions and transformations, including the hodograph
transformation and dependent variable transformation.

###  2.1 A brief review of the gsG equation with $\nu=1$

Firstly, we give a brief review of the results in [4] about the bilinear form
of the gsG equation (1) with $\nu=1$

$\displaystyle u_{tx}=\left(1+\partial_{x}^{2}\right)\sin u.$ (2)

Through the new dependent variable r in accordance with the relation

$\displaystyle r^{2}=1-u_{x}^{2},\qquad(0<r<1),$ (3)

the gsG equation can be rewritten as

$\displaystyle r_{t}-(r\cos u)_{x}=0,$ (4)

which is exactly a conservation law of (2). Then we define the hodograph
transformation $(x,t)\rightarrow(y,\tau)$ by

$\displaystyle\mathrm{d}y=\lambda r\mathrm{d}x+\lambda r\cos
u\mathrm{d}t,\qquad\mathrm{d}\tau=\lambda^{-1}\mathrm{d}t,$ (5)

where $\lambda\in\mathbb{R}$ is a constant. The derivatives for $x$ and $t$
are then rewritten in terms of $y$ and $\tau$ as

$\displaystyle\frac{\partial}{\partial x}=\lambda r\frac{\partial}{\partial
y},\quad\frac{\partial}{\partial
t}=\lambda^{-1}\frac{\partial}{\partial\tau}+\lambda r\cos
u\frac{\partial}{\partial y}.$ (6)

With the new variables $y$ and $\tau$, (3) and (4) are recast into the form

$\displaystyle r^{2}=1-\lambda^{2}r^{2}u_{y}^{2},$ (7)
$\displaystyle\lambda^{-1}\left(\frac{1}{r}\right)_{\tau}+\lambda(\cos
u)_{y}=0,$ (8)

respectively. Further reduction is possible if one defines the variable
$\varphi$ by

$\displaystyle\lambda u_{y}=\sinh\varphi,\quad\varphi=\varphi(y,\tau).$ (9)

It follows from (7) and (9) that

$\displaystyle\frac{1}{r}=\cosh\varphi.$ (10)

Substituting (9) and (10) into equation (8), we find

$\displaystyle\varphi_{\tau}=\lambda\sin u.$ (11)

Let $\sigma$ and $\sigma^{\prime}$ be solutions of sG equation

$\displaystyle\sigma_{\tau y}=\sin\sigma,\quad\sigma=\sigma(\tau,y),$ (12)
$\displaystyle\sigma_{\tau
y}^{\prime}=\sin\sigma^{\prime},\quad\sigma^{\prime}=\sigma^{\prime}(\tau,y).$
(13)

Then we put

$\displaystyle u=\frac{1}{2}(\sigma+\sigma^{\prime}),$ (14)
$\displaystyle\varphi=\frac{1}{2\mathrm{i}}(\sigma-\sigma^{\prime}).$ (15)

In terms of $\sigma$ and $\sigma^{\prime}$, equations (9) and (11) can be
written as

$\displaystyle\frac{1}{2}(\sigma+\sigma^{\prime})_{y}=\lambda^{-1}\sinh\left(\frac{1}{2\mathrm{i}}(\sigma-\sigma^{\prime})\right)=-\mathrm{i}\lambda^{-1}\sin\frac{1}{2}(\sigma-\sigma^{\prime}),$
(16)
$\displaystyle\frac{1}{2}(\sigma-\sigma^{\prime})_{\tau}=\mathrm{i}\lambda\sin\frac{1}{2}(\sigma+\sigma^{\prime}).$
(17)

Introducing the dependent variable transformation

$\displaystyle\sigma=2\mathrm{i}\ln\frac{\bar{g}}{f},$ (18)
$\displaystyle\sigma^{\prime}=2\mathrm{i}\ln\frac{\bar{f}}{g},$ (19)

where $\bar{f}$ and $\bar{g}$ denote the complex conjugate of $f$ and $g$,
respectively. From (12) and (13), one can obtain

$\displaystyle D_{\tau}D_{y}f\cdot f=\frac{1}{2}(f^{2}-\bar{g}^{2}),$ (20)
$\displaystyle D_{\tau}D_{y}g\cdot g=\frac{1}{2}(g^{2}-\bar{f}^{2}),$ (21)

and their complex conjugates. And from (16)-(17), we have

$\displaystyle\mathrm{i}\left(\ln\frac{\bar{f}\bar{g}}{fg}\right)_{y}=\frac{1}{2\lambda}\left(\frac{g\bar{g}}{f\bar{f}}-\frac{f\bar{f}}{g\bar{g}}\right),$
(22)
$\displaystyle\left(\ln\frac{g\bar{g}}{f\bar{f}}\right)_{\tau}=\frac{\mathrm{i}\lambda}{2}\left(\frac{\bar{f}\bar{g}}{fg}-\frac{fg}{\bar{f}\bar{g}}\right).$
(23)

By using

$\displaystyle\frac{D_{\tau}f\cdot g}{fg}=\left(\ln\frac{f}{g}\right)_{\tau},$
(24)

Eqs. (22) and (23) can be represented as

$\displaystyle\mathrm{i}\lambda
D_{y}f\cdot\bar{f}-\frac{1}{2}(f\bar{f}-g\bar{g})=0,$ (25)
$\displaystyle\mathrm{i}\lambda
D_{y}g\cdot\bar{g}+\frac{1}{2}(g\bar{g}-f\bar{f})=0,$ (26)
$\displaystyle\mathrm{i}\lambda^{-1}D_{\tau}f\cdot
g+\frac{1}{2}(fg-\bar{f}\bar{g})=0,$ (27)
$\displaystyle\mathrm{i}\lambda^{-1}D_{\tau}\bar{g}\cdot\bar{f}+\frac{1}{2}(\bar{f}\bar{g}-fg)=0,$
(28)

here Eq. (28) is the complex conjugate of Eq. (27). In addition, the
definition of $u$ and $\varphi$ can be expressed as

$\displaystyle u=\mathrm{i}\ln\frac{\bar{f}\bar{g}}{fg},$ (29)
$\displaystyle\varphi=\ln\frac{g\bar{g}}{f\bar{f}}.$ (30)

###  2.2 From the 2DTL equation and its Bäcklund transformation to the
bilinear form of the gsG equation (2)

We give two sets of bilinear equations of the 2DTL equation with
$\tau$-functions $\tau_{n}$ and $\tau_{n}^{\prime}$ and a Bäcklund
transformation(BT) between them, respectively,

$\displaystyle\left(\frac{1}{2}D_{x_{-1}}D_{x_{1}}-1\right)\tau_{n}\cdot\tau_{n}=-\tau_{n+1}\tau_{n-1},$
(31)
$\displaystyle\left(\frac{1}{2}D_{x_{-1}}D_{x_{1}}-1\right)\tau_{n}^{\prime}\cdot\tau_{n}^{\prime}=-\tau_{n+1}^{\prime}\tau_{n-1}^{\prime},$
(32)
$\displaystyle\left(c^{-1}D_{x_{-1}}-1\right)\tau_{n}\cdot\tau_{n}^{\prime}+\tau_{n+1}\tau_{n-1}^{\prime}=0,$
(33)
$\displaystyle\left(cD_{x_{1}}-1\right)\tau_{n+1}\cdot\tau_{n}^{\prime}+\tau_{n}\tau_{n+1}^{\prime}=0.$
(34)

Here $c$ is a constant. As mentioned in [22], we can apply the BT recursively
and denote the $\tau$-functions of the $m$-th 2DTL equation as $\tau_{n,m}$.
Then we rewrite Eqs. (31)-(34) as

$\displaystyle\left(\frac{1}{2}D_{x_{-1}}D_{x_{1}}-1\right)\tau_{n,m}\cdot\tau_{n,m}=-\tau_{n+1,m}\tau_{n-1,m},$
(35)
$\displaystyle\left(c^{-1}D_{x_{-1}}-1\right)\tau_{n,m}\cdot\tau_{n,m+1}+\tau_{n+1,m}\tau_{n-1,m+1}=0,$
(36)
$\displaystyle\left(cD_{x_{1}}-1\right)\tau_{n+1,m}\cdot\tau_{n,m+1}+\tau_{n,m}\tau_{n+1,m+1}=0,$
(37)

where we have the following correspondence: $\tau_{n}\equiv\tau_{n,m}$,
$\tau^{\prime}_{n}\equiv\tau_{n,m+1}$. The above equations (35)-(37) have
exact solutions in Casorati determinant form with arbitrary parameters as
follows.

######  Lemma 2.1.

The bilinear equations (35)-(37) have the following Casorati-type determinant
solution

$\displaystyle\tau_{n,m}(x_{-1},x_{1})$
$\displaystyle=\left|\begin{array}[]{cccc}\phi_{n}^{(1)}(m)&\phi_{n+1}^{(1)}(m)&\cdots&\phi_{n+N-1}^{(1)}(m)\\\
\phi_{n}^{(2)}(m)&\phi_{n+1}^{(2)}(m)&\cdots&\phi_{n+N-1}^{(2)}(m)\\\
\cdots&\cdots&\cdots&\cdots\\\
\phi_{n}^{(N)}(m)&\phi_{n+1}^{(N)}(m)&\cdots&\phi_{n+N-1}^{(N)}(m)\end{array}\right|,$
(42)

where

$\displaystyle\phi_{n}^{(i)}(m)=c_{i}p_{i}^{n}\left(1-cp_{i}\right)^{m}e^{\xi_{i}}+d_{i}q_{i}^{n}\left(1-cq_{i}\right)^{m}e^{\eta_{i}},$
(43)
$\displaystyle\xi_{i}=p_{i}x_{1}+p_{i}^{-1}x_{-1}+\xi_{i0},\quad\eta_{i}=q_{i}x_{1}+q_{i}^{-1}x_{-1}+\eta_{i0}.$
(44)

Here, $c_{i},d_{i},p_{i},q_{i},\xi_{i0}$ and $\eta_{i0}$ are the arbitrary
parameters that can take either real or complex values.

In addition to the Casorati determinant solution, the $\tau$-functions can
also be expressed by the Gram-type determinant, which is given by the
following lemma.

######  Lemma 2.2.

The following Gram-type determinants satisfy bilinear equations (35)-(37)

$\displaystyle\tau_{n,m}(x_{-1},x_{1})=\left|m_{ij}^{n,m}\right|_{N\times
N}=\left|c_{ij}+\frac{1}{p_{i}+q_{j}}\left(-\frac{p_{i}}{q_{j}}\right)^{n}\left(\frac{1-cp_{i}}{1+cq_{j}}\right)^{m}e^{\xi_{i}+\eta_{j}}\right|_{N\times
N},$ (45)

where

$\displaystyle\xi_{i}=p_{i}x_{1}+p_{i}^{-1}x_{-1}+\xi_{i0},\quad\eta_{j}=q_{j}x_{1}+q_{j}^{-1}x_{-1}+\eta_{j0}$
(46)

Here, $p_{i},q_{j},\xi_{i0}$ and $\eta_{j0}$ are the arbitrary parameters that
can take either real or complex values and $N\in\mathbb{R}$.

Next we are ready to obtain $N$-soliton solution of the gsG equation (2).
Firstly, we set $c=\lambda\mathrm{i}\ (\lambda\in\mathbb{R})$.

Case 1: for real $p_{i}$, $q_{i}$

We impose restrictions on the parameters

$\displaystyle
p_{i}=-q_{i},\,d_{i}=\mathrm{i}\left({\frac{1-cp_{i}}{1-cq_{i}}}\right)^{\frac{1}{2}}c_{i},$
(47)

for Casorati-type solution or

$\displaystyle
p_{i}=q_{i},\,c_{ij}=\mathrm{i}\left({\frac{1-cp_{i}}{1+cq_{j}}}\right)^{\frac{1}{2}}\delta_{ij},$
(48)

for Gram-type solution. In addition, we take variable transformations
$y=2x_{1},\,\tau=2x_{-1}$. As a result, for Casorati-type solution we have

$\displaystyle\phi_{n+1}^{(i)}(0)$
$\displaystyle=c_{i}p_{i}^{n+1}e^{\xi_{i}}+d_{i}q_{i}^{n+1}e^{\eta_{i}}$
$\displaystyle=c_{i}p_{i}^{n+1}e^{\xi_{i}}\left[1+\mathrm{i}(-1)^{n+1}\left({\frac{1-cp_{i}}{1-cq_{i}}}\right)^{\frac{1}{2}}e^{\eta_{i}-\xi_{i}}\right]$
$\displaystyle=c_{i}p_{i}^{n+1}e^{\xi_{i}}\left[1-\mathrm{i}(-1)^{n}\left({\frac{1-\mathrm{i}\lambda
p_{i}}{1+\mathrm{i}\lambda
p_{i}}}\right)^{\frac{1}{2}}e^{-p_{i}y-\tau/p_{i}+\eta_{i0}-\xi_{i0}}\right],$

and

$\displaystyle\phi_{n}^{(i)}(1)$
$\displaystyle=c_{i}p_{i}^{n}(1-cp_{i})e^{\xi_{i}}+d_{i}q_{i}^{n}(1-cq_{i})e^{\eta_{i}}$
$\displaystyle=c_{i}p_{i}^{n}(1-cp_{i})e^{\xi}_{i}\left[1+\mathrm{i}(-1)^{n}\left(\frac{1+\mathrm{i}\lambda
p_{i}}{1-\mathrm{i}\lambda
p_{i}}\right)^{\frac{1}{2}}e^{-p_{i}y-\tau/p_{i}+\eta_{i0}-\xi_{i0}}\right].$

Thus we know

$\displaystyle\phi^{(i)}_{n+1}(0)\Bumpeq\bar{\phi}^{(i)}_{n}(1),\,\phi^{(i)}_{n+1}(1)\Bumpeq\bar{\phi}^{(i)}_{n}(0),$
(49)

which imply the relations

$\displaystyle\tau_{n+1,0}\Bumpeq\bar{\tau}_{n,1},\qquad\tau_{n+1,1}\Bumpeq\bar{\tau}_{n,0}.$
(50)

Relations (50) means both $\tau_{n}(0)$ and $\tau_{n}(1)$ are 2-period
sequences. Here $\Bumpeq$ means two $\tau$ functions are equivalent up to a
constant multiple and $\bar{\tau}$ denotes complex conjugate of $\tau$. The
same result appears for the relations between $\tau$ functions with Gram-type
determinant form, which we omit here. In this case, the kink and anti-kink
solutions are obtained.

Case 2: for complex $p_{i}$, $q_{i}$

We impose restrictions on the parameters of $\tau$ functions

$\displaystyle
p_{i}=-q_{i},\,d_{i}=\mathrm{i}\left({\frac{1-cp_{i}}{1-cq_{i}}}\right)^{\frac{1}{2}}c_{i},\,p_{2i-1}=\bar{p}_{2i},\,N=2M,$
(51)

for Casorati-type solution or

$\displaystyle
p_{i}=q_{i},\,c_{ij}=\mathrm{i}\left({\frac{1-cp_{i}}{1+cq_{j}}}\right)^{\frac{1}{2}}\delta_{ij},\
p_{2i-1}=\bar{p}_{2i},\,N=2M,$ (52)

for Gram-type solution. By taking $y=2x_{1},\,\tau=2x_{-1}$, from Case 1, we
know that

$\displaystyle\phi_{n+1}^{(i)}(0)=p_{i}^{n+1}e^{\xi_{i}}\left[1-\mathrm{i}(-1)^{n}\sqrt{\frac{1-\mathrm{i}\lambda
p_{i}}{1+\mathrm{i}\lambda
p_{i}}}e^{-p_{i}y-\tau/p_{i}+\eta_{i0}-\xi_{i0}}\right],$
$\displaystyle\phi_{n}^{(j)}(1)=p_{j}^{n}(1-cp_{j})e^{\xi_{j}}\left[1+\mathrm{i}(-1)^{n}\sqrt{\frac{1+\mathrm{i}\lambda
p_{j}}{1-\mathrm{i}\lambda
p_{j}}}e^{-p_{j}y-\tau/p_{j}+\eta_{j0}-\xi_{j0}}\right].$

Thus we can obtain

$\displaystyle\phi_{n+1}^{(2i-1)}(0)\Bumpeq\bar{\phi}_{n}^{(2i)}(1),\,\phi_{n+1}^{(2i)}(0)\Bumpeq\bar{\phi}_{n}^{(2i-1)}(1),$
(53)

and

$\displaystyle\phi_{n+1}^{(2i-1)}(1)\Bumpeq\bar{\phi}_{n}^{(2i)}(0),\,\phi_{n+1}^{(2i)}(1)\Bumpeq\bar{\phi}_{n}^{(2i-1)}(0),$
(54)

which also correspond to the relation (50). In this case, the breather
solutions are obtained.

######  Remark 2.1.

Reductions from (35)-(37) to the bilinear form of the gsG equation with
$\nu=1$ differ from the case with $\nu=-1$ we proposed in [5], in which
$\tau_{n+1,0}\Bumpeq\bar{\tau}_{n,0},\ \tau_{n+1,1}\Bumpeq\bar{\tau}_{n,1}$.
Actually, reductions we introduce here are relatively rare in other equations.

Naturally, we can get kink-breather solutions by mixing Case 1 and Case 2.
Moreover, by substitution

$\displaystyle\tau_{00}=f,\,\tau_{01}=g,$ (55)

equations (35)-(37) can be recast into

$\displaystyle D_{\tau}D_{y}f\cdot f=\frac{1}{2}(f^{2}-\bar{g}^{2}),$ (56)
$\displaystyle D_{\tau}D_{y}g\cdot g=\frac{1}{2}(g^{2}-\bar{f}^{2}),$ (57)
$\displaystyle\mathrm{i}\lambda
D_{y}f\cdot\bar{f}-\frac{1}{2}(f\bar{f}-g\bar{g})=0,$ (58)
$\displaystyle\mathrm{i}\lambda
D_{y}g\cdot\bar{g}+\frac{1}{2}(g\bar{g}-f\bar{f})=0,$ (59)
$\displaystyle\mathrm{i}\lambda^{-1}D_{\tau}f\cdot
g+\frac{1}{2}(fg-\bar{f}\bar{g})=0,$ (60)
$\displaystyle\mathrm{i}\lambda^{-1}D_{\tau}\bar{g}\cdot\bar{f}+\frac{1}{2}(\bar{f}\bar{g}-fg)=0,$
(61)

which are nothing but the bilinear equations of the gsG equation (2). From
equation (58)-(61), we have

$\displaystyle\mathrm{i}\lambda\left(\ln\frac{f}{\bar{f}}\right)_{y}=\frac{1}{2}-\frac{1}{2}\frac{g\bar{g}}{f\bar{f}},$
(62)
$\displaystyle\mathrm{i}\lambda\left(\ln\frac{g}{\bar{g}}\right)_{y}=-\frac{1}{2}+\frac{1}{2}\frac{f\bar{f}}{g\bar{g}},$
(63)
$\displaystyle\mathrm{i}\lambda^{-1}\left(\ln\frac{f}{g}\right)_{\tau}=-\frac{1}{2}+\frac{1}{2}\frac{\bar{f}\bar{g}}{fg},$
(64)
$\displaystyle\mathrm{i}\lambda^{-1}\left(\ln\frac{\bar{f}}{\bar{g}}\right)_{\tau}=\frac{1}{2}-\frac{1}{2}\frac{fg}{\bar{f}\bar{g}},$
(65)

that lead to

$\displaystyle x_{y}=\frac{1}{\lambda
r}=\lambda^{-1}\cosh\varphi=\frac{1}{2\lambda}\left(\frac{g\bar{g}}{f\bar{f}}+\frac{f\bar{f}}{g\bar{g}}\right)=\lambda^{-1}+\mathrm{i}\left(\ln\frac{\bar{f}g}{f\bar{g}}\right)_{y},$
(66) $\displaystyle x_{\tau}=-\lambda^{2}rx_{y}\cos u=-\lambda\cos
u=-\frac{\lambda}{2}\left(\frac{\bar{f}\bar{g}}{fg}+\frac{fg}{\bar{f}\bar{g}}\right)=-\lambda+\mathrm{i}\left(\ln\frac{\bar{f}g}{f\bar{g}}\right)_{\tau}.$
(67)

Thus we obtain the expression for $x$ by tau functions

$\displaystyle
x=\lambda^{-1}y-\lambda\tau+\mathrm{i}\ln\frac{\bar{f}g}{f\bar{g}},$ (68)

Summarizing the above results, the determinant ($N$-soliton) solution of the
gsG equation is given by the following theorem.

######  Theorem 2.1.

The parametric form for the soliton and breather solution of the gsG equation
(2) is

$\displaystyle u=\mathrm{i}\ln\frac{\bar{f}\bar{g}}{fg},$ (69) $\displaystyle
x=\lambda^{-1}y-\lambda\tau+\mathrm{i}\ln\frac{\bar{f}g}{f\bar{g}},\
t=\lambda\tau,\ \lambda\in\mathbb{R},$ (70)

where $f$ and $g$ are the determinants given by (55) and $\tau_{n,m}$ can be
written either as a Casorati-type determinant

$\displaystyle\tau_{n,m}$
$\displaystyle=\left|\begin{array}[]{cccc}\phi_{n}^{(1)}(m)&\phi_{n+1}^{(1)}(m)&\cdots&\phi_{n+N-1}^{(1)}(m)\\\
\phi_{n}^{(2)}(m)&\phi_{n+1}^{(2)}(m)&\cdots&\phi_{n+N-1}^{(2)}(m)\\\
\cdots&\cdots&\cdots&\cdots\\\
\phi_{n}^{(N)}(m)&\phi_{n+1}^{(N)}(m)&\cdots&\phi_{n+N-1}^{(N)}(m)\end{array}\right|,$
(75)

where

$\displaystyle\phi_{n}^{(i)}(m)=p_{i}^{n}\left(1-\mathrm{i}\lambda
p_{i}\right)^{m}e^{\frac{p_{i}}{2}y+\frac{1}{2p_{i}}\tau+\xi_{i0}}+\mathrm{i}\sqrt{\frac{1-\mathrm{i}\lambda
p_{i}}{1+\mathrm{i}\lambda p_{i}}}(-p_{i})^{n}\left(1+\mathrm{i}\lambda
p_{i}\right)^{m}e^{-\frac{p_{i}}{2}y-\frac{1}{2p_{i}}\tau+\eta_{i0}},\quad
i=1,2,\cdots,N,$ (76) $\displaystyle p_{2i-1}=\bar{p}_{2i},\quad
i=1,2,\cdots,M,$ (77) $\displaystyle
p_{i}\in\mathbb{R},\,i=2M+1,2M+2,\cdots,N,$ (78)

or a Gram-type determinant

$\displaystyle\tau_{n,m}=\left|m_{ij}^{n,m}\right|_{N\times
N}=\left|\mathrm{i}\left({\frac{1-\mathrm{i}\lambda p_{i}}{1+\mathrm{i}\lambda
p_{j}}}\right)^{\frac{1}{2}}\delta_{ij}+\frac{1}{p_{i}+p_{j}}\left(-\frac{p_{i}}{p_{j}}\right)^{n}\left(\frac{1-\mathrm{i}\lambda
p_{i}}{1+\mathrm{i}\lambda
p_{j}}\right)^{m}e^{\xi_{i}+\eta_{j}}\right|_{N\times N},$ (79)

where

$\displaystyle\xi_{i}=\frac{p_{i}}{2}y+\frac{1}{2p_{i}}\tau+\xi_{i0},\quad\eta_{j}=\frac{p_{j}}{2}y+\frac{1}{2p_{j}}\tau+\eta_{j0},$
(80) $\displaystyle p_{2i-1}=\bar{p}_{2i},\quad i=1,2,\cdots,M,$ (81)
$\displaystyle p_{i}\in\mathbb{R},\,i=2M+1,2M+2,\cdots,N.$ (82)

######  Remark 2.2.

Similar to the deduction of the gsG equation with $\nu=1$, one can generalize
the Theorem 1 in [5]. The parametric form for the $N$-soliton solution of the
gsG equation (1) with $\nu=-1$ is

$\displaystyle u=\mathrm{i}\ln\frac{\bar{f}\bar{g}}{fg},$ (83) $\displaystyle
x=c^{-1}y+c\tau+\ln\frac{\bar{g}g}{\bar{f}f},\ t=c\tau,\ c\in\mathbb{R},$ (84)
$\displaystyle f=\tau_{00},\ g=\tau_{01},$ (85)

and $\tau_{n,m}$ can be written as a Casorati-type determinant

$\displaystyle\tau_{n,m}$
$\displaystyle=\left|\begin{array}[]{cccc}\phi_{n}^{(1)}(m)&\phi_{n+1}^{(1)}(m)&\cdots&\phi_{n+N-1}^{(1)}(m)\\\
\phi_{n}^{(2)}(m)&\phi_{n+1}^{(2)}(m)&\cdots&\phi_{n+N-1}^{(2)}(m)\\\
\cdots&\cdots&\cdots&\cdots\\\
\phi_{n}^{(N)}(m)&\phi_{n+1}^{(N)}(m)&\cdots&\phi_{n+N-1}^{(N)}(m)\end{array}\right|,$
(90)

with

$\displaystyle\phi_{n}^{(i)}(m)=p_{i}^{n}\left(1-cp_{i}\right)^{m}e^{\frac{p_{i}}{2}y+\frac{1}{2p_{i}}\tau+\xi_{i0}}+\mathrm{i}(-p_{i})^{n}\left(1+cp_{i}\right)^{m}e^{-\frac{p_{i}}{2}y-\frac{1}{2p_{i}}\tau+\eta_{i0}},\quad
i=1,2,\cdots,N.$ (91)

Obviously, the result in this remark is equivalent to that in Theorem 1 of [5]
when $c=1$.

######  Remark 2.3.

The determinant solutions we obtained above are consistent with the solutions
given in [4].

###  2.3 Reduction to the sG and SP equation

In [3] and [4], Matsuno demonstrated that the gsG equation is reduced to the
sG equation in the long wave limit and to the SP equation in the short wave
limit. Here we introduce the scaling parameter $c$ (or $\lambda$) in the
hodograph transformation and the $\tau$-function, and give another kind of
reduction.

####  2.3.1 Reduction to the sine-Gordon equation

The sG equation

$\displaystyle u_{xt}=\sin u,\ \quad u\equiv u(x,t)\in\mathbb{R},\
\quad(x,t)\in\mathbb{R}^{2},$ (92)

is a fundamental model in the integrable system, which appears in a number of
disciplines of physics including magnetic flux propagation [23, 24], one-
dimensional classical field theory [25, 26], and nonlinear optics [27]. In
this part, we take into account the reductions from the gsG equation with
$\nu=\pm 1$ to the sG equation in the continuous case through some scaling
transformations.

(I) From the gsG equation with $\nu=-1$ to the sG equation. In this part of
reduction, we take $c\in\mathbb{R}$ as a small parameter. The matrix elements
of the $\tau$-function for the gsG equation we proposed in [5] can be written
as

$\displaystyle\phi_{n}^{(i)}(1)$
$\displaystyle=p_{i}^{n}(1-cp_{i})e^{\frac{p_{i}}{2}y+\frac{1}{2p_{i}}\tau+\xi_{i0}}+\mathrm{i}(-p_{i})^{n}\left(1+cp_{i}\right)e^{-\frac{p_{i}}{2}y-\frac{1}{2p_{i}}\tau+\eta_{i0}}$
$\displaystyle=\phi_{n}^{(i)}(0)-c\phi_{n+1}^{(i)}(0),$ (93)

which means

$\displaystyle g=\tau_{0}(1)=f+\mathnormal{O}(c),\
\bar{g}=\tau_{1}(1)=\bar{f}+\mathnormal{O}(c).$ (94)

Then we can rewrite the dependent variable transformations and introduce the
scaling transformation

$\displaystyle
u=\mathrm{i}\ln\frac{\bar{f}\bar{g}}{fg}=2\mathrm{i}\ln\frac{\bar{f}}{f}+\mathnormal{O}(c),$
(95)
$\displaystyle\phi=\mathrm{i}\ln\frac{\bar{f}g}{f\bar{g}}=\mathnormal{O}(c),$
(96)
$\displaystyle\hat{x}=cx=y+c^{2}\tau+c\ln\frac{g\bar{g}}{f\bar{f}}=y+\mathnormal{O}(c^{2}),$
(97) $\displaystyle\hat{t}=c^{-1}t=\tau.$ (98)

In addition, the gsG equation (1) with $\nu=-1$ can be recast into

$\displaystyle u_{\hat{x}\hat{t}}=(1-c^{2}\partial_{\hat{x}}^{2})\sin u.$ (99)

With the scaling limit $c\rightarrow 0$, equation (99) becomes

$\displaystyle u_{\hat{x}\hat{t}}=\sin u,$ (100)

which is the well-known sG equation. And the dependent variable
transformation, as well as the $\tau$-function $f$, also reduces to the usual
form of the $N$-soliton solutions of the sG equation [28, 29, 30].

It should be point out that the transformation (95)-(98) are agree with the
scaled variables introducing by Matsuno in [3]. The difference is that we
introduce the parameter $c$ in the hodograph transformation, thus the
$\tau$-function can be transformed naturally without the scaling of variables
$y,\ \tau$ and the parameter $p$.

(II) From the gsG equation with $\nu=1$ to the sG equation. In this part of
reduction, we take $\lambda\in\mathbb{R}$ as a small parameter. Similar to the
case $\nu=-1$, recall that

$\displaystyle\phi_{n}^{(i)}(1)$
$\displaystyle=p_{i}^{n}\left(1-\mathrm{i}\lambda
p_{i}\right)e^{\frac{p_{i}}{2}y+\frac{1}{2p_{i}}\tau+\xi_{i0}}+\mathrm{i}\sqrt{\frac{1-\mathrm{i}\lambda
p_{i}}{1+\mathrm{i}\lambda p_{i}}}(-p_{i})^{n}\left(1+\mathrm{i}\lambda
p_{i}\right)e^{-\frac{p_{i}}{2}y-\frac{1}{2p_{i}}\tau+\eta_{i0}},$
$\displaystyle=p_{i}^{n}e^{\frac{p_{i}}{2}y+\frac{1}{2p_{i}}\tau+\xi_{i0}}+\mathrm{i}(-p_{i})^{n}e^{-\frac{p_{i}}{2}y-\frac{1}{2p_{i}}\tau+\eta_{i0}}+\mathnormal{O}(\lambda),$
$\displaystyle=\hat{\phi}_{n}^{(i)}(0)+\mathnormal{O}(\lambda),$ (101)
$\displaystyle\phi_{n}^{(i)}(0)$
$\displaystyle=p_{i}^{n}e^{\frac{p_{i}}{2}y+\frac{1}{2p_{i}}\tau+\xi_{i0}}+\mathrm{i}\sqrt{\frac{1-\mathrm{i}\lambda
p_{i}}{1+\mathrm{i}\lambda
p_{i}}}(-p_{i})^{n}e^{-\frac{p_{i}}{2}y-\frac{1}{2p_{i}}\tau+\eta_{i0}},$
$\displaystyle=\hat{\phi}_{n}^{(i)}(0)+\mathnormal{O}(\lambda),$ (102)

which lead to

$\displaystyle f=\hat{f}+\mathnormal{O}(\lambda),\
g=\hat{f}+\mathnormal{O}(\lambda),\
\bar{f}=\bar{\hat{f}}+\mathnormal{O}(\lambda),\
\bar{g}=\bar{\hat{f}}+\mathnormal{O}(\lambda)$ (103)

here $\hat{f}$ is the $\tau$-function of the sG equation. Thus we have

$\displaystyle
u=2\mathrm{i}\ln\frac{\bar{\hat{f}}}{\hat{f}}+\mathnormal{O}(\lambda),\
\varphi=\mathnormal{O}(\lambda),\ \hat{x}=\lambda
x=y+\mathnormal{O}(\lambda^{2}),\ \hat{t}=\lambda^{-1}t=\tau,$ (104)

And the gsG equation (1) with $\nu=1$ becomes

$\displaystyle u_{\hat{x}\hat{t}}=(1+\lambda^{2}\partial_{\hat{x}}^{2})\sin
u.$ (105)

The gsG equation is converted into the sG equation with the scaling limit
$\lambda\rightarrow 0$, as well as its solutions. Here those transformations
are agree with transformations introduced in [4].

####  2.3.2 Reduction to the short pulse equation

The short pulse (SP) equation

$\displaystyle u_{xt}=u-\frac{\sigma}{6}\left(u^{3}\right)_{xx},$ (106)

was derived to describe the propagation of ultra-short optical pulses in
nonlinear media by Schäfer and Wayne when $\sigma=-1$ [34]. Here, the real-
valued function $u=u(x,t)$ represent the magnitude of the electric field, and
the subscripts $t$ and $x$ signify partial differentiation. The SP equation
has also been developed as an integrable differential equation linked to
pseudospherical surfaces outside of the context of nonlinear optics [35]. When
$\sigma=1$, equation (106)

$\displaystyle u_{xt}=u-\frac{1}{6}\left(u^{3}\right)_{xx},$ (107)

was shown to model the evolution of ultra-short pulses in the band gap of
nonlinear metamaterials [36]. Here, we show that applying the right scaling
limit and variable transformations results in the gsG equation with $\nu=\pm
1$ being reduced to the SP equation (106) with $\sigma=\pm 1$ in the
continuous case.

(I) From the gsG equation with $\nu=-1$ to the SP equation with $\sigma=-1$.
In this part of reduction, we take $c$ as a big parameter (or
$\epsilon=\frac{1}{c}$ small). The matrix elements of the $\tau$-function for
the gsG equation with $\nu=-1$ in [5] are

$\displaystyle\phi_{n}^{(i)}(1)$
$\displaystyle=p_{i}^{n}(1-cp_{i})e^{\frac{p_{i}}{2}y+\frac{1}{2p_{i}}\tau+\xi_{i0}}+\mathrm{i}(-p_{i})^{n}\left(1+cp_{i}\right)e^{-\frac{p_{i}}{2}y-\frac{1}{2p_{i}}\tau+\eta_{i0}}$
$\displaystyle\propto\phi_{n+1}^{(i)}(0)-2\epsilon\frac{\partial}{\partial\tau}\phi_{n+1}^{(i)}(0),$
(108)

from which we can obtain

$\displaystyle
g=\tau_{0}(1)\propto\bar{f}-2\epsilon\frac{\partial}{\partial\tau}\bar{f}+\mathnormal{O}(\epsilon^{2}),$
(109)
$\displaystyle\bar{g}\propto{f}-2\epsilon\frac{\partial}{\partial\tau}{f}+\mathnormal{O}(\epsilon^{2}).$
(110)

Then we rewrite the dependent variable transformations and introduce new
variable as

$\displaystyle\epsilon\hat{u}=u=\mathrm{i}\ln\frac{\bar{f}\bar{g}}{{f}{g}}=\mathrm{i}\ln\frac{\bar{f}(f-2\epsilon\frac{\partial}{\partial\tau}{f})}{{f}(\bar{f}-2\epsilon\frac{\partial}{\partial\tau}\bar{f})}+\mathnormal{O}(\epsilon^{2}),$
(111)
$\displaystyle\phi=\mathrm{i}\ln\frac{\bar{f}{g}}{{f}\bar{g}}=\mathrm{i}\ln\frac{\bar{f}(\bar{f}-2\epsilon\frac{\partial}{\partial\tau}\bar{f})}{{f}(f-2\epsilon\frac{\partial}{\partial\tau}{f})}+\mathnormal{O}(\epsilon^{2}),$
(112) $\displaystyle\epsilon\hat{x}=x-t=\epsilon
y+\ln\frac{g\bar{g}}{f\bar{f}}=\epsilon
y+\ln\frac{(\bar{f}-2\epsilon\frac{\partial}{\partial\tau}\bar{f})(f-2\epsilon\frac{\partial}{\partial\tau}{f})}{f\bar{f}}+\mathnormal{O}(\epsilon^{2}),$
(113) $\displaystyle\hat{t}=\epsilon t=\tau.$ (114)

The gsG equation (1) with $\nu=-1$ can be recast into

$\displaystyle\epsilon\hat{u}_{\hat{t}\hat{x}}=\epsilon\left(\hat{u}+\frac{1}{6}\left(\hat{u}^{3}\right)_{\hat{x}\hat{x}}\right)+\mathnormal{O}\left(\epsilon^{3}\right).$
(115)

Dividing both sides of (115) by $\epsilon$ and taking $\epsilon\rightarrow 0$
in (111)-(115), we arrive at

$\displaystyle\hat{u}=2\mathrm{i}\left(\ln\frac{\bar{f}}{f}\right)_{\tau},\
\phi=2\mathrm{i}\ln\frac{\bar{f}}{f}$ $\displaystyle,\
\hat{x}=y-2(\ln(f\bar{f}))_{\tau},\ \hat{t}=\tau,$ (116)
$\displaystyle\hat{u}_{\hat{t}\hat{x}}=\hat{u}$
$\displaystyle+\frac{1}{6}\left(\hat{u}^{3}\right)_{\hat{x}\hat{x}}.$ (117)

Here the SP equation (106) is derived and its parametric representation
$\tau$-function are equivalent to those in [7].

(II) From the gsG equation with $\nu=1$ to the SP equation with $\sigma=1$.
Similar to the case with $\nu=-1$, we take $\epsilon=\frac{1}{\lambda}$ as a
small parameter. The $\tau$-function in Theorem 2.1 can be written as

$\displaystyle\phi_{n}^{(i)}(0)$
$\displaystyle=p_{i}^{n}e^{\frac{p_{i}}{2}y+\frac{1}{2p_{i}}\tau+\xi_{i0}}+\mathrm{i}\sqrt{\frac{1-\mathrm{i}\lambda
p_{i}}{1+\mathrm{i}\lambda
p_{i}}}(-p_{i})^{n}e^{-\frac{p_{i}}{2}y-\frac{1}{2p_{i}}\tau+\eta_{i0}},$
$\displaystyle\propto\hat{\phi}_{n+1}^{(i)}-{\mathrm{i}\epsilon}\frac{\partial}{\partial\tau}\hat{\phi}_{n+1}^{(i)}+\mathnormal{O}(\epsilon^{2}),$
(118) $\displaystyle\phi_{n}^{(i)}(1)$
$\displaystyle=p_{i}^{n}\left(1-\mathrm{i}\lambda
p_{i}\right)e^{\frac{p_{i}}{2}y+\frac{1}{2p_{i}}\tau+\xi_{i0}}+\mathrm{i}\sqrt{\frac{1-\mathrm{i}\lambda
p_{i}}{1+\mathrm{i}\lambda p_{i}}}(-p_{i})^{n}\left(1+\mathrm{i}\lambda
p_{i}\right)e^{-\frac{p_{i}}{2}y-\frac{1}{2p_{i}}\tau+\eta_{i0}},$
$\displaystyle\propto\hat{\phi}_{n}^{(i)}-{\mathrm{i}\epsilon}\frac{\partial}{\partial\tau}\hat{\phi}_{n}^{(i)}+\mathnormal{O}(\epsilon^{2}),$
(119)

if we define

$\displaystyle\hat{\phi}_{n}^{(i)}=p_{i}^{n}e^{\frac{p_{i}}{2}y+\frac{1}{2p_{i}}\tau+\xi_{i0}}+(-p_{i})^{n}e^{-\frac{p_{i}}{2}y-\frac{1}{2p_{i}}\tau+\eta_{i0}},$
(120)

and

$\displaystyle\hat{f}=\left|\begin{array}[]{cccc}\hat{\phi}_{n+1}^{(1)}&\hat{\phi}_{n+2}^{(1)}&\cdots&\hat{\phi}_{n+N}^{(1)}\\\
\hat{\phi}_{n+1}^{(2)}&\hat{\phi}_{n+2}^{(2)}&\cdots&\hat{\phi}_{n+N}^{(2)}\\\
\cdots&\cdots&\cdots&\cdots\\\
\hat{\phi}_{n+1}^{(N)}&\hat{\phi}_{n+2}^{(N)}&\cdots&\hat{\phi}_{n+N}^{(N)}\end{array}\right|,\
\hat{g}=\left|\begin{array}[]{cccc}\hat{\phi}_{n}^{(1)}&\hat{\phi}_{n+1}^{(1)}&\cdots&\hat{\phi}_{n+N-1}^{(1)}\\\
\hat{\phi}_{n}^{(2)}&\hat{\phi}_{n+1}^{(2)}&\cdots&\hat{\phi}_{n+N-1}^{(2)}\\\
\cdots&\cdots&\cdots&\cdots\\\
\hat{\phi}_{n}^{(N)}&\hat{\phi}_{n+1}^{(N)}&\cdots&\hat{\phi}_{n+N-1}^{(N)}\end{array}\right|.$
(129)

Furthermore, one obtains

$\displaystyle
f=\hat{f}-{\mathrm{i}\epsilon}\frac{\partial}{\partial\tau}\hat{f}+\mathnormal{O}(\epsilon^{2}),\
\bar{f}=\hat{f}+{\mathrm{i}\epsilon}\frac{\partial}{\partial\tau}\hat{f}+\mathnormal{O}(\epsilon^{2}),$
(130) $\displaystyle
g=\hat{g}+{\mathrm{i}\epsilon}\frac{\partial}{\partial\tau}\hat{g}+\mathnormal{O}(\epsilon^{2}),\
\bar{g}=\hat{g}-{\mathrm{i}\epsilon}\frac{\partial}{\partial\tau}\hat{g}+\mathnormal{O}(\epsilon^{2}).$
(131)

Here we introduce new variables as

$\displaystyle\epsilon\hat{u}=u=\mathrm{i}\ln\frac{\bar{f}\bar{g}}{{f}{g}}=\mathrm{i}\ln\frac{(\hat{f}+{\mathrm{i}\epsilon}\hat{f}_{\tau})(\hat{g}-{\mathrm{i}\epsilon}\hat{g}_{\tau})}{(\hat{f}-{\mathrm{i}\epsilon}\hat{f}_{\tau})(\hat{g}+{\mathrm{i}\epsilon}\hat{g}_{\tau})}+\mathnormal{O}(\epsilon^{2}),$
(132)
$\displaystyle\varphi=\ln\frac{\bar{g}g}{f\bar{f}}=\ln\frac{(\hat{g}-{\mathrm{i}\epsilon}\hat{g}_{\tau})(\hat{g}+{\mathrm{i}\epsilon}\hat{g}_{\tau})}{(\hat{f}-{\mathrm{i}\epsilon}\hat{f}_{\tau})(\hat{f}+{\mathrm{i}\epsilon}\hat{f}_{\tau})}+\mathnormal{O}(\epsilon^{2}),$
(133) $\displaystyle{\epsilon\hat{x}=x+t=\epsilon
y+\mathrm{i}\ln\frac{(\hat{f}+{\mathrm{i}\epsilon}\hat{f}_{\tau})(\hat{g}+{\mathrm{i}\epsilon}\hat{g}_{\tau})}{(\hat{f}-{\mathrm{i}\epsilon}{f}_{\tau})(\hat{g}-{\mathrm{i}\epsilon}\hat{g}_{\tau})}+\mathnormal{O}(\epsilon^{2})},$
(134) $\displaystyle\hat{t}=\tau,$ (135)

then the scaling limit $\epsilon\rightarrow 0$ leads to

$\displaystyle\hat{u}=2\left(\ln\frac{\hat{g}}{\hat{f}}\right)_{\tau},\
\varphi=2\ln\frac{\hat{g}}{\hat{f}},\ \hat{x}=y-2(\ln\hat{f}\hat{g})_{\tau},\
\hat{t}=\tau,$ (136)

and

$\displaystyle\hat{u}_{\hat{t}\hat{x}}=\hat{u}-\frac{1}{6}(\hat{u}^{3})_{\hat{x}\hat{x}}.$
(137)

Note that the $N$-soliton solutions of the SP equation with $\sigma=1$
exhibits the singular nature since $u$ diverges when $|x|\rightarrow\infty$ as
shown in [4].

##  3 Integrable fully discretization of the gsG equation

To construct a fully discrete analogue of the gsG equation, we introduce two
discrete variables, $k$ and $l$, which correspond to the discrete spartial and
time variables, respectively. We start with the following fully discrete
bilinear equations.

$\displaystyle\tau_{n}(k,l+1,m)\tau_{n}(k,l,m-1)-bc\tau_{n+1}(k,l,m-1)\tau_{n-1}(k,l+1,m)$
$\displaystyle=(1-bc)\tau_{n}(k,l,m)\tau_{n}(k,l+1,m-1),$ (138)
$\displaystyle\tau_{n+1}(k,l,m-1)\tau_{n}(k+1,l,m)-ac^{-1}\tau_{n+1}(k+1,l,m)\tau_{n}(k,l,m-1)$
$\displaystyle=(1-ac^{-1})\tau_{n}(k,l,m)\tau_{n+1}(k+1,l,m-1).$ (139)

Here $n$, $k$, $l$, $m$ are integers, $a$, $b$, $c$ are parameters.

######  Proposition 1.

The bilinear equations (138)-(139) admit either the following Gram-type
determinant solutions

$\tau_{n}(k,l,m)=\left|m_{ij}^{n}(k,l,m)\right|_{1\leqslant i,j\leqslant N},$

with

$\displaystyle m_{ij}^{(n)}(k,l,m)$
$\displaystyle=c_{ij}+\frac{d_{ij}}{p_{i}+q_{j}}\left(-\frac{p_{i}}{q_{j}}\right)^{n}\left(\frac{1-ap_{i}}{1+aq_{j}}\right)^{-k}\left(\frac{1-bp_{i}^{-1}}{1+bq_{j}^{-1}}\right)^{-l}\left(\frac{1-cp_{i}}{1+cq_{j}}\right)^{m},$
(140)

or the Casorati-type determinant solutions

$\displaystyle\tau_{n}(k,l,m)=\left|\phi_{(n+j-1)}^{(i)}(k,l,m)\right|_{1\leqslant
i,j\leqslant N},$ (141)

with

$\displaystyle\phi_{(n)}^{(i)}(k,l,m)=c_{i}p_{i}^{n}\left(1-ap_{i}\right)^{-k}\left({1-bp_{i}^{-1}}\right)^{-l}\left(1-cp_{i}\right)^{m}+d_{i}q_{i}^{n}\left(1-aq_{i}\right)^{-k}\left({1+bq_{i}^{-1}}\right)^{-l}\left(1-cq_{i}\right)^{m}.$

###### Proof.

Here we give a proof for the Gram-type determinant solution. The proof for the
Casorati-type determinant solution is similar. The discrete Kadomtsev-
Petviashvili (dKP) equation was proposed independently by Hirota [19] and Miwa
[20] in early 1980s so it is also called Hirota-Miwa (HW) equation. Discrete
KP hierarchy is an infinite number of bilinear equations with
$(k_{i},k_{j},k_{m})$ taken from $(k_{1},k_{2},k_{3},\cdots)$, among which,
let us choose two triples: $(k_{1},k_{3},k_{4})$, $(k_{1},k_{2},k_{4})$ so
that the following two bilinear equations follow

$\displaystyle(a^{-1}_{3}-a^{-1}_{4})\tau(k_{1}+1,k_{2},k_{3},k_{4})\tau(k_{1},k_{2},k_{3}+1,k_{4}+1)$
$\displaystyle+(a^{-1}_{4}-a^{-1}_{1})\tau(k_{1},k_{2},k_{3}+1,k_{4})\tau(k_{1}+1,k_{2},k_{3},k_{4}+1)$
$\displaystyle+(a^{-1}_{1}-a^{-1}_{3})\tau(k_{1},k_{2},k_{3},k_{4}+1)\tau(k_{1}+1,k_{2},k_{3}+1,k_{4})=0\,,$
(142)
$\displaystyle(a^{-1}_{2}-a^{-1}_{4})\tau(k_{1}+1,k_{2},k_{3},k_{4})\tau(k_{1},k_{2}+1,k_{3},k_{4}+1)$
$\displaystyle+(a^{-1}_{4}-a^{-1}_{1})\tau(k_{1},k_{2}+1,k_{3},k_{4})\tau(k_{1}+1,k_{2},k_{3},k_{4}+1)$
$\displaystyle+(a^{-1}_{1}-a^{-1}_{2})\tau(k_{1},k_{2},k_{3},k_{4}+1)\tau(k_{1}+1,k_{2}+1,k_{3},k_{4})=0\,.$
(143)

As is shown by Ohta et al. [21], above two bilinear equations admit Gram type
solution

$\displaystyle\tau(k_{1},\cdots,k_{4})=\left|c_{ij}+\frac{1}{p_{i}+q_{j}}\prod_{l=1}^{4}\left(\frac{1-a_{l}p_{i}}{1+a_{l}q_{j}}\right)^{-k_{l}}\right|_{1\leqslant
i,j\leqslant N}.$ (144)

By taking $a_{1}\to\infty$, and redefined $a_{2}=a$, $a_{3}=b^{-1}$,
$a_{4}=c$, $k_{2}=k$, $k_{3}=l,k_{4}=-m$, $k_{1}+k_{3}=-n-1$ and
$\tau({k_{1},\cdots,k_{4}})=\tau_{n}(k,l,m)$, we have the bilinear equation
(138) together with its Gram-type determinant solution from (142).

On the other hand, by taking the same limit $a_{1}\to\infty$, and redefined
$a_{2}=a$, $a_{3}=b^{-1}$, $a_{4}=c$, $k_{2}=k,k_{3}=l$, $k_{4}=-m$,
$k_{1}=-n-1$ and $\tau({k_{1},\cdots,k_{4}})=\tau_{n}(k,l,m)$, we have the
bilinear equation (139) together with its Gram-type determinant solution from
(143). The proof is complete. ∎

######  Remark 3.1.

Eq. (138) is actually the fully discrete 2DTL equation, while eq. (139) is
discrete modified KP equation. As shown from above proof, they are equivalent
to discrete KP equation via reparameterization. As shown in this section, the
discrete analog of the gsG equation is constructed from the combination of
(138) and(139).

By applying a 2-reduction condition: $q_{i}=p_{i}$ for Gram-type determinant
solution, or $q_{i}=-p_{i}$ for Casorati-type, we have
$\tau_{n}\Bumpeq\tau_{n+2}$. Here $\Bumpeq$ means two $\tau$-functions are
equivalent up to a constant multiple. In addition, by defining

$\displaystyle
f_{k}^{l}=\tau_{0}(k,l,0),\,\tilde{f}_{k}^{l}=\tau_{1}(k,l,0),\,g_{k}^{l}=\tau_{0}(k,l,1),\,\tilde{g}_{k}^{l}=\tau_{1}(k,l,1),$
(145)

we can obtain the following equations

$\displaystyle f_{k}^{l}g_{k}^{l+1}-bc\tilde{f}_{k}^{l}\tilde{g}_{k}^{l+1}$
$\displaystyle=(1-bc)f_{k}^{l+1}g_{k}^{l},$ (146)
$\displaystyle\tilde{f}_{k}^{l}\tilde{g}_{k}^{l+1}-bc{f}_{k}^{l}{g}_{k}^{l+1}$
$\displaystyle=(1-bc)\tilde{f}_{k}^{l+1}\tilde{g}_{k}^{l},$ (147)
$\displaystyle
f_{k}^{l}\tilde{g}_{k+1}^{l}-ac^{-1}\tilde{f}_{k}^{l}g_{k+1}^{l}$
$\displaystyle=(1-ac^{-1})f_{k+1}^{l}\tilde{g}_{k}^{l},$ (148)
$\displaystyle\tilde{f}_{k}^{l}{g}_{k+1}^{l}-ac^{-1}{f}_{k}^{l}\tilde{g}_{k+1}^{l}$
$\displaystyle=(1-ac^{-1})\tilde{f}_{k+1}^{l}{g}_{k}^{l}.$ (149)

Introducing four intermediate variable transformations

$\displaystyle\sigma_{k}^{l}=2\mathrm{i}\ln\frac{\tilde{f}_{k}^{l}}{f_{k}^{l}},\,\tilde{\sigma}_{k}^{l}=2\mathrm{i}\ln\frac{\tilde{g}_{k}^{l}}{g_{k}^{l}},$
(150) $\displaystyle\theta_{k}^{l}=\ln\frac{f_{k}^{l}}{g_{k}^{l}},\
\tilde{\theta}_{k}^{l}=\ln\frac{\tilde{f}_{k}^{l}}{\tilde{g}_{k}^{l}},$ (151)

and then dividing (146) and (147) by $\tilde{f}_{k}^{l}\tilde{g}_{k}^{l+1}$
and $\tilde{f}_{k}^{l+1}\tilde{g}_{k}^{l}$, respectively, lead to

$\displaystyle
e^{\frac{\mathrm{i}}{2}(\sigma_{k}^{l}+\tilde{\sigma}_{k}^{l+1})}-bc=(1-bc)\frac{\tilde{f}_{k}^{l+1}\tilde{g}_{k}^{l}}{\tilde{f}_{k}^{l}\tilde{g}_{k}^{l+1}}e^{\frac{\mathrm{i}}{2}(\sigma_{k}^{l+1}+\tilde{\sigma}_{k}^{l})},$
(152)
$\displaystyle\frac{\tilde{f}_{k}^{l}\tilde{g}_{k}^{l+1}}{\tilde{f}_{k}^{l+1}\tilde{g}_{k}^{l}}\left(1-bce^{\frac{\mathrm{i}}{2}(\sigma_{k}^{l}+\tilde{\sigma}_{k}^{l+1})}\right)=1-bc.$
(153)

Eliminating
$\frac{\tilde{f}_{k}^{l+1}\tilde{g}_{k}^{l}}{\tilde{f}_{k}^{l}\tilde{g}_{k}^{l+1}}$,
we get

$\displaystyle\frac{1}{bc}\sin\frac{\sigma_{k}^{l+1}-\tilde{\sigma}_{k}^{l+1}-\sigma_{k}^{l}+\tilde{\sigma}_{k}^{l}}{4}=\sin\frac{\sigma_{k}^{l+1}+\tilde{\sigma}_{k}^{l+1}+\sigma_{k}^{l}+\tilde{\sigma}_{k}^{l}}{4}.$
(154)

Meanwhile, if we dividing (146) and (147) by ${f}_{k}^{l}{g}_{k}^{{l+1}}$ and
$\tilde{f}_{k}^{l}\tilde{g}_{k}^{l+1}$, respectively, we know that

$\displaystyle
1-bc\frac{\tilde{f}_{k}^{l}\tilde{g}_{k}^{l+1}}{f_{k}^{l}g_{k}^{l+1}}=(1-bc)e^{\theta_{k}^{l+1}-\theta_{k}^{l}},$
(155) $\displaystyle
1-bc\frac{f_{k}^{l}g_{k}^{l+1}}{\tilde{f}_{k}^{l}\tilde{g}_{k}^{l+1}}=(1-bc)e^{\tilde{\theta}_{k}^{l+1}-\tilde{\theta}_{k}^{l}},$
(156)

which can be transformed into

$\displaystyle\cosh\frac{\theta_{k}^{l+1}-\theta_{k}^{l}+\tilde{\theta}_{k}^{l+1}-\tilde{\theta}_{k}^{l}}{2}-bc\sinh\frac{\theta_{k}^{l+1}-\theta_{k}^{l}+\tilde{\theta}_{k}^{l+1}-\tilde{\theta}_{k}^{l}}{2}=\cosh\frac{\theta_{k}^{l+1}-\theta_{k}^{l}+\tilde{\theta}_{k}^{l}-\tilde{\theta}_{k}^{l+1}}{2}.$
(157)

Similarly, from (148) and (149), we know that

$\displaystyle
ac^{-1}\sin\frac{\sigma_{k+1}^{l}-\tilde{\sigma}_{k+1}^{l}+\sigma_{k}^{l}-\tilde{\sigma}_{k}^{l}}{4}=\sin\frac{\sigma_{k+1}^{l}+\tilde{\sigma}_{k+1}^{l}-\sigma_{k}^{l}-\tilde{\sigma}_{k}^{l}}{4},$
(158) $\displaystyle
ac^{-1}\cosh\frac{\theta_{k+1}^{l}-\theta_{k}^{l}+\tilde{\theta}_{k+1}^{l}-\tilde{\theta}_{k}^{l}}{2}-\sinh\frac{\theta_{k+1}^{l}-\theta_{k}^{l}+\tilde{\theta}_{k+1}^{l}-\tilde{\theta}_{k}^{l}}{2}=ac^{-1}\cosh\frac{\theta_{k+1}^{l}+\theta_{k}^{l}-\tilde{\theta}_{k}^{l}-\tilde{\theta}_{k+1}^{l}}{2}.$
(159)

###  3.1 Fully discretization of the gsG equation with $\nu=-1$

By choosing particular values in phase constants

$\displaystyle c_{ij}=\mathrm{i}\delta_{ij},$ (160)

for Gram-type solution, or

$\displaystyle d_{i}=\mathrm{i}c_{i},$ (161)

for Casorati-type solution, we can make $\tau_{n}$ and $\tau_{n+1}$ complex
conjugate to each other, which means

$\displaystyle\tilde{f}_{k}^{l}\Bumpeq\bar{f}_{k}^{l},\
\tilde{g}_{k}^{l}\Bumpeq\bar{g}_{k}^{l}.$ (162)

Here $\bar{f}$ denotes complex conjugate of $f$. Then, similar to the
continuous case, we introduce discrete hodograph transformation and dependent
variable transformation

$\displaystyle u_{k}^{l}=\frac{1}{2}(\sigma_{k}^{l}+\tilde{\sigma}_{k}^{l})$
$\displaystyle=\mathrm{i}\ln\frac{\bar{f}_{k}^{l}\bar{g}_{k}^{l}}{{f}_{k}^{l}{g}_{k}^{l}},$
(163)
$\displaystyle\phi_{k}^{l}=\frac{1}{2}(\sigma_{k}^{l}-\tilde{\sigma}_{k}^{l})$
$\displaystyle=\mathrm{i}\ln\frac{\bar{f}_{k}^{l}{g}_{k}^{l}}{{f}_{k}^{l}\bar{g}_{k}^{l}}.$
(164) $\displaystyle
x_{k}^{l}=2kac^{-1}+2lbc-\theta_{k}^{l}-\tilde{\theta}_{k}^{l}$
$\displaystyle=2kac^{-1}+2lbc+\ln\frac{g_{k}^{l}\bar{g}_{k}^{l}}{f_{k}^{l}\bar{f}_{k}^{l}}.$
(165)

The fully discrete analogue of the gsG equation with $\nu=-1$ is given by the
following theorem:

######  Theorem 3.1.

The fully discrete analogue of the gsG equation with $\nu=-1$ is of the form

$\displaystyle\frac{1}{b}\sin\frac{u_{k+1}^{l+1}-u_{k+1}^{l}-u_{k}^{l+1}+u_{k}^{l}}{4}=\Delta_{k}^{l}\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}+u_{k}^{l+1}+u_{k}^{l}}{4},$
(166)
$\displaystyle({b^{2}c^{2}-1})\sinh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}+x_{k}^{l+1}-x_{k}^{l}-4bc+4\chi_{1}}{2}\sinh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}-x_{k}^{l+1}+x_{k}^{l}}{2}$
$\displaystyle=-b^{2}c^{2}\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}+u_{k}^{l+1}+u_{k}^{l}}{2}\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}-u_{k}^{l+1}-u_{k}^{l}}{2},$
(167)

with

$\displaystyle\Delta_{k}^{l}=\frac{\sqrt{c^{2}-a^{2}}\sinh\frac{x_{k+1}^{l+1}-x_{k}^{l+1}+x_{k+1}^{l}-x_{k}^{l}-4ac^{-1}+4\chi_{2}}{4}}{\sqrt{b^{2}c^{2}-1}\sinh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}+x_{k}^{l+1}-x_{k}^{l}-4bc+4\chi_{1}}{4}},$
(168) $\displaystyle\sinh\chi_{1}=\frac{1}{\sqrt{b^{2}c^{2}-1}},\
\sinh\chi_{2}=\frac{a}{\sqrt{c^{2}-a^{2}}}.$ (169)

Here $u_{k}^{l},\ x_{k}^{l}$ are defined in (163) and (165). The
$\tau$-functions $f_{k}^{l}$ and $g_{k}^{l}$ are defined in (145) with
restriction (160) for Gram-type determinants, or with restriction (161) for
Casorati-type determinants in Proposition 1. Moreover, two conserved
quantities in the full-discrete gsG equation read as

$\displaystyle
I_{k}^{l}=({c^{2}-a^{2}})\sinh^{2}\frac{x_{k+1}^{l}-x_{k}^{l}-2ac^{-1}+2\chi_{2}}{2}+c^{2}\sin^{2}\frac{u_{k+1}^{l}-u_{k}^{l}}{2}=a^{2},$
(170) $\displaystyle
J_{k}^{l}=\frac{{b^{2}c^{2}-1}}{b^{2}}\sinh^{2}\frac{x_{k}^{l+1}-x_{k}^{l}-2bc+2\chi_{1}}{2}+c^{2}\sin^{2}\frac{u_{k}^{l}+u_{k}^{l+1}}{2}=\frac{1}{b^{2}}.$
(171)

###### Proof.

Note that we can rewrite (154), (157), (158) and (159) as

$\displaystyle\frac{1}{b}\sin\frac{\phi_{k}^{l+1}-\phi_{k}^{l}}{2}$
$\displaystyle=c\sin\frac{u_{k}^{l+1}+u_{k}^{l}}{2},$ (172)
$\displaystyle\frac{1}{b}\cos\frac{\phi_{k}^{l+1}-\phi_{k}^{l}}{2}$
$\displaystyle=\frac{\sqrt{b^{2}c^{2}-1}}{b}\sinh\frac{x_{k}^{l+1}-x_{k}^{l}-2bc+2\chi_{1}}{2},\
\sinh\chi_{1}=\frac{1}{\sqrt{b^{2}c^{2}-1}},$ (173)
$\displaystyle{a}\sin\frac{\phi_{k}^{l}+\phi_{k+1}^{l}}{2}$
$\displaystyle={c}\sin\frac{u_{k+1}^{l}-u_{k}^{l}}{2},$ (174)
$\displaystyle{a}\cos\frac{\phi_{k}^{l}+\phi_{k+1}^{l}}{2}$
$\displaystyle=\sqrt{c^{2}-a^{2}}\sinh\frac{x_{k+1}^{l}-x_{k}^{l}-2ac^{-1}+2\chi_{2}}{2},\
\sinh\chi_{2}=\frac{a}{\sqrt{c^{2}-a^{2}}}.$ (175)

By making a shift of $k\rightarrow k+1$ in (172), then adding and subtracting
with (172), we obtain, respectively

$\displaystyle\sin\frac{\phi_{k+1}^{l+1}-\phi_{k+1}^{l}+\phi_{k}^{l+1}-\phi_{k}^{l}}{4}\cos\frac{\phi_{k+1}^{l+1}-\phi_{k+1}^{l}-\phi_{k}^{l+1}+\phi_{k}^{l}}{4}$
$\displaystyle=bc\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}+u_{k}^{l+1}+u_{k}^{l}}{4}\cos\frac{u_{k+1}^{l+1}+u_{k+1}^{l}-u_{k}^{l+1}-u_{k}^{l}}{4},$
(176)
$\displaystyle\cos\frac{\phi_{k+1}^{l+1}-\phi_{k+1}^{l}+\phi_{k}^{l+1}-\phi_{k}^{l}}{4}\sin\frac{\phi_{k+1}^{l+1}-\phi_{k+1}^{l}-\phi_{k}^{l+1}+\phi_{k}^{l}}{4}$
$\displaystyle=bc\cos\frac{u_{k+1}^{l+1}+u_{k+1}^{l}+u_{k}^{l+1}+u_{k}^{l}}{4}\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}-u_{k}^{l+1}-u_{k}^{l}}{4}.$
(177)

Similarly, from (173)-(175), one can obtain

$\displaystyle\sqrt{b^{2}c^{2}-1}\sinh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}+x_{k}^{l+1}-x_{k}^{l}-4bc+4\chi_{1}}{4}\cosh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}-x_{k}^{l+1}+x_{k}^{l}}{4}$
$\displaystyle=\cos\frac{\phi_{k+1}^{l+1}-\phi_{k+1}^{l}+\phi_{k}^{l+1}-\phi_{k}^{l}}{4}\cos\frac{\phi_{k+1}^{l+1}-\phi_{k+1}^{l}-\phi_{k}^{l+1}+\phi_{k}^{l}}{4},$
(178)
$\displaystyle\sqrt{b^{2}c^{2}-1}\cosh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}+x_{k}^{l+1}-x_{k}^{l}-4bc+4\chi_{1}}{4}\sinh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}-x_{k}^{l+1}+x_{k}^{l}}{4}$
$\displaystyle=-\sin\frac{\phi_{k+1}^{l+1}-\phi_{k+1}^{l}+\phi_{k}^{l+1}-\phi_{k}^{l}}{4}\sin\frac{\phi_{k+1}^{l+1}-\phi_{k+1}^{l}-\phi_{k}^{l+1}+\phi_{k}^{l}}{4},$
(179) $\displaystyle
a\sin\frac{\phi_{k+1}^{l+1}+\phi_{k+1}^{l}+\phi_{k}^{l+1}+\phi_{k}^{l}}{4}\cos\frac{\phi_{k+1}^{l+1}-\phi_{k+1}^{l}+\phi_{k}^{l+1}-\phi_{k}^{l}}{4}$
$\displaystyle=c\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}-u_{k}^{l+1}-u_{k}^{l}}{4}\cos\frac{u_{k+1}^{l+1}-u_{k+1}^{l}-u_{k}^{l+1}+u_{k}^{l}}{4},$
(180) $\displaystyle
a\cos\frac{\phi_{k+1}^{l+1}+\phi_{k+1}^{l}+\phi_{k}^{l+1}+\phi_{k}^{l}}{4}\sin\frac{\phi_{k+1}^{l+1}-\phi_{k+1}^{l}+\phi_{k}^{l+1}-\phi_{k}^{l}}{4}$
$\displaystyle=c\cos\frac{u_{k+1}^{l+1}+u_{k+1}^{l}-u_{k}^{l+1}-u_{k}^{l}}{4}\sin\frac{u_{k+1}^{l+1}-u_{k+1}^{l}-u_{k}^{l+1}+u_{k}^{l}}{4},$
(181)

and

$\displaystyle\sqrt{c^{2}-a^{2}}\sinh\frac{x_{k+1}^{l+1}-x_{k}^{l+1}+x_{k+1}^{l}-x_{k}^{l}-4ac^{-1}+4\chi_{2}}{4}\cosh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}-x_{k}^{l+1}+x_{k}^{l}}{4}$
$\displaystyle=a\cos\frac{\phi_{k+1}^{l+1}+\phi_{k+1}^{l}+\phi_{k}^{l+1}+\phi_{k}^{l}}{4}\cos\frac{\phi_{k+1}^{l+1}-\phi_{k+1}^{l}+\phi_{k}^{l+1}-\phi_{k}^{l}}{4},$
(182)
$\displaystyle\sqrt{c^{2}-a^{2}}\cosh\frac{x_{k+1}^{l+1}-x_{k}^{l+1}+x_{k+1}^{l}-x_{k}^{l}-4ac^{-1}+4\chi_{2}}{4}\sinh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}-x_{k}^{l+1}+x_{k}^{l}}{4}$
$\displaystyle=-a\sin\frac{\phi_{k+1}^{l+1}+\phi_{k+1}^{l}+\phi_{k}^{l+1}+\phi_{k}^{l}}{4}\sin\frac{\phi_{k+1}^{l+1}-\phi_{k+1}^{l}+\phi_{k}^{l+1}-\phi_{k}^{l}}{4},$
(183)

respectively. Thus, eqs. (176) and (181) give

$\displaystyle\frac{1}{ab}\sin\frac{u_{k+1}^{l+1}-u_{k+1}^{l}-u_{k}^{l+1}+u_{k}^{l}}{4}\cos\frac{\phi_{k+1}^{l+1}-\phi_{k+1}^{l}-\phi_{k}^{l+1}+\phi_{k}^{l}}{4}$
$\displaystyle=\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}+u_{k}^{l+1}+u_{k}^{l}}{4}\cos\frac{\phi_{k+1}^{l+1}+\phi_{k+1}^{l}+\phi_{k}^{l+1}+\phi_{k}^{l}}{4}.$
(184)

Eqs. (178) and (182) lead to

$\displaystyle
a\sqrt{b^{2}c^{2}-1}\sinh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}+x_{k}^{l+1}-x_{k}^{l}-4bc+4\chi_{1}}{4}\cos\frac{\phi_{k+1}^{l+1}+\phi_{k+1}^{l}+\phi_{k}^{l+1}+\phi_{k}^{l}}{4}$
$\displaystyle=\sqrt{c^{2}-a^{2}}\sinh\frac{x_{k+1}^{l+1}-x_{k}^{l+1}+x_{k+1}^{l}-x_{k}^{l}-4ac^{-1}+4\chi_{2}}{4}\cos\frac{\phi_{k+1}^{l+1}-\phi_{k+1}^{l}-\phi_{k}^{l+1}+\phi_{k}^{l}}{4}.$
(185)

A substitution of (185) into (184) leads to

$\displaystyle\frac{1}{b}\sin\frac{u_{k+1}^{l+1}-u_{k+1}^{l}-u_{k}^{l+1}+u_{k}^{l}}{4}=\Delta_{k}^{l}\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}+u_{k}^{l+1}+u_{k}^{l}}{4},$
(186)

with

$\displaystyle\Delta_{k}^{l}=\frac{\sqrt{c^{2}-a^{2}}\sinh\frac{x_{k+1}^{l+1}-x_{k}^{l+1}+x_{k+1}^{l}-x_{k}^{l}-4ac^{-1}+4\chi_{2}}{4}}{\sqrt{b^{2}c^{2}-1}\sinh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}+x_{k}^{l+1}-x_{k}^{l}-4bc+4\chi_{1}}{4}}.$
(187)

On the other hand, by multiplying (176) and (177), (178) and (179), we obtain,
respectively

$\displaystyle
b^{2}c^{2}\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}+u_{k}^{l+1}+u_{k}^{l}}{2}\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}-u_{k}^{l+1}-u_{k}^{l}}{2}$
$\displaystyle=\sin\frac{\phi_{k+1}^{l+1}-\phi_{k+1}^{l}+\phi_{k}^{l+1}-\phi_{k}^{l}}{2}\sin\frac{\phi_{k+1}^{l+1}-\phi_{k+1}^{l}-\phi_{k}^{l+1}+\phi_{k}^{l}}{2},$
(188)
$\displaystyle({b^{2}c^{2}-1})\sinh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}+x_{k}^{l+1}-x_{k}^{l}-4bc+4\chi_{1}}{2}\sinh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}-x_{k}^{l+1}+x_{k}^{l}}{2}$
$\displaystyle=-\sin\frac{\phi_{k+1}^{l+1}-\phi_{k+1}^{l}+\phi_{k}^{l+1}-\phi_{k}^{l}}{2}\sin\frac{\phi_{k+1}^{l+1}-\phi_{k+1}^{l}-\phi_{k}^{l+1}+\phi_{k}^{l}}{2},$
(189)

which leads to exactly (167) by eliminating the right side of the equations.
Meanwhile, from (172)-(175), we have

$\displaystyle
J_{k}^{l}=\frac{{b^{2}c^{2}-1}}{b^{2}}\sinh^{2}\frac{x_{k}^{l+1}-x_{k}^{l}-2bc+2\chi_{1}}{2}+c^{2}\sin^{2}\frac{u_{k}^{l}+u_{k}^{l+1}}{2}=\frac{1}{b^{2}},$
(190) $\displaystyle
I_{k}^{l}=({c^{2}-a^{2}})\sinh^{2}\frac{x_{k+1}^{l}-x_{k}^{l}-2ac^{-1}+2\chi_{2}}{2}+c^{2}\sin^{2}\frac{u_{k+1}^{l}-u_{k}^{l}}{2}=a^{2}.$
(191)

Here $a^{2}$ and $\frac{1}{b^{2}}$ are constants, thus equations (190) and
(191) actually give conserved quantities. The proof is complete. ∎

###  3.2 Fully discretization of the gsG equation with $\nu=1$

In order to construct the fully discrete analogue of the gsG equation with
$\nu=1$, we choose the restriction as

$\displaystyle
c=\lambda\mathrm{i},\,c_{ij}=\mathrm{i}\sqrt{\frac{1-cp_{i}}{1+cq_{j}}}\delta_{ij},$
(192)

for Gram-type solution, or

$\displaystyle
c=\lambda\mathrm{i},\,d_{i}=\mathrm{i}{\sqrt{\frac{1-cp_{i}}{1-cq_{i}}}}c_{i},$
(193)

for Casorati-type, which implies

$\displaystyle\tilde{f}_{k}^{l}\Bumpeq\bar{g}_{k}^{l},\
\tilde{g}_{k}^{l}\Bumpeq\bar{f}_{k}^{l}.$ (194)

Next we introduce dependent variable transformations

$\displaystyle
u_{k}^{l}=\frac{1}{2}(\sigma_{k}^{l}+\tilde{\sigma}_{k}^{l})=\mathrm{i}\ln\frac{\bar{f}_{k}^{l}\bar{g}_{k}^{l}}{{f}_{k}^{l}{g}_{k}^{l}},$
(195)
$\displaystyle\varphi_{k}^{l}=\frac{1}{2\mathrm{i}}(\sigma_{k}^{l}-\tilde{\sigma}_{k}^{l})=\ln\frac{\bar{g}_{k}^{l}{g}_{k}^{l}}{{f}_{k}^{l}\bar{f}_{k}^{l}},$
(196)

and

$\displaystyle\tilde{x}_{k}^{l}=2ka\lambda^{-1}-2lb\lambda-\mathrm{i}(\theta_{k}^{l}+\tilde{\theta}_{k}^{l})=2ka\lambda^{-1}-2lb\lambda+\mathrm{i}\ln\frac{\bar{f}_{k}^{l}g_{k}^{l}}{f_{k}^{l}\bar{g}_{k}^{l}},$
(197) $\displaystyle\tilde{t}_{k}^{l}=2lb\lambda.$ (198)

We can construct the fully discrete analogue of the gsG equation with $\nu=1$
through the following theorem.

######  Theorem 3.2.

The fully discrete analogue of the gsG equation with $\nu=1$ is of the form

$\displaystyle\frac{1}{b}\sin\frac{u_{k+1}^{l+1}-u_{k+1}^{l}-u_{k}^{l+1}+u_{k}^{l}}{4}=\tilde{\Delta}_{k}^{l}\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}+u_{k}^{l+1}+u_{k}^{l}}{4},$
(199)
$\displaystyle({1+b^{2}\lambda^{2}})\sin\frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^{l}+\tilde{x}_{k}^{l+1}-\tilde{x}_{k}^{l}+4b\lambda+4\omega_{1}}{2}\sin\frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^{l}-\tilde{x}_{k}^{l+1}+\tilde{x}_{k}^{l}}{2}$
$\displaystyle=b^{2}\lambda^{2}\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}+u_{k}^{l+1}+u_{k}^{l}}{2}\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}-u_{k}^{l+1}-u_{k}^{l}}{2},$
(200)

with

$\displaystyle\tilde{\Delta}_{k}^{l}=\frac{\sqrt{a^{2}+\lambda^{2}}\sin\frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k}^{l+1}+\tilde{x}_{k+1}^{l}-\tilde{x}_{k}^{l}-4a\lambda^{-1}+4\omega_{2}}{4}}{\sqrt{1+b^{2}\lambda^{2}}\sin\frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^{l}+\tilde{x}_{k}^{l+1}-\tilde{x}_{k}^{l}+4b\lambda+4\omega_{1}}{4}},$
(201) $\displaystyle\sin\omega_{1}=\frac{1}{\sqrt{b^{2}\lambda^{2}+1}},\
\sin\omega_{2}=\frac{a}{\sqrt{a^{2}+\lambda^{2}}}.$ (202)

Here $u_{k}^{l},\ x_{k}^{l}$ are defined in (195) and (197). The
$\tau$-functions $f_{k}^{l}$ and $g_{k}^{l}$ are defined in (145) with
restriction (192) for Gram-type determinants, or with restriction (193) for
Casorati-type determinants in Proposition 1. Moreover, there are two conserved
quantities in the full-discrete gsG equation read as

$\displaystyle\tilde{I}_{k}^{l}=\left(a\cos\frac{\tilde{x}_{k+1}^{l}-\tilde{x}_{k}^{l}-2a\lambda^{-1}}{2}+\lambda\sin\frac{\tilde{x}_{k+1}^{l}-\tilde{x}_{k}^{l}-2a\lambda^{-1}}{2}\right)^{2}-\lambda^{2}\sin^{2}\frac{u_{k+1}^{l}-u_{k}^{l}}{2}=a^{2},$
(203)
$\displaystyle\tilde{J}_{k}^{l}=\left(\frac{1}{\lambda}\cos\frac{\tilde{x}_{k}^{l+1}-\tilde{x}_{k}^{l}+2b\lambda}{2}+\lambda\sin\frac{\tilde{x}_{k}^{l+1}-\tilde{x}_{k}^{l}+2b\lambda}{2}\right)^{2}-\lambda^{2}\sin^{2}\frac{u_{k}^{l+1}+u_{k}^{l}}{2}=\frac{1}{b^{2}}.$
(204)

######  Remark 3.2.

Fully discrete analogues of the generalized sG equation with $\nu=1$ can also
be transformed into the case with $\nu=-1$ through

$\displaystyle\phi_{k}^{l}=\mathrm{i}\varphi_{k}^{l},\
\tilde{x}_{k}^{l}=\mathrm{i}x_{k}^{l},\ \tilde{t}=-\mathrm{i}t,\
c=\lambda\mathrm{i}.$ (205)

The detail proof of Theorem 3.2 is similar to the case with $\nu=-1$, which is
given in Appendix A.

###  3.3 Reduction to the discrete sG and discrete SP equation

In this section, we mainly investigate the reduction from the discrete gsG
equation to the discrete sG equation [16, 17, 18] and the discrete SP equation
[7]. The $\tau$-functions, as well as the variable transformations of the
discrete sG equation and the discrete SP equation, can be derived from those
of the discrete gsG equation. The parameter $c$ (or $\lambda$) also plays an
important role in the reduction of the discrete case.

####  3.3.1 Reduction to the discrete sG equation

(I) From the discrete gsG equation with $\nu=-1$ to the discrete sG equation.
In this part of reduction, we take $c\in\mathbb{R}$ as a small parameter. Then
the elements of the $\tau$-function in Theorem 3.1 can be written as

$\displaystyle m_{ij}^{(n)}(k,l,1)$
$\displaystyle=\mathrm{i}\delta_{ij}+\frac{1}{p_{i}+p_{j}}\left(-\frac{p_{i}}{p_{j}}\right)^{n}\left(\frac{1-ap_{i}}{1+ap_{j}}\right)^{-k}\left(\frac{1-bp_{i}^{-1}}{1+bp_{j}^{-1}}\right)^{-l}\left(\frac{1-cp_{i}}{1+cp_{j}}\right)$
$\displaystyle=\mathrm{i}\delta_{ij}+\frac{1}{p_{i}+p_{j}}\left(-\frac{p_{i}}{p_{j}}\right)^{n}\left(\frac{1-ap_{i}}{1+ap_{j}}\right)^{-k}\left(\frac{1-bp_{i}^{-1}}{1+bp_{j}^{-1}}\right)^{-l}(1-(p_{i}+p_{j})c)+\mathnormal{O}(c^{2})$
$\displaystyle=m_{ij}^{(n)}(k,l,0)-(p_{i}+p_{j})m_{ij}^{(n)}(k,l,0)c+\mathnormal{O}(c^{2}).$
(206)

We can rewrite the dependent variable transformations and define new variable
as

$\displaystyle
u_{k}^{l}=\mathrm{i}\ln\frac{\bar{f}_{k}^{l}\bar{g}_{k}^{l}}{{f}_{k}^{l}{g}_{k}^{l}}=2\mathrm{i}\ln\frac{\bar{f}_{k}^{l}}{{f}_{k}^{l}}+\mathnormal{O}(c),$
(207)
$\displaystyle\phi_{k}^{l}=\mathrm{i}\ln\frac{\bar{f}_{k}^{l}{g}_{k}^{l}}{{f}_{k}^{l}\bar{g}_{k}^{l}}=\mathrm{i}\ln\frac{\bar{f}_{k}^{l}{f}_{k}^{l}}{f_{k}^{l}\bar{f}_{k}^{l}}+\mathnormal{O}(c)=\mathnormal{O}(c),$
(208)
$\displaystyle\hat{x}_{k}^{l}=cx_{k}^{l}=2ka+2lbc^{2}+c\ln\frac{g_{k}^{l}\bar{g}_{k}^{l}}{f_{k}^{l}\bar{f}_{k}^{l}}=2ka+\mathnormal{O}(c^{2}),$
(209) $\displaystyle\hat{t}_{k}^{l}=c^{-1}{t_{k}^{l}}=2lb.$ (210)

In the limit of $c\rightarrow 0$, (207)-(209) lead to

$\displaystyle u_{k}^{l}=2\mathrm{i}\ln\frac{\bar{f}_{k}^{l}}{{f}_{k}^{l}},\
\phi_{k}^{l}=0,\ \hat{x}_{k}^{l}=2ka.$ (211)

The definition of $u_{k}^{l}$ corresponds to the dependent transformation of
the discrete sG equation. Substituting (207)-(209) into (166) and (168) and
taking $c\rightarrow 0$, one can obtain

$\displaystyle\frac{1}{b}\sin\frac{u_{k+1}^{l+1}-u_{k+1}^{l}-u_{k}^{l+1}+u_{k}^{l}}{4}=a\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}+u_{k}^{l+1}+u_{k}^{l}}{4},$
(212)

which is just the discrete sG equation [16, 17, 18].

(II) From the discrete gsG equation with $\nu=1$ to the discrete sG equation.
In this part of reduction, we take $\lambda\in\mathbb{R}$ as a small
parameter. Here we have

$\displaystyle m_{ij}^{(n)}(k,l,1)$
$\displaystyle=\mathrm{i}\sqrt{\frac{1-\mathrm{i}\lambda
p_{i}}{1+\mathrm{i}\lambda
q_{j}}}\delta_{ij}+\frac{1}{p_{i}+p_{j}}\left(-\frac{p_{i}}{p_{j}}\right)^{n}\left(\frac{1-ap_{i}}{1+ap_{j}}\right)^{-k}\left(\frac{1-bp_{i}^{-1}}{1+bp_{j}^{-1}}\right)^{-l}\left(\frac{1-\mathrm{i}\lambda
p_{i}}{1+\mathrm{i}\lambda p_{j}}\right)$
$\displaystyle=\mathrm{i}\delta_{ij}+\frac{1}{p_{i}+p_{j}}\left(-\frac{p_{i}}{p_{j}}\right)^{n}\left(\frac{1-ap_{i}}{1+ap_{j}}\right)^{-k}\left(\frac{1-bp_{i}^{-1}}{1+bp_{j}^{-1}}\right)^{-l}+\mathnormal{O}(\lambda)$
$\displaystyle=\hat{m}_{ij}^{(n)}(k,l,0)+\mathnormal{O}(\lambda),$ (213)
$\displaystyle m_{ij}^{(n)}(k,l,0)$
$\displaystyle=\mathrm{i}\sqrt{\frac{1-\mathrm{i}\lambda
p_{i}}{1+\mathrm{i}\lambda
q_{j}}}\delta_{ij}+\frac{1}{p_{i}+p_{j}}\left(-\frac{p_{i}}{p_{j}}\right)^{n}\left(\frac{1-ap_{i}}{1+ap_{j}}\right)^{-k}\left(\frac{1-bp_{i}^{-1}}{1+bp_{j}^{-1}}\right)^{-l}$
$\displaystyle=\hat{m}_{ij}^{(n)}(k,l,0)+\mathnormal{O}(\lambda).$ (214)

We can rewrite the dependent variable transformations and define new variable
as

$\displaystyle
u_{k}^{l}=2\mathrm{i}\ln\frac{\bar{\hat{f}}_{k}^{l}}{{\hat{f}}_{k}^{l}}+\mathnormal{O}(\lambda),\
\varphi_{k}^{l}=\mathnormal{O}(\lambda),\
\hat{x}_{k}^{l}=\lambda\tilde{x}_{k}^{l}=2ka+\mathnormal{O}(\lambda^{2}),\
\hat{t}_{k}^{l}=\frac{\tilde{t}_{k}^{l}}{\lambda}=2lb.$ (215)

In the limit of $\lambda\rightarrow 0$, eqs. (215) and (199) converge to

$\displaystyle
u_{k}^{l}=2\mathrm{i}\ln\frac{\bar{\hat{f}}_{k}^{l}}{{\hat{f}}_{k}^{l}},\
\varphi_{k}^{l}=0,\ $ $\displaystyle\hat{x}_{k}^{l}=2ka,\
\hat{t}_{k}^{l}=2lb,$ (216)
$\displaystyle\frac{1}{ab}\sin\frac{u_{k+1}^{l+1}-u_{k+1}^{l}-u_{k}^{l+1}+u_{k}^{l}}{4}$
$\displaystyle=\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}+u_{k}^{l+1}+u_{k}^{l}}{4},$
(217)

respectively. The latter is exactly the fully discrete sG equation.

####  3.3.2 Reduction to the discrete SP equation

(I) From the discrete gsG equation with $\nu=-1$ to the discrete SP equation
with $\sigma=-1$. In this part of reduction, we take $c$ as a big parameter
(or $\epsilon=\frac{1}{c}$ small). Let us introduce a new auxiliary parameter
$s$, and redefine the matrix elements of the $\tau$ function as

$\displaystyle\tilde{m}_{ij}^{(n)}(k,l,m)=\mathrm{i}\delta_{ij}+\frac{1}{p_{i}+p_{j}}\left(-\frac{p_{i}}{p_{j}}\right)^{n}\left(\frac{1-ap_{i}}{1+ap_{j}}\right)^{-k}\left(\frac{1-bp_{i}^{-1}}{1+bp_{j}^{-1}}\right)^{-l}\left(\frac{1-cp_{i}}{1+cp_{j}}\right)^{m}e^{\left(\frac{1}{2p_{i}}+\frac{1}{2p_{j}}\right)s}.$
(218)

It is obviously that
$\tau_{n}(k,l,m)=\left|\tilde{m}_{ij}^{n}(k,l,m)\right|_{1\leqslant
i,j\leqslant N}$ is still the solution of (138)-(139). If we take $c$ as a big
parameter (or $\epsilon=\frac{1}{c}$ small), the elements of the $\tau$
function can be written as

$\displaystyle\tilde{m}_{ij}^{(n)}(k,l,1)$
$\displaystyle=\mathrm{i}\delta_{ij}+\frac{1}{p_{i}+p_{j}}\left(-\frac{p_{i}}{p_{j}}\right)^{n}\left(\frac{1-ap_{i}}{1+ap_{j}}\right)^{-k}\left(\frac{1-bp_{i}^{-1}}{1+bp_{j}^{-1}}\right)^{-l}\left(\frac{1-cp_{i}}{1+cp_{j}}\right)^{1}e^{\left(\frac{1}{2p_{i}}+\frac{1}{2p_{j}}\right)s}$
$\displaystyle=\mathrm{i}\delta_{ij}+\frac{1}{p_{i}+p_{j}}\left(-\frac{p_{i}}{p_{j}}\right)^{n}\left(\frac{1-ap_{i}}{1+ap_{j}}\right)^{-k}\left(\frac{1-bp_{i}^{-1}}{1+bp_{j}^{-1}}\right)^{-l}e^{\left(\frac{1}{2p_{i}}+\frac{1}{2p_{j}}\right)s}\left(-\frac{p_{i}}{p_{j}}+\frac{p_{i}+p_{j}}{p_{j}^{2}}\epsilon\right)+\mathnormal{O}(\epsilon^{2})$
$\displaystyle=m_{ij}^{(n+1)}(k,l,0)-2\epsilon\frac{\mathrm{d}}{\mathrm{d}s}m_{ij}^{(n+1)}(k,l,0)+\mathnormal{O}(\epsilon^{2}),$
(219)

from which one can obtain

$\displaystyle
g_{k}^{l}=\tau_{0}(k,l,1)=\bar{f}_{k}^{l}-2\epsilon\frac{\mathrm{d}}{\mathrm{d}s}\bar{f}_{k}^{l}+\mathnormal{O}(\epsilon^{2}),$
(220)
$\displaystyle\bar{g}_{k}^{l}=\tau_{1}(k,l,1)={f}_{k}^{l}-2\epsilon\frac{\mathrm{d}}{\mathrm{d}s}{f}_{k}^{l}+\mathnormal{O}(\epsilon^{2}).$
(221)

We rewrite the dependent variable transformations and introduce new variable
as

$\displaystyle\epsilon\hat{u}_{k}^{l}=u_{k}^{l}=\mathrm{i}\ln\frac{\bar{f}_{k}^{l}\bar{g}_{k}^{l}}{{f}_{k}^{l}{g}_{k}^{l}}=\mathrm{i}\ln\frac{\bar{f}_{k}^{l}(f_{k}^{l}-2\epsilon\frac{\mathrm{d}}{\mathrm{d}s}{f}_{k}^{l})}{{f}_{k}^{l}(\bar{f}_{k}^{l}-2\epsilon\frac{\mathrm{d}}{\mathrm{d}s}\bar{f}_{k}^{l})}+\mathnormal{O}(\epsilon^{2}),$
(222)
$\displaystyle\phi_{k}^{l}=\mathrm{i}\ln\frac{\bar{f}_{k}^{l}{g}_{k}^{l}}{{f}_{k}^{l}\bar{g}_{k}^{l}}=\mathrm{i}\ln\frac{\bar{f}_{k}^{l}(\bar{f}_{k}^{l}-2\epsilon\frac{\mathrm{d}}{\mathrm{d}s}\bar{f}_{k}^{l})}{{f}_{k}^{l}(f_{k}^{l}-2\epsilon\frac{\mathrm{d}}{\mathrm{d}s}{f}_{k}^{l})}+\mathnormal{O}(\epsilon^{2}),$
(223)
$\displaystyle\epsilon\hat{x}_{k}^{l}+\frac{2lb}{\epsilon}=x_{k}^{l}=2ka\epsilon+\frac{2lb}{\epsilon}+\ln\frac{(\bar{f}_{k}^{l}-2\epsilon\frac{\mathrm{d}}{\mathrm{d}s}\bar{f}_{k}^{l})(f_{k}^{l}-2\epsilon\frac{\mathrm{d}}{\mathrm{d}s}{f}_{k}^{l})}{f_{k}^{l}\bar{f}_{k}^{l}}+\mathnormal{O}(\epsilon^{2}),$
(224) $\displaystyle\hat{t}_{k}^{l}=\epsilon t_{k}^{l}=2lb.$ (225)

In the limit of $\epsilon\rightarrow 0$, (222)-(224) lead to

$\displaystyle\hat{u}_{k}^{l}=2\mathrm{i}\left(\ln\frac{\bar{f}_{k}^{l}}{f_{k}^{l}}\right)_{s},$
(226)
$\displaystyle\phi_{k}^{l}=2\mathrm{i}\ln\frac{\bar{f}_{k}^{l}}{f_{k}^{l}},$
(227) $\displaystyle\hat{x}_{k}^{l}=2ka-2(\ln(f_{k}^{l}\bar{f}_{k}^{l}))_{s}.$
(228)

Substituting (222)-(224) into (166)-(168) and taking $\epsilon\rightarrow 0$,
one can obtain

$\displaystyle({\hat{x}_{k+1}^{l+1}-\hat{x}_{k+1}^{l}+\hat{x}_{k}^{l+1}-\hat{x}_{k}^{l}}+\frac{4}{b})({\hat{u}_{k+1}^{l+1}-\hat{u}_{k+1}^{l}-\hat{u}_{k}^{l+1}+\hat{u}_{k}^{l}})$
$\displaystyle=(\hat{x}_{k+1}^{l+1}-\hat{x}_{k}^{l+1}+\hat{x}_{k+1}^{l}-\hat{x}_{k}^{l})(\hat{u}_{k+1}^{l+1}+\hat{u}_{k+1}^{l}+\hat{u}_{k}^{l+1}+\hat{u}_{k}^{l}),$
(229)
$\displaystyle({\hat{x}_{k+1}^{l+1}-\hat{x}_{k+1}^{l}+\hat{x}_{k}^{l+1}-\hat{x}_{k}^{l}}+\frac{4}{b})({\hat{x}_{k+1}^{l+1}-\hat{x}_{k+1}^{l}-\hat{x}_{k}^{l+1}+\hat{x}_{k}^{l}})$
$\displaystyle=-(\hat{u}_{k+1}^{l+1}+\hat{u}_{k+1}^{l}+\hat{u}_{k}^{l+1}+\hat{u}_{k}^{l})({\hat{u}_{k+1}^{l+1}+\hat{u}_{k+1}^{l}-\hat{u}_{k}^{l+1}-\hat{u}_{k}^{l}}),$
(230)

which lead to the discrete SP equations. In addition, conserved quantities
$I_{k}^{l}$ and $J_{k}^{l}$ are recast into

$\displaystyle
J_{k}^{l}=\left(\frac{1}{b}+\frac{\hat{x}_{k}^{l+1}-\hat{x}_{k}^{l}}{2}\right)^{2}+\left(\frac{\hat{u}_{k}^{l}+\hat{u}_{k}^{l+1}}{2}\right)^{2},$
(231) $\displaystyle
I_{k}^{l}=\left(\frac{\hat{x}_{k+1}^{l}-\hat{x}_{k}^{l}}{2}\right)^{2}+\left(\frac{\hat{u}_{k+1}^{l}-\hat{u}_{k}^{l}}{2}\right)^{2},$
(232)

which correspond to the conserved quantities of the discrete SP equation
derived in [7].

(II) From the discrete gsG equation with $\nu=1$ to the discrete SP equation
with $\sigma=1$. Here we give a discrete analog of the SP equation with
$\sigma=1$ through the variable transformation and the scaling limit from the
discrete gsG equation with $\nu=1$. Similar to the case with $\nu=-1$, we take
$\epsilon=\frac{1}{\lambda}$ as a small parameter. Introducing an auxiliary
parameter $s$, and redefining the matrix elements of the $\tau$ function as
shown in (218), we have

$\displaystyle\tilde{m}_{ij}^{(n)}(k,l,0)\propto\hat{m}_{ij}^{(n+1)}(k,l)-\mathrm{i}\epsilon\frac{\mathrm{d}}{\mathrm{d}s}\hat{m}_{ij}^{(n+1)}(k,l)+\mathnormal{O}(\epsilon^{2}),$
(233)
$\displaystyle\tilde{m}_{ij}^{(n+1)}(k,l,0)\propto\hat{m}_{ij}^{(n)}(k,l)-\mathrm{i}\epsilon\frac{\mathrm{d}}{\mathrm{d}s}\hat{m}_{ij}^{(n)}(k,l)+\mathnormal{O}(\epsilon^{2}),$
(234)

where

$\displaystyle\hat{m}_{ij}^{(n)}(k,l)=\delta_{ij}+\frac{1}{p_{i}+p_{j}}\left(-\frac{p_{i}}{p_{j}}\right)^{n}\left(\frac{1-ap_{i}}{1+ap_{j}}\right)^{-k}\left(\frac{1-bp_{i}^{-1}}{1+bp_{j}^{-1}}\right)^{-l}e^{\left(\frac{1}{2p_{i}}+\frac{1}{2p_{j}}\right)s}.$
(235)

Thus one can obtain

$\displaystyle
f_{k}^{l}=\hat{f}_{k}^{l}-{\mathrm{i}\epsilon}\frac{\mathrm{d}}{\mathrm{d}\tau}\hat{f}_{k}^{l}+\mathnormal{O}(\epsilon^{2}),\
\bar{f}_{k}^{l}=\hat{f}_{k}^{l}+{\mathrm{i}\epsilon}\frac{\mathrm{d}}{\mathrm{d}\tau}\hat{f}_{k}^{l}+\mathnormal{O}(\epsilon^{2}),$
(236) $\displaystyle
g_{k}^{l}=\hat{g}_{k}^{l}+{\mathrm{i}\epsilon}\frac{\mathrm{d}}{\mathrm{d}\tau}\hat{g}_{k}^{l}+\mathnormal{O}(\epsilon^{2}),\
\bar{g}_{k}^{l}=\hat{g}_{k}^{l}-{\mathrm{i}\epsilon}\frac{\partial}{\partial\tau}\hat{g}_{k}^{l}+\mathnormal{O}(\epsilon^{2}),$
(237)
$\displaystyle\hat{f}_{k}^{l}=\left|\hat{m}_{ij}^{(n+1)}(k,l)\right|_{1\leqslant
i,j\leqslant N},\
\hat{g}_{k}^{l}=\left|\hat{m}_{ij}^{(n)}(k,l)\right|_{1\leqslant i,j\leqslant
N}.$ (238)

Similar to the continuou, we rewrite the dependent variable transformations
and introduce new variables, then under the limit $\epsilon\rightarrow 0$, we
have

$\displaystyle\hat{u}_{k}^{l}=2\left(\ln\frac{\hat{g}_{k}^{l}}{\hat{f}_{k}^{l}}\right)_{s},\
\varphi_{k}^{l}=2\ln\frac{\hat{g}_{k}^{l}}{\hat{f}_{k}^{l}},\
\hat{x}_{k}^{l}=2ka-2(\ln(\hat{f}_{k}^{l}\hat{g}_{k}^{l}))_{s},\
\hat{t}_{k}^{l}=2lb.$ (239)

Moreover, one can propose a fully discrete analogue of the SP equation with
$\sigma=1$

$\displaystyle({\hat{x}_{k+1}^{l+1}-\hat{x}_{k+1}^{l}+\hat{x}_{k}^{l+1}-\hat{x}_{k}^{l}}+\frac{4}{b})({\hat{u}_{k+1}^{l+1}-\hat{u}_{k+1}^{l}-\hat{u}_{k}^{l+1}+\hat{u}_{k}^{l}})$
$\displaystyle=(\hat{x}_{k+1}^{l+1}-\hat{x}_{k}^{l+1}+\hat{x}_{k+1}^{l}-\hat{x}_{k}^{l})(\hat{u}_{k+1}^{l+1}+\hat{u}_{k+1}^{l}+\hat{u}_{k}^{l+1}+\hat{u}_{k}^{l}),$
(240)
$\displaystyle({\hat{x}_{k+1}^{l+1}-\hat{x}_{k+1}^{l}+\hat{x}_{k}^{l+1}-\hat{x}_{k}^{l}}+\frac{4}{b})({\hat{x}_{k+1}^{l+1}-\hat{x}_{k+1}^{l}-\hat{x}_{k}^{l+1}+\hat{x}_{k}^{l}})$
$\displaystyle=(\hat{u}_{k+1}^{l+1}+\hat{u}_{k+1}^{l}+\hat{u}_{k}^{l+1}+\hat{u}_{k}^{l})({\hat{u}_{k+1}^{l+1}+\hat{u}_{k+1}^{l}-\hat{u}_{k}^{l+1}-\hat{u}_{k}^{l}}),$
(241)

and conserved quantities $\tilde{I}_{k}^{l}$ and $\tilde{J}_{k}^{l}$ are
recast into

$\displaystyle\tilde{J}_{k}^{l}=\left(\frac{1}{b}+\frac{\hat{x}_{k}^{l+1}-\hat{x}_{k}^{l}}{2}\right)^{2}-\left(\frac{\hat{u}_{k}^{l}+\hat{u}_{k}^{l+1}}{2}\right)^{2},$
(242)
$\displaystyle\tilde{I}_{k}^{l}=(\frac{\hat{x}_{k+1}^{l}-\hat{x}_{k}^{l}}{2})^{2}-\left(\frac{\hat{u}_{k+1}^{l}-\hat{u}_{k}^{l}}{2}\right)^{2}.$
(243)

Here $N$-soliton solutions of the full-discrete SP equation with $\sigma=1$
also exhibits the singular nature.

##  4 The semi-discrete gsG equation

###  4.1 From the fully discrete gsG equation to the semi-discrete gsG
equation

In this section, we demonstrate that the proposed fully discrete gsG equation
with $\nu=-1$ converges to the semi-discrete equation we obtained in [5] in
the continuous limit $b\rightarrow 0$. Moreover, we give a semi-discrete gsG
equation with $\nu=1$ through the same continuous limit from the fully
discrete gsG equation with $\nu=1$.

####  4.1.1 The semi-discrete gsG equation with $\nu=-1$

Recall that

$\displaystyle\Delta_{k}^{l}$
$\displaystyle=\frac{\sqrt{c^{2}-a^{2}}\sinh\frac{x_{k+1}^{l+1}-x_{k}^{l+1}+x_{k+1}^{l}-x_{k}^{l}-4ac^{-1}+4\chi_{2}}{4}}{\sqrt{b^{2}c^{2}-1}\sinh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}+x_{k}^{l+1}-x_{k}^{l}-4bc+4\chi_{1}}{4}}$
$\displaystyle=\frac{\sqrt{c^{2}-a^{2}}\sinh\frac{x_{k+1}^{l+1}-x_{k}^{l+1}+x_{k+1}^{l}-x_{k}^{l}-4ac^{-1}+4\chi_{2}}{4}}{\cosh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}+x_{k}^{l+1}-x_{k}^{l}-4bc}{4}+bc\sinh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}+x_{k}^{l+1}-x_{k}^{l}-4bc}{4}}.$
(244)

Obviously, as $b\rightarrow 0$, we have

$\displaystyle\cosh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}+x_{k}^{l+1}-x_{k}^{l}-4bc}{4}+bc\sinh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}+x_{k}^{l+1}-x_{k}^{l}-4bc}{4}\rightarrow
1,$ (245)
$\displaystyle\sqrt{c^{2}-a^{2}}\sinh\frac{x_{k+1}^{l+1}-x_{k}^{l+1}+x_{k+1}^{l}-x_{k}^{l}-4ac^{-1}+4\chi_{2}}{4}\rightarrow\sqrt{c^{2}-a^{2}}\sinh\frac{x_{k+1}-x_{k}-2ac^{-1}+2\chi_{2}}{2}.$
(246)

Note the relation (170) holds. Thus we have

$\displaystyle\Delta_{k}^{l}\rightarrow\sqrt{a^{2}-c^{2}\sin^{2}\frac{u_{k+1}-u_{k}}{2}}=\frac{\Delta_{k}}{2}.$
(247)

Furthermore, we can easily verify that

$\displaystyle\frac{1}{b}\sin\frac{u_{k+1}^{l+1}-u_{k+1}^{l}-u_{k}^{l+1}+u_{k}^{l}}{4}\rightarrow\frac{\mathrm{d}}{\mathrm{d}\tau}\frac{u_{k+1}-u_{k}}{2},$
(248)

and

$\displaystyle\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}+u_{k}^{l+1}+u_{k}^{l}}{4}\rightarrow\sin\frac{u_{k+1}+u_{k}}{2}.$
(249)

Thus equation (166) converges to

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\tau}\left(u_{k+1}-u_{k}\right)=\Delta_{k}\sin\frac{u_{k+1}+u_{k}}{2},$
(250)
$\displaystyle\Delta_{k}=\sqrt{4a^{2}-4c^{2}\sin^{2}\frac{u_{k+1}-u_{k}}{2}},$
(251)

which are actually part of the semi-discrete gsG equation with $\nu=-1$ (see
eqs. (3.47) and (3.49) in [5]). Here we used
$\frac{f^{l+1}-f^{l}}{2b}\rightarrow\frac{\mathrm{d}}{\mathrm{d}\tau}f$ as
$b\rightarrow 0$. For equation (167), the continuous limit $b\rightarrow 0$
leads to

$\displaystyle({b^{2}c^{2}-1})\sinh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}+x_{k}^{l+1}-x_{k}^{l}-4bc+4\chi_{1}}{2}$
$\displaystyle=$
$\displaystyle({b^{2}c^{2}+1})\sinh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}+x_{k}^{l+1}-x_{k}^{l}-4bc}{2}+2bc\cosh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}+x_{k}^{l+1}-x_{k}^{l}-4bc}{2}$
$\displaystyle\sim$
$\displaystyle({b^{2}c^{2}+1})\left[\left(\frac{\mathrm{d}}{\mathrm{d}\tau}(x_{k}+x_{k+1})-2c\right)b+\mathcal{O}(b^{2})\right]+2bc+\mathcal{O}(b^{3})$
$\displaystyle\sim$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\tau}(x_{k}+x_{k+1})b+\mathcal{O}(b^{3}),$
(252)
$\displaystyle\sinh\frac{x_{k+1}^{l+1}-x_{k+1}^{l}-x_{k}^{l+1}+x_{k}^{l}}{2}\sim\frac{\mathrm{d}}{\mathrm{d}\tau}(x_{k+1}-x_{k})b+\mathcal{O}(b^{3}).$
(253)

Note that

$\displaystyle\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}+u_{k}^{l+1}+u_{k}^{l}}{2}\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}-u_{k}^{l+1}-u_{k}^{l}}{2}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\cos(u_{k}^{l+1}+u_{k}^{l})-\cos(u_{k+1}^{l+1}+u_{k+1}^{l})\right)$
$\displaystyle=$
$\displaystyle\cos^{2}\frac{u_{k}^{l+1}+u_{k}^{l}}{2}-\cos^{2}\frac{u_{k+1}^{l+1}+u_{k+1}^{l}}{2}$
$\displaystyle=$
$\displaystyle(\cos\frac{u_{k}^{l+1}+u_{k}^{l}}{2}+\cos\frac{u_{k+1}^{l+1}+u_{k+1}^{l}}{2})(\cos\frac{u_{k}^{l+1}+u_{k}^{l}}{2}-\cos\frac{u_{k+1}^{l+1}+u_{k+1}^{l}}{2}),$
(254)

and

$\displaystyle
bc\left(\cos\frac{u_{k}^{l+1}+u_{k}^{l}}{2}+\cos\frac{u_{k+1}^{l+1}+u_{k+1}^{l}}{2}\right)$
$\displaystyle=$
$\displaystyle\sqrt{b^{2}c^{2}-1+(b^{2}c^{2}-1)\sinh^{2}\frac{x_{k}^{l+1}-x_{k}^{l}-2bc+2\chi_{1}}{2}}+\sqrt{b^{2}c^{2}-1+(b^{2}c^{2}-1)\sinh^{2}\frac{x_{k+1}^{l+1}-x_{k+1}^{l}-2bc+2\chi_{1}}{2}}$
$\displaystyle\sim$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\tau}(x_{k}+x_{k+1})b+\mathcal{O}(b^{3}).$
(255)

If we divide both sides of (167) by
$b^{2}\frac{\mathrm{d}}{\mathrm{d}\tau}(x_{k}+x_{k+1})$ and take $b\rightarrow
0$, we arrive at

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\tau}(x_{k+1}-x_{k})=c(\cos
u_{k+1}-\cos u_{k}),$ (256)

Eqs. (250), (251) and (256) are the semi-discrete analog of the gsG equation
we proposed in [5]. It can be easily verified that the $\tau$-function and the
variable transformations converge to those in the semi-discrete gsG equation
with $\nu=-1$

$\displaystyle
u_{k}(\tau)=\mathrm{i}\ln\frac{\bar{f}_{k}\bar{g}_{k}}{f_{k}g_{k}},\
\phi_{k}(\tau)=\mathrm{i}\ln\frac{g_{k}\bar{f}_{k}}{f_{k}\bar{g}_{k}},$ (257)
$\displaystyle
x_{k}(\tau)=2kac^{-1}+c\tau+\ln\frac{\bar{g}_{k}g_{k}}{f_{k}\bar{f}_{k}},\
t=c\tau,$ (258) $\displaystyle f_{k}=\tau_{00}(k),\,g_{k}=\tau_{01}(k),$ (259)

with

$\displaystyle\tau_{n,m}(\frac{\tau}{2},k)=\left|m_{ij}^{n,m}(k)\right|_{N\times
N}=\left|\mathrm{i}\delta_{ij}+\frac{1}{p_{i}+p_{j}}\left(-\frac{p_{i}}{p_{j}}\right)^{n}\left(\frac{1-cp_{i}}{1+cp_{j}}\right)^{m}\left(\frac{1-ap_{i}}{1+ap_{j}}\right)^{-k}e^{\xi_{i}+\eta_{j}}\right|_{N\times
N},$
$\displaystyle\xi_{i}=\frac{\tau}{2p_{i}}+\xi_{i0},\quad\eta_{j}=\frac{\tau}{2p_{j}}+\eta_{j0},$

or

$\displaystyle\tau_{nm}(\frac{\tau}{2},k)=\left|\phi_{(n+j-1)}^{(i)}(k,l,m)\right|_{1\leqslant
i,j\leqslant N},$ (260)

with

$\displaystyle\phi_{(n)}^{(i)}(k,l,m)=p_{i}^{n}\left(1-ap_{i}\right)^{-k}\left(1-cp_{i}\right)^{m}e^{\frac{1}{2p_{i}}\tau+\xi_{i0}}+\mathrm{i}(-p_{i})^{n}\left(1+ap_{i}\right)^{-k}\left(1+cp_{i}\right)^{m}e^{-\frac{1}{2p_{i}}\tau+\eta_{i0}}.$

####  4.1.2 The semi-discrete gsG equation with $\nu=1$

With the continuous limit $b\rightarrow 0$ and the similar procedure in 4.1.1,
one can obtain the following theorem.

######  Theorem 4.1.

An integrable semi-discrete analogue of the gsG equation with $\nu=1$ is of
the form

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\tau}\left(u_{k+1}-u_{k}\right)=\tilde{\Delta}_{k}\sin\frac{u_{k+1}+u_{k}}{2},$
(261) $\displaystyle\frac{\mathrm{d}\delta_{k}}{\mathrm{d}\tau}=\lambda(\cos
u_{k}-\cos u_{k+1}),$ (262)

where the lattice parameter $\tilde{\Delta}_{k}$ is a function depending on
$(k,\tau)$ defined by

$\displaystyle\tilde{\Delta}_{k}=\sqrt{4a^{2}+4\lambda^{2}\sin^{2}\frac{u_{k+1}-u_{k}}{2}}.$
(263)

Moreover, the $N$-soliton solution is given by

$\displaystyle u_{k}(\tau)$
$\displaystyle=\mathrm{i}\ln\frac{\bar{f}_{k}\bar{g}_{k}}{f_{k}g_{k}},$ (264)
$\displaystyle x_{k}(\tau)$
$\displaystyle=2ka\lambda^{-1}-\lambda\tau+\mathrm{i}\ln\frac{\bar{f}_{k}g_{k}}{f_{k}\bar{g}_{k}},\
t=\lambda\tau,$ (265) $\displaystyle\varphi_{k}(\tau)$
$\displaystyle=\ln\frac{g_{k}\bar{g}_{k}}{f_{k}\bar{f}_{k}},$ (266)

where $f_{k},g_{k},\bar{f}_{k}$ and $\bar{g}_{k}$ are $\tau$-functions defined
by

$\displaystyle f_{k}=\tau_{00}(k),\,g_{k}=\tau_{01}(k),$ (267)

either with Gram-type determinant

$\displaystyle\tau_{n,m}(\frac{\tau}{2},k)=\left|m_{ij}^{n,m}(k)\right|_{N\times
N}=\left|\mathrm{i}\sqrt{\frac{1-\lambda\mathrm{i}p_{i}}{1+\lambda\mathrm{i}q_{j}}}\delta_{ij}+\frac{1}{p_{i}+p_{j}}\left(-\frac{p_{i}}{p_{j}}\right)^{n}\left(\frac{1-\lambda\mathrm{i}p_{i}}{1+\lambda\mathrm{i}p_{j}}\right)^{m}\left(\frac{1-ap_{i}}{1+ap_{j}}\right)^{-k}e^{\xi_{i}+\eta_{j}}\right|_{N\times
N},$

and

$\displaystyle\xi_{i}=\frac{\tau}{2p_{i}}+\xi_{i0},\quad\eta_{j}=\frac{\tau}{2p_{j}}+\eta_{j0},$

or with Casorati-type determinant

$\displaystyle\tau_{nm}(\frac{\tau}{2},k)=\left|\phi_{(n+j-1)}^{(i)}(k,m)\right|_{1\leqslant
i,j\leqslant N},$ (268)
$\displaystyle\phi_{(n)}^{(i)}(k,m)=p_{i}^{n}\left(1-ap_{i}\right)^{-k}\left(1-cp_{i}\right)^{m}e^{\frac{1}{2p_{i}}\tau+\xi_{i0}}+\mathrm{i}\left({\frac{1-cp_{i}}{1+cp_{i}}}\right)^{\frac{1}{2}}(-p_{i})^{n}\left(1+ap_{i}\right)^{-k}\left(1+cp_{i}\right)^{m}e^{-\frac{1}{2p_{i}}\tau+\eta_{i0}}.$

######  Remark 4.1.

Semi-discrete analogue of the generalized sG equation with $\nu=-1$ can be
transformed into the case with $\nu=1$ we proposed here through

$\displaystyle\phi_{k}=\mathrm{i}\varphi_{k},\ \tilde{x}_{k}=\mathrm{i}x_{k},\
\tilde{t}=-\mathrm{i}t,\ c=\lambda\mathrm{i}.$ (269)

According to a proof similar to that in [5], we know that the continuous limit
of this semi-discrete analogue we proposed in this paper is the gsG equation
with $\nu=1$. However, their solutions are pretty dissimilar which we will
illustrate in Section 5.

###  4.2 From the semi-discrete 2DTL equation to the semi-discrete gsG
equation with $\nu=1$

We proposed two integrable semi-discrete gsG equations with $\nu=-1$ from the
semi-discrete 2DTL equation in [5]. In this part, we show that the semi-
discrete gsG equation with $\nu=1$ can also be obtained from the semi-discrete
2DTL equation through some reductions and appropriate definitions of discrete
hodograph transformation.

We start with the following semi-discrete 2DTL equations

$\displaystyle\left(\frac{1}{a}D_{x_{-1}}-1\right)\tau_{n,m}(k+1)\cdot\tau_{n,m}(k)+\tau_{n+1,m}(k+1)\tau_{n-1,m}(k)=0,$
(270)
$\displaystyle\left(\frac{1}{c}D_{x_{-1}}-1\right)\tau_{n,m}(k)\cdot\tau_{n,m+1}(k)+\tau_{n+1,m}(k)\tau_{n-1,m+1}(k)=0.$
(271)

Here $a$ is a spatial discrete step. Eqs. (270) can also be viewed as Bäcklund
transformations for the 2DTL equations in the sense that if $\tau_{n,m}(k)$ is
a solution to the 2DTL equations (35), so is $\tau_{n,m}(k+1)$, while Eq.
(271) is the BT linking the solution $\tau_{n,m}(k)$ of Eq. (270) to
$\tau_{n,m+1}(k)$.

######  Lemma 4.1.

The bilinear equations (270)-(271) admit the following Casorati determinant
solutions

$\displaystyle\tau_{n,m}\left(x_{-1},k\right)=\left|\begin{array}[]{cccc}\phi_{n,m}^{(1)}(k)&\phi_{n+1,m}^{(1)}(k)&\cdots&\phi_{n+N-1,m}^{(1)}(k)\\\
\phi_{n,m}^{(2)}(k)&\phi_{n+1,m}^{(2)}(k)&\cdots&\phi_{n+N-1,m}^{(2)}(k)\\\
\cdots&\cdots&\cdots&\cdots\\\
\phi_{n,m}^{(N)}(k)&\phi_{n+1,m}^{(N)}(k)&\cdots&\phi_{n+N-1,m}^{(N)}(k)\end{array}\right|,$
(276)

where

$\displaystyle\phi_{n,m}^{(i)}(k)=c_{i}p_{i}^{n}\left(1-cp_{i}\right)^{m}\left(1-ap_{i}\right)^{-k}e^{\xi_{i}}+d_{i}q_{i}^{n}\left(1-cq_{i}\right)^{m}\left(1-aq_{i}\right)^{-k}e^{\eta_{i}},$
(277)

with

$\xi_{i}=p_{i}{}^{-1}x_{-1}+\xi_{i0},\quad\eta_{i}=q_{i}{}^{-1}x_{-1}+\eta_{i0},$

and Gram-type determinant solutions

$\displaystyle\tau_{n,m}(x_{-1},k)=\left|m_{ij}^{n,m}(k)\right|_{N\times
N}=\left|c_{ij}+\frac{1}{p_{i}+q_{j}}\left(-\frac{p_{i}}{q_{j}}\right)^{n}\left(\frac{1-cp_{i}}{1+cq_{j}}\right)^{m}\left(\frac{1-ap_{i}}{1+aq_{j}}\right)^{-k}e^{\xi_{i}+\eta_{j}}\right|_{N\times
N},$ (278)

with

$\displaystyle\xi_{i}=p_{i}^{-1}x_{-1}+\xi_{i0},\quad\eta_{j}=q_{j}^{-1}x_{-1}+\eta_{j0}.$
(279)

Here $p_{i},q_{i},\xi_{i0}$ and $\eta_{i0}$ are arbitrary parameters which can
take either real or complex values.

Applying reductions similar to continuous case, then we could have each of the
$\tau$ functions satisfies the following relations

$\displaystyle\tau_{n+1,0}\Bumpeq\bar{\tau}_{n,1},\,\tau_{n+1,1}\Bumpeq\bar{\tau}_{n,0}.$
(280)

By putting $\tau=2x_{-1},\,\tau_{00}(k)=f_{k},\ \tau_{01}(k)=g_{k}$,
(270)-(271) can be converted into

$\displaystyle\left(\frac{2}{a}D_{\tau}-1\right)f_{k+1}\cdot
f_{k}+\bar{g}_{k+1}\bar{g}_{k}=0,$ (281)
$\displaystyle\left(\frac{2}{a}D_{\tau}-1\right)\bar{f}_{k+1}\cdot\bar{f}_{k}+{g}_{k+1}{g}_{k}=0,$
(282)
$\displaystyle\left(\frac{2}{a}D_{\tau}-1\right){g}_{k+1}\cdot{g}_{k}+\bar{f}_{k+1}\cdot\bar{f}_{k}=0,$
(283)
$\displaystyle\left(\frac{2}{a}D_{\tau}-1\right)\bar{g}_{k+1}\cdot\bar{g}_{k}+{f}_{k+1}\cdot{f}_{k}=0,$
(284) $\displaystyle(-2\mathrm{i}\lambda^{-1}D_{\tau}-1)f_{k}\cdot
g_{k}+\bar{f}_{k}\bar{g}_{k}=0,$ (285)
$\displaystyle(2\mathrm{i}\lambda^{-1}D_{\tau}-1)\bar{f}_{k}\cdot\bar{g}_{k}+f_{k}g_{k}=0.$
(286)

Now we can rewrite the bilinear equations (281)-(286) as

$\displaystyle\frac{2}{a}\left(\ln\frac{f_{k+1}}{f_{k}}\right)_{\tau}-1=-\frac{\bar{g}_{k+1}\bar{g}_{k}}{f_{k+1}f_{k}},$
(287)
$\displaystyle\frac{2}{a}\left(\ln\frac{\bar{f}_{k+1}}{\bar{f}_{k}}\right)_{\tau}-1=-\frac{g_{k+1}g_{k}}{\bar{f}_{k+1}\bar{f}_{k}},$
(288)
$\displaystyle\frac{2}{a}\left(\ln\frac{g_{k+1}}{g_{k}}\right)_{\tau}-1=-\frac{\bar{f}_{k+1}\bar{f}_{k}}{g_{k+1}g_{k}},$
(289)
$\displaystyle\frac{2}{a}\left(\ln\frac{\bar{g}_{k+1}}{\bar{g}_{k}}\right)_{\tau}-1=-\frac{f_{k+1}f_{k}}{\bar{g}_{k+1}\bar{g}_{k}},$
(290) $\displaystyle
2\mathrm{i}\lambda^{-1}\left(\ln\frac{f_{k}}{g_{k}}\right)_{\tau}+1$
$\displaystyle=\frac{\bar{f}_{k}\bar{g}_{k}}{f_{k}g_{k}},$ (291)
$\displaystyle
2\mathrm{i}\lambda^{-1}\left(\ln\frac{\bar{f}_{k}}{\bar{g}_{k}}\right)_{\tau}-1$
$\displaystyle=-\frac{f_{k}g_{k}}{\bar{f}_{k}\bar{g}_{k}}.$ (292)

Introducing two intermediate variables

$\displaystyle\sigma_{k}(\tau)$
$\displaystyle=2\mathrm{i}\ln\frac{\bar{g}_{k}(s)}{f_{k}(s)},$ (293)
$\displaystyle\sigma_{k}^{\prime}(\tau)$
$\displaystyle=2\mathrm{i}\ln\frac{\bar{f}_{k}(s)}{g_{k}(s)},$ (294)

one arrives at a pair of semi-discrete sG equation

$\displaystyle\frac{1}{2a}\left(\sigma_{k+1}-\sigma_{k}\right)_{\tau}=\sin\left(\frac{\sigma_{k+1}+\sigma_{k}}{2}\right),$
(295)
$\displaystyle\frac{1}{2a}\left(\sigma_{k+1}^{\prime}-\sigma_{k}^{\prime}\right)_{\tau}=\sin\left(\frac{\sigma_{k+1}^{\prime}+\sigma_{k}^{\prime}}{2}\right).$
(296)

Next we introduce dependent variable transformations

$\displaystyle
u_{k}(\tau)=\frac{1}{2}\left(\sigma_{k}+\sigma_{k}^{\prime}\right)=\mathrm{i}\ln\frac{\bar{f}_{k}\bar{g}_{k}}{f_{k}g_{k}},\
\varphi_{k}(\tau)=\frac{1}{2\mathrm{i}}\left(\sigma_{k}-\sigma_{k}^{\prime}\right)=\ln\frac{g_{k}\bar{g}_{k}}{f_{k}\bar{f}_{k}},$
(297)

######  Proposition 2.

$\displaystyle\frac{\mathrm{d}\varphi_{k}}{\mathrm{d}\tau}=\lambda\sin u_{k}$
(298) $\displaystyle
a\sinh\left(\frac{\varphi_{k+1}+\varphi_{k}}{2}\right)=\lambda\sin\left(\frac{u_{k+1}-u_{k}}{2}\right)$
(299)

One can find a similar proof of this proposition in [5] for details, which we
omit here. We define a discrete hodograph transformation

$\displaystyle
x_{k}=2ka\lambda^{-1}-\lambda\tau+\mathrm{i}\ln\frac{\bar{f}_{k}g_{k}}{f_{k}\bar{g}_{k}},$
(300)

then the nonuniform mesh can be derived as

$\displaystyle\delta_{k}=x_{k+1}-x_{k}=2a\lambda^{-1}+\mathrm{i}\ln\frac{\bar{f}_{k+1}g_{k+1}f_{k}\bar{g}_{k}}{f_{k+1}\bar{g}_{k+1}\bar{f}_{k}g_{k}}.$
(301)

Taking the derivative with respect to $\tau$ results in

$\displaystyle\frac{\mathrm{d}\delta_{k}}{\mathrm{d}\tau}$
$\displaystyle=\mathrm{i}\left(\ln\frac{\bar{f}_{k+1}}{\bar{g}_{k+1}}+\ln\frac{f_{k}}{g_{k}}-\ln\frac{f_{k+1}}{g_{k+1}}-\ln\frac{\bar{f}_{k}}{\bar{g}_{k}}\right)_{\tau}$
$\displaystyle=\frac{\lambda}{2}\left(\frac{f_{k}g_{k}}{\bar{f}_{k}\bar{g}_{k}}+\frac{\bar{f}_{k}\bar{g}_{k}}{f_{k}g_{k}}-\frac{f_{k+1}g_{k+1}}{\bar{f}_{k+1}\bar{g}_{k+1}}-\frac{\bar{f}_{k+1}\bar{g}_{k+1}}{f_{k+1}g_{k+1}}\right)$
$\displaystyle=\lambda(\cos u_{k}-\cos u_{k+1}).$ (302)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\tau}\left(u_{k+1}-u_{k}\right)$
$\displaystyle=\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}\tau}\left(\sigma_{k+1}+\sigma^{\prime}_{k+1}-\sigma_{k}-\sigma^{\prime}_{k}\right)$
$\displaystyle=2a\sin\frac{u_{k}+u_{k+1}}{2}\cosh\frac{\varphi_{k}+\varphi_{k+1}}{2}$
$\displaystyle=\sin\frac{u_{k}+u_{k+1}}{2}\sqrt{4a^{2}+4\lambda^{2}\sin^{2}\frac{u_{k+1}-u_{k}}{2}}.$
(303)

Therefore, we can propose an integrable semi-discrete gsG equation with
$\nu=1$ which is just the same as the analog we obtained from the fully
discrete gsG equation in Theorem 4.1.

####  4.2.1 Another integrable semi-discrete generalized sine-Gordon equation
with $\nu=1$

If we define a discrete hodograph transformation

$\displaystyle\delta_{k}=2a\lambda^{-1}\cosh\left(\frac{\varphi_{k}+\varphi_{k+1}}{2}\right).$
(304)

Then, we have

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\tau}(u_{k+1}-u_{k})$
$\displaystyle=\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}\tau}(\sigma_{k+1}+\sigma_{k+1}^{\prime}-\sigma_{k}-\sigma_{k}^{\prime})$
$\displaystyle=a\left[\sin\left(\frac{\sigma_{k}+\sigma_{k+1}}{2}\right)+\sin\left(\frac{\sigma_{k}^{\prime}+\sigma_{k+1}^{\prime}}{2}\right)\right]$
$\displaystyle=2a\sin\left(\frac{u_{k}+u_{k+1}}{2}\right)\cos\frac{\mathrm{i}(\varphi_{k}+\varphi_{k+1})}{2}$
$\displaystyle=\lambda\delta_{k}\sin\left(\frac{u_{k}+u_{k+1}}{2}\right).$

On the other hand, taking the derivative with respect to $\delta_{k}$ results
in

$\displaystyle\frac{\mathrm{d}\delta_{k}}{\mathrm{d}\tau}$
$\displaystyle=a\lambda^{-1}\sinh\left(\frac{\varphi_{k}+\varphi_{k+1}}{2}\right)(\varphi_{k}+\varphi_{k+1})_{\tau}$
$\displaystyle=\lambda\sin\left(\frac{u_{k+1}-u_{k}}{2}\right)(\sin u_{k}+\sin
u_{k+1})$
$\displaystyle=\frac{\lambda}{2}\cos\left(\frac{3u_{k}-u_{k+1}}{2}\right)-\frac{\lambda}{2}\cos\left(\frac{3u_{k+1}-u_{k}}{2}\right)$
$\displaystyle=-\lambda\cos\left(\frac{u_{k+1}-u_{k}}{2}\right)(\cos
u_{k+1}-\cos u_{k}).$

Summarizing the above results, we obtain an alternative integrable semi-
discrete analogue of the gsG equation (2) by the following theorem.

######  Theorem 4.2.

An alternative semi-discrete analogue of the gsG equation (2) is of the form

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\tau}(u_{k+1}-u_{k})=\lambda\delta_{k}\sin\left(\frac{u_{k}+u_{k+1}}{2}\right)$
(305)
$\displaystyle\frac{\mathrm{d}\delta_{k}}{\mathrm{d}\tau}=\lambda\cos\left(\frac{u_{k+1}-u_{k}}{2}\right)(\cos
u_{k}-\cos u_{k+1})$ (306)

where the lattice parameter $\delta_{k}$ is a function depending on $(k,\tau)$
defined by

$\displaystyle\delta_{k}=2a\lambda^{-1}\cosh\left(\frac{\varphi_{k}+\varphi_{k+1}}{2}\right),$
(307)

and $t=\lambda\tau.$

###  4.3 Reduction to the semi-discrete sG and semi-discrete SP equation

Similar to the continuous case and the discrete case, we demonstrate that the
semi-discrete gsG equations reduce to the semi-discrete sG equation [15] and
the semi-discrete SP equation [7] with the variable transformations and the
scaling limit.

####  4.3.1 Reduction to the semi-discrete sG equation

(I) From the semi-discrete gsG equation with $\nu=-1$ to the semi-discrete sG
equation. Similar to the continuous case and the discrete case, we rewrite the
elements of the $\tau$-function as

$\displaystyle\phi_{n}^{(i)}(k,1)$
$\displaystyle=p_{i}^{n}(1-ap_{i})^{-k}(1-cp_{i})e^{\frac{1}{2p_{i}}\tau+\xi_{i0}}+\mathrm{i}(-p_{i})^{n}(1+ap_{i})^{-k}\left(1+cp_{i}\right)e^{-\frac{1}{2p_{i}}\tau+\eta_{i0}}$
$\displaystyle=\phi_{n}^{(i)}(k,0)-c\phi_{n+1}^{(i)}(k,0).$ (308)

Thus we have

$\displaystyle
u_{k}=2\mathrm{i}\ln\frac{\bar{f}_{k}}{f_{k}}+\mathnormal{O}(c),$ (309)
$\displaystyle\phi_{k}=\mathnormal{O}(c),$ (310)
$\displaystyle\hat{x}_{k}=cx_{k}=2ka+\mathnormal{O}(c^{2}),$ (311)
$\displaystyle\hat{t}=c^{-1}t=\tau.$ (312)

We can develop the semi-discrete analogue of the gsG equation with $\nu=-1$
proposed in [5] to

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\hat{t}}(u_{k+1}-u_{k})=\sqrt{4a^{2}-4c^{2}\sin^{2}\left(\frac{u_{k+1}-u_{k}}{2}\right)}\sin\left(\frac{u_{k+1}+u_{k}}{2}\right),$
(313)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\hat{t}}(\hat{x}_{k+1}-\hat{x}_{k})=c^{2}(\cos
u_{k+1}-\cos u_{k}).$ (314)

With the scaling limit $c\rightarrow 0$, equation (313)-(314) lead to

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\hat{t}}(u_{k+1}-u_{k})=2a\sin\left(\frac{u_{k+1}+u_{k}}{2}\right),$
(315)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\hat{t}}(\hat{x}_{k+1}-\hat{x}_{k})=0.$
(316)

Here equation (315) is just the semi-discrete sG equation [15]. Additionally,
the solutions also reduce to those for the semi-discrete sG equation [31, 32,
33].

(II) From the semi-discrete gsG equation with $\nu=1$ to the semi-discrete sG
equation. In this case, the relations between $\tau$-functions are the same as
in the semi-discrete gsG equation with $\nu=-1$, then we have

$\displaystyle
u_{k}=2\mathrm{i}\ln\frac{\bar{\hat{f}}_{k}}{\hat{f}_{k}}+\mathnormal{O}(\lambda),\
\phi_{k}=\mathnormal{O}(\lambda),\ \hat{x}_{k}=\lambda
x_{k}=2ka+\mathnormal{O}(\lambda^{2}),\ \hat{t}=\lambda^{-1}t=\tau.$ (317)

One can rewrite the semi-discrete analogue of the gsG equation with $\nu=1$ as

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\hat{t}}\left(u_{k+1}-u_{k}\right)=\sqrt{4a^{2}+4\lambda^{2}\sin^{2}\frac{u_{k+1}-u_{k}}{2}}\sin\frac{u_{k+1}+u_{k}}{2},$
(318)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\hat{t}}\left(\hat{x}_{k+1}-\hat{x}_{k}\right)=\lambda^{2}(\cos
u_{k}-\cos u_{k+1}).$ (319)

Taking $\lambda\rightarrow 0$, one can also arrive at the semi-discrete sG
equation and its determinant solutions.

####  4.3.2 Reduction to the semi-discrete SP equation

(I) From the gsG equation with $\nu=-1$ to the SP equation with $\sigma=-1$.
We have

$\displaystyle\phi_{n}^{(i)}(k,1)$
$\displaystyle=p_{i}^{n}(1-ap_{i})^{-k}(1-cp_{i})e^{\frac{1}{2p_{i}}\tau+\xi_{i0}}+\mathrm{i}(-p_{i})^{n}(1+ap_{i})^{-k}\left(1+cp_{i}\right)e^{-\frac{1}{2p_{i}}\tau+\eta_{i0}}$
$\displaystyle\propto\phi_{n+1}^{(i)}(k,0)-2\epsilon\frac{\mathrm{d}}{\mathrm{d}\tau}\phi_{n+1}^{(i)}(k,0),$
(320)

and

$\displaystyle
g_{k}\propto\bar{f}_{k}-2\epsilon\frac{\mathrm{d}}{\mathrm{d}\tau}\bar{f}_{k}+\mathnormal{O}(\epsilon^{2}),\
\bar{g}_{k}\propto{f}_{k}-2\epsilon\frac{\mathrm{d}}{\mathrm{d}\tau}{f}_{k}+\mathnormal{O}(\epsilon^{2}),$
(321)
$\displaystyle\hat{u}_{k}=\frac{u_{k}}{\epsilon}=2\mathrm{i}\left(\ln\frac{\bar{f}_{k}}{f_{k}}\right)_{\tau}+\mathnormal{O}(\epsilon),\
\phi_{k}=2\mathrm{i}\ln\frac{\bar{f}_{k}}{f_{k}}+\mathnormal{O}(\epsilon),$
(322)
$\displaystyle\hat{x}_{k}=2ka-2(\ln(f_{k}\bar{f}_{k}))_{\tau}+\mathnormal{O}(\epsilon),\
\hat{t}=\tau.$ (323)

The semi-discrete gsG equation with $\nu=-1$ can be recast to

$\displaystyle{\epsilon}\frac{\mathrm{d}}{\mathrm{d}\hat{t}}(\hat{u}_{k+1}-\hat{u}_{k})=\sqrt{4a^{2}-\frac{4}{\epsilon^{2}}\sin^{2}\left(\frac{\epsilon(\hat{u}_{k+1}-\hat{u}_{k})}{2}\right)}\sin\left(\frac{\epsilon(\hat{u}_{k+1}+\hat{u}_{k})}{2}\right),$
(324)
$\displaystyle\epsilon\frac{\mathrm{d}}{\mathrm{d}\hat{t}}(\hat{x}_{k+1}-\hat{x}_{k})=\frac{1}{\epsilon}(\cos(\epsilon\hat{u}_{k+1})-\cos(\epsilon\hat{u}_{k})).$
(325)

Dividing both sides of (324)-(325) by $\epsilon$ and taking
$\epsilon\rightarrow 0$ in (321)-(325), we arrive at

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\hat{t}}(\hat{u}_{k+1}-\hat{u}_{k})=\sqrt{4a^{2}-(\hat{u}_{k+1}-\hat{u}_{k})^{2}}\frac{\hat{u}_{k+1}+\hat{u}_{k}}{2},$
(326)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\hat{t}}(\hat{x}_{k+1}-\hat{x}_{k})=\frac{\hat{u}_{k}^{2}-\hat{u}_{k+1}^{2}}{2},$
(327)

which is nothing but the semi-discrete SP equation proposed in [7] by taking
$X_{k}=\hat{x}_{k}/2$.

(II) From the gsG equation with $\nu=1$ to the SP equation with $\sigma=1$.
With the similar analysis in the cotinuous case and the discrete case, we can
construct the semi-discrete SP equation with $\sigma=1$. By defining

$\displaystyle\hat{\phi}_{n}^{(i)}(k)=p_{i}^{n}(1-ap_{i})^{-k}e^{\frac{1}{2p_{i}}\tau+\xi_{i0}}+(-p_{i})^{n}(1+ap_{i})^{-k}e^{-\frac{1}{2p_{i}}\tau+\eta_{i0}},$
(328)

and

$\displaystyle\hat{f}_{k}=\left|\begin{array}[]{cccc}\hat{\phi}_{1}^{(1)}(k)&\hat{\phi}_{2}^{(1)}(k)&\cdots&\hat{\phi}_{N}^{(1)}(k)\\\
\hat{\phi}_{1}^{(2)}(k)&\hat{\phi}_{2}^{(2)}(k)&\cdots&\hat{\phi}_{N}^{(2)}(k)\\\
\cdots&\cdots&\cdots&\cdots\\\
\hat{\phi}_{1}^{(N)}(k)&\hat{\phi}_{2}^{(N)}(k)&\cdots&\hat{\phi}_{n+N}^{(N)}(k)\end{array}\right|,\
\hat{g}_{k}=\left|\begin{array}[]{cccc}\hat{\phi}_{0}^{(1)}(k)&\hat{\phi}_{1}^{(1)}(k)&\cdots&\hat{\phi}_{N-1}^{(1)}(k)\\\
\hat{\phi}_{0}^{(2)}(k)&\hat{\phi}_{1}^{(2)}(k)&\cdots&\hat{\phi}_{N-1}^{(2)}(k)\\\
\cdots&\cdots&\cdots&\cdots\\\
\hat{\phi}_{0}^{(N)}(k)&\hat{\phi}_{1}^{(N)}(k)&\cdots&\hat{\phi}_{N-1}^{(N)}(k)\end{array}\right|,$
(337)

we know the expansion of the $\tau$-functions for the semi-discrete gsG
equation with $\nu=1$ are

$\displaystyle\phi_{n}^{(i)}(k,0)\propto\hat{\phi}_{n+1}^{(i)}(k)-\mathrm{i}\epsilon\frac{\mathrm{d}}{\mathrm{d}\tau}\hat{\phi}_{n+1}^{(i)}(k)+\mathnormal{O}(\epsilon^{2}),$
(338)
$\displaystyle\phi_{n}^{(i)}(k,1)\propto\hat{\phi}_{n}^{(i)}(k)-\mathrm{i}\epsilon\frac{\mathrm{d}}{\mathrm{d}\tau}\hat{\phi}_{n}^{(i)}(k)+\mathnormal{O}(\epsilon^{2}),$
(339) $\displaystyle
f_{k}=\hat{f}_{k}-{\mathrm{i}\epsilon}\frac{\mathrm{d}}{\mathrm{d}\tau}\hat{f}_{k}+\mathnormal{O}(\epsilon^{2}),\
\bar{f}_{k}=\hat{f}_{k}+{\mathrm{i}\epsilon}\frac{\mathrm{d}}{\mathrm{d}\tau}\hat{f}_{k}+\mathnormal{O}(\epsilon^{2}),$
(340) $\displaystyle
g_{k}=\hat{g}_{k}+{\mathrm{i}\epsilon}\frac{\mathrm{d}}{\mathrm{d}\tau}\hat{g}_{k}+\mathnormal{O}(\epsilon^{2}),\
\bar{g}_{k}=\hat{g}_{k}-{\mathrm{i}\epsilon}\frac{\partial}{\partial\tau}\hat{g}_{k}+\mathnormal{O}(\epsilon^{2}).$
(341)

The semi-discrete dependent transformation and the corresponding variables can
be defined similar to the continuous case as

$\displaystyle\hat{u}_{k}=2\left(\ln\frac{\hat{g}_{k}}{\hat{f}_{k}}\right)_{\tau},\
\varphi_{k}=2\ln\frac{\hat{g}_{k}}{\hat{f}_{k}},\
\hat{x}_{k}=2ka-2(\ln\hat{f}_{k}\hat{g}_{k})_{\tau},\ \hat{t}=\tau,$ (342)

under the scaling limit $\epsilon\rightarrow 0$. In addition, the semi-
discrete SP equation with $\sigma=1$ can be derived from the semi-discrete gsG
equation with $\nu=1$

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\hat{t}}(\hat{u}_{k+1}-\hat{u}_{k})=\sqrt{4a^{2}+(\hat{u}_{k+1}-\hat{u}_{k})^{2}}\frac{\hat{u}_{k+1}+\hat{u}_{k}}{2},$
(343)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\hat{t}}(\hat{x}_{k+1}-\hat{x}_{k})=\frac{\hat{u}_{k+1}^{2}-\hat{u}_{k}^{2}}{2}.$
(344)

Similarly to the continuous equation (107), solutions of the semi-discrete SP
equation with $\sigma=1$ develops singularities. Here we take the one-soliton
solution as an example. The $\tau$-functions for the one-soliton solutions are
given by

$\displaystyle\hat{f}_{k}=p_{1}(1-ap_{1})^{-k}e^{\frac{1}{2p_{1}}\tau+\xi_{10}}-p_{1}(1+ap_{1})^{-k}e^{-\frac{1}{2p_{1}}\tau+\eta_{10}}\propto
1-\left(\frac{1-ap_{1}}{1+ap_{1}}\right)^{-k}e^{p_{1}^{-1}\tau+\xi_{10}-\eta_{10}},$
(345)
$\displaystyle\hat{g}_{k}=(1-ap_{1})^{-k}e^{\frac{1}{2p_{1}}\tau+\xi_{10}}+(1+ap_{1})^{-k}e^{-\frac{1}{2p_{1}}\tau+\eta_{10}}\propto
1+\left(\frac{1-ap_{1}}{1+ap_{1}}\right)^{-k}e^{p_{1}^{-1}\tau+\xi_{10}-\eta_{10}}.$
(346)

One can express the one-soliton solution as

$\displaystyle\hat{u}_{k}=-\frac{2}{p_{1}}\frac{1}{\sinh\left(p_{1}^{-1}\tau+\xi_{10}-\eta_{10}+k\ln\frac{1+ap_{1}}{1-ap_{1}}\right)},$
(347)
$\displaystyle\hat{x}_{k}=2ka-\frac{2}{p_{1}}\frac{1}{\tanh\left(p_{1}^{-1}\tau+\xi_{10}-\eta_{10}+k\ln\frac{1+ap_{1}}{1-ap_{1}}\right)}-\frac{2}{p_{1}}.$
(348)

When $|\hat{x}_{k}|$ tends to infinity, $\hat{u}_{k}$ diverges. The more
detailed description of $N$-soliton solutions will be reported elsewhere.

##  5 One- and two-soliton solutions

###  5.1 One-soliton solution

The $\tau$-functions for the one-soliton solutions of the gsG equation (2) are
given by

$\displaystyle
f=e^{\xi_{1}}+\mathrm{i}\sqrt{\frac{1-\mathrm{i}p_{1}}{1+\mathrm{i}p_{1}}}e^{\eta_{1}}\propto
1+\mathrm{i}\sqrt{\frac{1-\mathrm{i}p}{1+\mathrm{i}p}}e^{-\zeta},$ (349)
$\displaystyle
g=\left(1-\mathrm{i}p_{1}\right)e^{\xi_{1}}+\mathrm{i}\sqrt{\frac{1-\mathrm{i}p_{1}}{1+\mathrm{i}p_{1}}}\left(1+\mathrm{i}p_{1}\right)e^{\eta_{1}}\propto
1+\mathrm{i}\sqrt{\frac{1+\mathrm{i}p}{1-\mathrm{i}p}}e^{-\zeta},$ (350)

with

$\displaystyle\zeta=py+\frac{\tau}{p}+\zeta_{0}.$ (351)

Here we set $p=p_{1}$ and $\lambda=1$ for simplicity. Thus, we are able to
obtain the parametric form of the one-soliton solution

$\displaystyle
u=\mathrm{i}\ln\frac{\bar{f}\bar{g}}{fg}=-2\arctan(\sqrt{1+p^{2}}\sinh\zeta)+\pi,$
(352) $\displaystyle X=x+c_{1}t+2\arctan
p=\frac{\zeta}{p}+2\arctan(p\tanh\zeta),$ (353)

with $c_{1}=1+\frac{1}{p^{2}}$ stands for the velocity of the soliton. By
setting $p\rightarrow-p$, one can see that the one-soliton solution presented
above is equivalent to the solution in [4]. This relates to the fact that if
$u$ solves equation (2), then so do the functions $\pm u+2\pi n$ ($n$:
integer). A detailed analysis of the solution can be seen in [4], which we
omit here.

For the semi-discrete gsG equation with $\nu=1$, the $\tau$-functions are

$\displaystyle f_{k}\propto
1+\mathrm{i}\sqrt{\frac{1-\mathrm{i}p}{1+\mathrm{i}p}}\left(\frac{1-ap}{1+ap}\right)^{k}e^{-\theta},\quad
g_{k}\propto
1+\mathrm{i}\sqrt{\frac{1+\mathrm{i}p}{1-\mathrm{i}p}}\left(\frac{1-ap}{1+ap}\right)^{k}e^{-\theta},$
(354)

with $\theta=p^{-1}\tau+\theta_{0}$. Then the one-soliton solution can be
expressed as

$\displaystyle u_{k}=-2\arctan(\sqrt{1+p^{2}}\sinh\zeta(k))+\pi,$ (355)
$\displaystyle X_{k}=x_{k}+c_{1}t+2\arctan
p=2ka+\frac{1}{p^{2}}t+2\arctan(p\tanh\zeta(k)),$ (356)

where $\zeta(k)=k\ln\frac{1+ap}{1-ap}+\theta$. By taking $a=0.5$, Figure 1
displays kink and anti-kink solutions for the semi-discrete gsG equation and
Figure 2 shows the solutions under different $p$ values. Figure 3 compares the
semi-discrete gsG equation’s kink and anti-kink solutions to those for the gsG
equation. Similar to the continuous case, if $p<0(>0)$, the solution $u_{k}$
represents an kink(anti-kink) solution. The value of $|p|$ has a positive
correlation with the amplitude of
$v_{k}\equiv\frac{u_{k+1}-u_{k}}{\delta_{k}}$.

Fig. 1: Kink and anti-kink solution $u_{k}$ for semi-discrete gsG equation.
(a)$p=-0.5$, (b)$p=0.5$.

Fig. 2: Kink and anti-kink solution $u_{k}$ for semi-discrete gsG equation and
corresponding $v_{k}\equiv\frac{u_{k+1}-u_{k}}{\delta_{k}}$ with different $p$
at $t=0$; dot: $u_{k}$, dashed line: $v_{k}$.

Fig. 3: Comparison between the kink and anti-kink solution of the gsG
equation(solid line) and the semi-discrete gsG equation(dot) at $t=0$.
(a)$p=-0.5$, (b)$p=0.5$.

For the fully discrete gsG equation with $\nu=1$, the $\tau$-functions are

$\displaystyle f_{k}^{l}\propto
1+\mathrm{i}\sqrt{\frac{1-\mathrm{i}p}{1+\mathrm{i}p}}\left(\frac{1-ap}{1+ap}\right)^{k}\left(\frac{1-bp^{-1}}{1+bp^{-1}}\right)^{l},\
g_{k}^{l}\propto
1+\mathrm{i}\sqrt{\frac{1+\mathrm{i}p}{1-\mathrm{i}p}}\left(\frac{1-ap}{1+ap}\right)^{k}\left(\frac{1-bp^{-1}}{1+bp^{-1}}\right)^{l}.$
(357)

Then the one-soliton solution can be expressed as

$\displaystyle u_{k}^{l}=-2\arctan(\sqrt{1+p^{2}}\sinh\zeta(k,l))+\pi,$ (358)
$\displaystyle x_{k}=2ka-2lb+2\arctan(p\tanh\zeta(k,l))-2\arctan p,$ (359)

where $\zeta(k,l)=k\ln\frac{1+ap}{1-ap}+l\ln\frac{1+bp^{-1}}{1-bp^{-1}}$. For
$a=0.5$ and $b=0.1$, Figure 4 shows kink and anti-kink solutions for the fully
discrete gsG equation with $\nu=1$.

Fig. 4: Kink and anti-kink solution $u_{k}^{l}$ for fully discrete gsG
equation with $\nu=1$. (a)$p=-0.5$, (b)$p=0.5$.

For $\nu=-1$, one-soliton solutions of the fully discrete gsG equation may be
multi-valued. The $\tau$-functions can be expressed as

$\displaystyle f_{k}^{l}\propto
1+\mathrm{i}\left(\frac{1-ap}{1+ap}\right)^{k}\left(\frac{1-bp^{-1}}{1+bp^{-1}}\right)^{l},\
g_{k}^{l}\propto
1+\mathrm{i}\frac{1+p}{1-p}\left(\frac{1-ap}{1+ap}\right)^{k}\left(\frac{1-bp^{-1}}{1+bp^{-1}}\right)^{l}.$
(360)

The one-soliton solution can therefore be written as

$\displaystyle u_{k}^{l}=-2\arctan(\sinh\zeta(k,l)-p\cosh\zeta(k,l))+\pi,$
(361) $\displaystyle
x_{k}^{l}=2ka+2lb+\ln\left(\frac{1+p^{2}}{\left(1-p\right)^{2}}-\frac{2p}{\left(1-p\right)^{2}}\tanh\zeta(k,l)\right),$
(362)

where $\zeta(k,l)=k\ln\frac{1+ap}{1-ap}+l\ln\frac{1+bp^{-1}}{1-bp^{-1}}$.
Figure 5 shows one-soliton solutions to the fully discrete gsG equation with
$\nu=-1$ for $a=0.1$ and $b=0.1$. Figure 5(a) illustrates that when $p=-0.4$,
$u$ is a single-valued kink solution because $x_{k+1}^{l}-x_{k}^{l}>0$. Figure
5(b) presents irregular kink solution from [4], which exhibits a three-valued
characteristic at $p=-0.9$, whereas for $p=-1.3$, $u$ becomes loop soliton(see
Figure 5(c)).

(a) regular kink

(b) irregular kink

(c) loop soliton

Fig. 5: Regular kink, irregular kink and loop soliton solution $u_{k}^{l}$ for
fully discrete gsG equation with $\nu=-1$. (a)$p=-0.4$, (b)$p=-0.9$,
(c)$p=-1.3$.

###  5.2 Two-soliton solutions

The tau-functions for the two-soliton solutions of the continuous and discrete
gsG equations are given by

(I) the gsG equation with $\nu=1$

$\displaystyle f\propto
1+\mathrm{i}\frac{p_{1}+p_{2}}{p_{2}-p_{1}}\sqrt{\frac{1-\mathrm{i}p_{1}}{1+\mathrm{i}p_{1}}}e^{-\zeta_{1}}-\mathrm{i}\frac{p_{1}+p_{2}}{p_{2}-p_{1}}\sqrt{\frac{1-\mathrm{i}p_{2}}{1+\mathrm{i}p_{2}}}e^{-\zeta_{2}}+\sqrt{\frac{1-\mathrm{i}p_{1}}{1+\mathrm{i}p_{1}}}\sqrt{\frac{1-\mathrm{i}p_{2}}{1+\mathrm{i}p_{2}}}e^{-\zeta_{1}-\zeta_{2}},$
(363) $\displaystyle g\propto
1+\mathrm{i}\frac{p_{1}+p_{2}}{p_{2}-p_{1}}\sqrt{\frac{1+\mathrm{i}p_{1}}{1-\mathrm{i}p_{1}}}e^{-\zeta_{1}}-\mathrm{i}\frac{p_{1}+p_{2}}{p_{2}-p_{1}}\sqrt{\frac{1+\mathrm{i}p_{2}}{1-\mathrm{i}p_{2}}}e^{-\zeta_{2}}+\sqrt{\frac{1+\mathrm{i}p_{1}}{1-\mathrm{i}p_{1}}}\sqrt{\frac{1+\mathrm{i}p_{2}}{1-\mathrm{i}p_{2}}}e^{-\zeta_{1}-\zeta_{2}},$
(364)

with

$\displaystyle\zeta_{i}=p_{i}y+\frac{\tau}{p_{i}}+\zeta_{i0},\ i=1,2,$ (365)

(II) the semi-discrete gsG equation with $\nu=1$

$\displaystyle f_{k}\propto
1+\mathrm{i}\frac{p_{1}+p_{2}}{p_{2}-p_{1}}\sqrt{\frac{1-\mathrm{i}p_{1}}{1+\mathrm{i}p_{1}}}z_{1}^{k}e^{-\theta_{1}}-\mathrm{i}\frac{p_{1}+p_{2}}{p_{2}-p_{1}}\sqrt{\frac{1-\mathrm{i}p_{2}}{1+\mathrm{i}p_{2}}}z_{2}^{k}e^{-\theta_{2}}+\sqrt{\frac{1-\mathrm{i}p_{1}}{1+\mathrm{i}p_{1}}}\sqrt{\frac{1-\mathrm{i}p_{2}}{1+\mathrm{i}p_{2}}}(z_{1}z_{2})^{k}e^{-\theta_{1}-\theta_{2}},$
(366) $\displaystyle g_{k}\propto
1+\mathrm{i}\frac{p_{1}+p_{2}}{p_{2}-p_{1}}\sqrt{\frac{1+\mathrm{i}p_{1}}{1-\mathrm{i}p_{1}}}z_{1}^{k}e^{-\theta_{1}}-\mathrm{i}\frac{p_{1}+p_{2}}{p_{2}-p_{1}}\sqrt{\frac{1+\mathrm{i}p_{2}}{1-\mathrm{i}p_{2}}}z_{2}^{k}e^{-\theta_{2}}+\sqrt{\frac{1+\mathrm{i}p_{1}}{1-\mathrm{i}p_{1}}}\sqrt{\frac{1+\mathrm{i}p_{2}}{1-\mathrm{i}p_{2}}}(z_{1}z_{2})^{k}e^{-\theta_{1}-\zeta_{2}},$
(367)

with

$\displaystyle z_{i}=\frac{1-ap_{i}}{1+ap_{i}},\
\theta_{i}=\frac{\tau}{p_{i}}+\theta_{i0},\ i=1,2,$ (368)

(III) the fully discrete gsG equation with $\nu=1$

$\displaystyle f_{k}^{l}\propto
1+\mathrm{i}\frac{p_{1}+p_{2}}{p_{2}-p_{1}}\sqrt{\frac{1-\mathrm{i}p_{1}}{1+\mathrm{i}p_{1}}}z_{1}^{k}w_{1}^{l}-\mathrm{i}\frac{p_{1}+p_{2}}{p_{2}-p_{1}}\sqrt{\frac{1-\mathrm{i}p_{2}}{1+\mathrm{i}p_{2}}}z_{2}^{k}w_{2}^{l}+\sqrt{\frac{1-\mathrm{i}p_{1}}{1+\mathrm{i}p_{1}}}\sqrt{\frac{1-\mathrm{i}p_{2}}{1+\mathrm{i}p_{2}}}(z_{1}z_{2})^{k}(w_{1}w_{2})^{l},$
(369) $\displaystyle g_{k}^{l}\propto
1+\mathrm{i}\frac{p_{1}+p_{2}}{p_{2}-p_{1}}\sqrt{\frac{1+\mathrm{i}p_{1}}{1-\mathrm{i}p_{1}}}z_{1}^{k}w_{1}^{l}-\mathrm{i}\frac{p_{1}+p_{2}}{p_{2}-p_{1}}\sqrt{\frac{1+\mathrm{i}p_{2}}{1-\mathrm{i}p_{2}}}z_{2}^{k}w_{2}^{l}+\sqrt{\frac{1+\mathrm{i}p_{1}}{1-\mathrm{i}p_{1}}}\sqrt{\frac{1+\mathrm{i}p_{2}}{1-\mathrm{i}p_{2}}}(z_{1}z_{2})^{k}(w_{1}w_{2})^{l},$
(370)

with

$\displaystyle z_{i}=\frac{1-ap_{i}}{1+ap_{i}},\
w_{i}=\frac{1-bp_{i}^{-1}}{1+bp_{i}^{-1}},\ i=1,2,$ (371)

(4) the fully discrete gsG equation with $\nu=-1$

$\displaystyle f_{k}^{l}\propto
1+\mathrm{i}\frac{p_{1}+p_{2}}{p_{2}-p_{1}}z_{1}^{k}w_{1}^{l}-\mathrm{i}\frac{p_{1}+p_{2}}{p_{2}-p_{1}}z_{2}^{k}w_{2}^{l}+(z_{1}z_{2})^{k}(w_{1}w_{2})^{l},$
(372) $\displaystyle g_{k}^{l}\propto
1+\mathrm{i}\frac{p_{1}+p_{2}}{p_{2}-p_{1}}{\frac{1+p_{1}}{1-p_{1}}}z_{1}^{k}w_{1}^{l}-\mathrm{i}\frac{p_{1}+p_{2}}{p_{2}-p_{1}}{\frac{1+p_{2}}{1-p_{2}}}z_{2}^{k}w_{2}^{l}+{\frac{1+p_{1}}{1-p_{1}}}{\frac{1+p_{2}}{1-p_{2}}}(z_{1}z_{2})^{k}(w_{1}w_{2})^{l},$
(373)

with

$\displaystyle z_{i}=\frac{1-ap_{i}}{1+ap_{i}},\
w_{i}=\frac{1-bp_{i}^{-1}}{1+bp_{i}^{-1}},\ i=1,2.$ (374)

In [3, 4], collisions between several types of one-soliton solutions for the
continuous gsG equation with $\nu=\pm 1$ are shown. Collisions of solutions
are similar in the discrete case, thus we omit here. And we demonstrate a
different type of solution known as breathers. As pointed out in [3, 4] and
Theorem 2.1, if we set $p_{1}=\bar{p}_{2}$, one can obtain breather solutions.
Figure 6 displays the comparison between the breather solutions of the gsG and
the semi-discrete gsG equation with $\nu=1$ for $a=0.1,\
p_{1}=-0.3+0.5\mathrm{i},\ p_{2}=-0.3-0.5\mathrm{i}$, in which one can find
that the breather solution of the semi-discrete gsG equation agrees with that
of the gsG equation very well. And Figure 7 shows such kind of breather
solutions also appear in fully discrete gsG equations with $\nu=\pm 1$.

(a) $t=0$

(b) $t=5$

(c) $t=10$

Fig. 6: Comparison between the breather solutions of the gsG(solid line) and
the semi-discrete gsG equation(dot) with $\nu=1$ for $a=0.1,\
p_{1}=-0.3+0.5\mathrm{i},\ p_{2}=-0.3-0.5\mathrm{i}$.

Fig. 7: Breather solutions $u_{k}^{l}$ for fully discrete gsG equation with
$a=b=0.2,\ p_{1}=-0.3+0.5\mathrm{i},\ p_{2}=-0.3-0.5\mathrm{i}$. (a)$\nu=1$,
(b)$\nu=-1$.

##  6 Conclusion

In this paper, we have successfully proposed integrable semi-discrete and
full-discrete analogues of a generalized sine-Gordon equation. Determinant
formulations for the $N$-soliton solutions, encompassing multi-kink solitons
and multi-breather solutions, have been derived for both the continuous and
discrete versions of the gsG equations. We have also investigated reductions
from the gsG equation to the sG equation and the SP equation, both in
continuous and discrete cases. Notably, we have demonstrated the essential
role of the Bäcklund transformation of bilinear equations and its parameters
in the construction and reduction processes. However, certain aspects still
remain unknown. Firstly, it is crucial to determine the Lax pairs associated
with the semi-discrete and full-discrete gsG equations presented in this
study. Given the close relation between the bilinear forms of these equations
and those of the 2DTL equation, derived from the discrete KP equation, it is
natural to explore their connections within the context of Lax pairs.
Identification of Lax pairs would enable further investigations into these
discrete gsG equations. Recent attention has been focused on multi-component
integrable systems, such as the integrable vector sine-Gordon equation [37,
38, 39, 40] and the multi-component short pulse equation [41, 8]. Given that
the gsG equation lies between the SP equation and the sG equation, it is
reasonable to propose multi-component gsG equations by establishing
connections with the SP equation and the sG equation. These intriguing
questions will be addressed in future studies.

## Acknowledgement

G. Yu is supported by National Natural Science Foundation of China (Grant no.
12175155), Shanghai Frontier Research Institute for Modern Analysis and the
Fundamental Research Funds for the Central Universities. B.F. Feng’s work is
supported by the U.S. Department of Defense (DoD), Air Force for Scientific
Research (AFOSR) under grant No. W911NF2010276.

##  Appendix A Proof of Theorem 3.2

###### Proof.

We can rewrite (154) and (157)-(159) as

$\displaystyle\frac{1}{b}\sinh\frac{\varphi_{k}^{l+1}-\varphi_{k}^{l}}{2}$
$\displaystyle=\lambda\sin\frac{u_{k}^{l+1}+u_{k}^{l}}{2},$ (375)
$\displaystyle\frac{1}{b}\cosh\frac{\varphi_{k}^{l+1}-\varphi_{k}^{l}}{2}$
$\displaystyle=\frac{\sqrt{b^{2}\lambda^{2}+1}}{b}\sin\frac{\tilde{x}_{k}^{l+1}-\tilde{x}_{k}^{l}+2b\lambda+2\omega_{1}}{2},\
\sin\omega_{1}=\frac{1}{\sqrt{b^{2}\lambda^{2}+1}},$ (376) $\displaystyle
a\sinh\frac{\varphi_{k}^{l}+\varphi_{k+1}^{l}}{2}$
$\displaystyle=d\sin\frac{u_{k+1}^{l}-u_{k}^{l}}{2},$ (377) $\displaystyle
a\cosh\frac{\varphi_{k}^{l}+\varphi_{k+1}^{l}}{2}$
$\displaystyle=\sqrt{a^{2}+\lambda^{2}}\sin\frac{\tilde{x}_{k+1}^{l}-\tilde{x}_{k}^{l}-2a\lambda^{-1}+2\omega_{2}}{2},\
\sin\omega_{2}=\frac{a}{\sqrt{a^{2}+\lambda^{2}}}.$ (378)

By making a shift of $k\rightarrow k+1$ in (375), then adding and subtracting
it and (375), one obtains

$\displaystyle\sinh\frac{\varphi_{k+1}^{l+1}-\varphi_{k+1}^{l}+\varphi_{k}^{l+1}-\varphi_{k}^{l}}{4}\cosh\frac{\varphi_{k+1}^{l+1}-\varphi_{k+1}^{l}-\varphi_{k}^{l+1}+\varphi_{k}^{l}}{4}$
$\displaystyle=bd\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}+u_{k}^{l+1}+u_{k}^{l}}{4}\cos\frac{u_{k+1}^{l+1}+u_{k+1}^{l}-u_{k}^{l+1}-u_{k}^{l}}{4},$
(379)
$\displaystyle\cosh\frac{\varphi_{k+1}^{l+1}-\varphi_{k+1}^{l}+\varphi_{k}^{l+1}-\varphi_{k}^{l}}{4}\sinh\frac{\varphi_{k+1}^{l+1}-\varphi_{k+1}^{l}-\varphi_{k}^{l+1}+\varphi_{k}^{l}}{4}$
$\displaystyle=bd\cos\frac{u_{k+1}^{l+1}+u_{k+1}^{l}+u_{k}^{l+1}+u_{k}^{l}}{4}\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}-u_{k}^{l+1}-u_{k}^{l}}{4}.$
(380)

Similarly, from equations (376)-(378), we arrive at

$\displaystyle\sqrt{1+b^{2}\lambda^{2}}\sin\frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^{l}+\tilde{x}_{k}^{l+1}-\tilde{x}_{k}^{l}+4b\lambda+4\omega_{1}}{4}\cos\frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^{l}-\tilde{x}_{k}^{l+1}+\tilde{x}_{k}^{l}}{4}$
$\displaystyle=\cosh\frac{\varphi_{k+1}^{l+1}-\varphi_{k+1}^{l}+\varphi_{k}^{l+1}-\varphi_{k}^{l}}{4}\cosh\frac{\varphi_{k+1}^{l+1}-\varphi_{k+1}^{l}-\varphi_{k}^{l+1}+\varphi_{k}^{l}}{4},$
(381)
$\displaystyle\sqrt{1+b^{2}\lambda^{2}}\cos\frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^{l}+\tilde{x}_{k}^{l+1}-\tilde{x}_{k}^{l}+4b\lambda+4\omega_{1}}{4}\sin\frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^{l}-\tilde{x}_{k}^{l+1}+\tilde{x}_{k}^{l}}{4}$
$\displaystyle=\sinh\frac{\varphi_{k+1}^{l+1}-\varphi_{k+1}^{l}+\varphi_{k}^{l+1}-\varphi_{k}^{l}}{4}\sinh\frac{\varphi_{k+1}^{l+1}-\varphi_{k+1}^{l}-\varphi_{k}^{l+1}+\varphi_{k}^{l}}{4},$
(382) $\displaystyle
a\sinh\frac{\varphi_{k+1}^{l+1}+\varphi_{k+1}^{l}+\varphi_{k}^{l+1}+\varphi_{k}^{l}}{4}\cosh\frac{\varphi_{k+1}^{l+1}-\varphi_{k+1}^{l}+\varphi_{k}^{l+1}-\varphi_{k}^{l}}{4}$
$\displaystyle=d\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}-u_{k}^{l+1}-u_{k}^{l}}{4}\cos\frac{u_{k+1}^{l+1}-u_{k+1}^{l}-u_{k}^{l+1}+u_{k}^{l}}{4},$
(383) $\displaystyle
a\cosh\frac{\varphi_{k+1}^{l+1}+\varphi_{k+1}^{l}+\varphi_{k}^{l+1}+\varphi_{k}^{l}}{4}\sinh\frac{\varphi_{k+1}^{l+1}-\varphi_{k+1}^{l}+\varphi_{k}^{l+1}-\varphi_{k}^{l}}{4}$
$\displaystyle=d\cos\frac{u_{k+1}^{l+1}+u_{k+1}^{l}-u_{k}^{l+1}-u_{k}^{l}}{4}\sin\frac{u_{k+1}^{l+1}-u_{k+1}^{l}-u_{k}^{l+1}+u_{k}^{l}}{4},$
(384)

and

$\displaystyle\sqrt{a^{2}+\lambda^{2}}\sin\frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k}^{l+1}+\tilde{x}_{k+1}^{l}-\tilde{x}_{k}^{l}-4a\lambda^{-1}+4\omega_{2}}{4}\cos\frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^{l}-\tilde{x}_{k}^{l+1}+\tilde{x}_{k}^{l}}{4}$
$\displaystyle=a\cosh\frac{\varphi_{k+1}^{l+1}+\varphi_{k+1}^{l}+\varphi_{k}^{l+1}+\varphi_{k}^{l}}{4}\cosh\frac{\varphi_{k+1}^{l+1}-\varphi_{k+1}^{l}+\varphi_{k}^{l+1}-\varphi_{k}^{l}}{4},$
(385)
$\displaystyle\sqrt{a^{2}+\lambda^{2}}\cos\frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k}^{l+1}+\tilde{x}_{k+1}^{l}-\tilde{x}_{k}^{l}-4a\lambda^{-1}+4\omega_{2}}{4}\sin\frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^{l}-\tilde{x}_{k}^{l+1}+\tilde{x}_{k}^{l}}{4}$
$\displaystyle=a\sinh\frac{\varphi_{k+1}^{l+1}+\varphi_{k+1}^{l}+\varphi_{k}^{l+1}+\varphi_{k}^{l}}{4}\sinh\frac{\varphi_{k+1}^{l+1}-\varphi_{k+1}^{l}+\varphi_{k}^{l+1}-\varphi_{k}^{l}}{4},$
(386)

respectively. Note that equation (379) and (384) give

$\displaystyle\frac{1}{ab}\sin\frac{u_{k+1}^{l+1}-u_{k+1}^{l}-u_{k}^{l+1}+u_{k}^{l}}{4}\cosh\frac{\varphi_{k+1}^{l+1}-\varphi_{k+1}^{l}-\varphi_{k}^{l+1}+\varphi_{k}^{l}}{4}$
$\displaystyle=\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}+u_{k}^{l+1}+u_{k}^{l}}{4}\cosh\frac{\varphi_{k+1}^{l+1}+\varphi_{k+1}^{l}+\varphi_{k}^{l+1}+\varphi_{k}^{l}}{4}.$
(387)

Equations (381) and (385) lead to

$\displaystyle
a\sqrt{1+b^{2}\lambda^{2}}\sin\frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^{l}+\tilde{x}_{k}^{l+1}-\tilde{x}_{k}^{l}+4b\lambda+4\omega_{1}}{4}\cosh\frac{\varphi_{k+1}^{l+1}+\varphi_{k+1}^{l}+\varphi_{k}^{l+1}+\varphi_{k}^{l}}{4}$
$\displaystyle=\sqrt{a^{2}+\lambda^{2}}\sin\frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k}^{l+1}+\tilde{x}_{k+1}^{l}-\tilde{x}_{k}^{l}-4a\lambda^{-1}+4\omega_{2}}{4}\cosh\frac{\varphi_{k+1}^{l+1}-\varphi_{k+1}^{l}-\varphi_{k}^{l+1}+\varphi_{k}^{l}}{4},$
(388)

Substituting (388) into (387), we have

$\displaystyle\frac{1}{b}\sin\frac{u_{k+1}^{l+1}-u_{k+1}^{l}-u_{k}^{l+1}+u_{k}^{l}}{4}=\tilde{\Delta}_{k}^{l}\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}+u_{k}^{l+1}+u_{k}^{l}}{4},$
(389)

with

$\displaystyle\tilde{\Delta}_{k}^{l}=\frac{\sqrt{a^{2}+\lambda^{2}}\sin\frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k}^{l+1}+\tilde{x}_{k+1}^{l}-\tilde{x}_{k}^{l}-4a\lambda^{-1}+4\omega_{2}}{4}}{\sqrt{1+b^{2}\lambda^{2}}\sin\frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^{l}+\tilde{x}_{k}^{l+1}-\tilde{x}_{k}^{l}+4b\lambda+4\omega_{1}}{4}}.$
(390)

Subsequently, by multiplying equation (379) and (380), (381) and (382), we
obtain

$\displaystyle
b^{2}\lambda^{2}\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}+u_{k}^{l+1}+u_{k}^{l}}{2}\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}-u_{k}^{l+1}-u_{k}^{l}}{2}$
$\displaystyle=\sinh\frac{\varphi_{k+1}^{l+1}-\varphi_{k+1}^{l}+\varphi_{k}^{l+1}-\varphi_{k}^{l}}{2}\sinh\frac{\varphi_{k+1}^{l+1}-\varphi_{k+1}^{l}-\varphi_{k}^{l+1}+\varphi_{k}^{l}}{2},$
(391)
$\displaystyle({1+b^{2}\lambda^{2}})\sin\frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^{l}+\tilde{x}_{k}^{l+1}-\tilde{x}_{k}^{l}+4b\lambda+4\omega_{1}}{2}\sin\frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^{l}-\tilde{x}_{k}^{l+1}+\tilde{x}_{k}^{l}}{2}$
$\displaystyle=\sinh\frac{\varphi_{k+1}^{l+1}-\varphi_{k+1}^{l}+\varphi_{k}^{l+1}-\varphi_{k}^{l}}{2}\sinh\frac{\varphi_{k+1}^{l+1}-\varphi_{k+1}^{l}-\varphi_{k}^{l+1}+\varphi_{k}^{l}}{2},$
(392)

which can be recast to

$\displaystyle({1+b^{2}\lambda^{2}})\sin\frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^{l}+\tilde{x}_{k}^{l+1}-\tilde{x}_{k}^{l}+4b\lambda+4\omega_{1}}{2}\sin\frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^{l}-\tilde{x}_{k}^{l+1}+\tilde{x}_{k}^{l}}{2}$
$\displaystyle=b^{2}\lambda^{2}\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}+u_{k}^{l+1}+u_{k}^{l}}{2}\sin\frac{u_{k+1}^{l+1}+u_{k+1}^{l}-u_{k}^{l+1}-u_{k}^{l}}{2}.$
(393)

Then we obtain the fully discrete gsG equation with $\nu=1$. In addition, from
(375)-(378), we know

$\displaystyle\tilde{J}_{k}^{l}=\left(\frac{1}{b}\cos\frac{\tilde{x}_{k}^{l+1}-\tilde{x}_{k}^{l}+2b\lambda}{2}+d\sin\frac{\tilde{x}_{k}^{l+1}-\tilde{x}_{k}^{l}+2b\lambda}{2}\right)^{2}-\lambda^{2}\sin^{2}\frac{u_{k}^{l+1}+u_{k}^{l}}{2}=\frac{1}{b^{2}},$
(394)
$\displaystyle\tilde{I}_{k}^{l}=\left(a\cos\frac{\tilde{x}_{k+1}^{l}-\tilde{x}_{k}^{l}-2a\lambda^{-1}}{2}+d\sin\frac{\tilde{x}_{k+1}^{l}-\tilde{x}_{k}^{l}-2a\lambda^{-1}}{2}\right)^{2}-\lambda^{2}\sin^{2}\frac{u_{k+1}^{l}-u_{k}^{l}}{2}={a^{2}}.$
(395)

$\tilde{I}_{k}^{l}$ and $\tilde{J}_{k}^{l}$ are conserved quantities because
$a^{2}$ and $\frac{1}{b^{2}}$ are constants. ∎

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