A Generalized Sine-Gordon Equation: Reductions and Integrable Discretizations

Abstract

In this paper, we propose fully discrete analogues of a generalized sine-Gordon (gsG) equation $u_{t x}=\left( 1+\nu \partial _x^2\right) \sin u$. The key points of the construction are based on the bilinear discrete KP hierarchy and appropriate definition of discrete reciprocal transformations. We derive semi-discrete analogues of the gsG equation from the fully discrete gsG equation by taking the temporal parameter limit $b\rightarrow 0$. In particular, one fully discrete gsG equation is reduced to a semi-discrete gsG equation in the case of $\nu =-1$ (Feng et al. in Numer Algorithms 94:351–370, 2023). Furthermore, N-soliton solutions to the semi- and fully discrete analogues of the gsG equation in the determinant form are presented. Dynamics of one- and two-soliton solutions for the discrete gsG equations are analyzed. By introducing a parameter c, we demonstrate that the gsG equation can reduce to the sine-Gordon equation and the short pulse at the levels of continuous, semi-discrete and fully discrete cases. The limiting forms of the N-soliton solutions to the gsG equation in each level also correspond to those of the sine-Gordon equation and the short pulse equation.

Appendices

A Proof of Theorem 1.1

Proof

Note that we can rewrite (144), (147), (148) and (149) as

$$\begin{aligned}&\frac{1}{b}\sin \frac{\phi _k^{l+1}-\phi _{k}^l}{2}=c\sin \frac{u_k^{l+1}+u_{k}^l}{2}, \end{aligned}$$

(319)

$$\begin{aligned}&\frac{1}{b}\cos \frac{\phi _k^{l\!+\!1}\!-\!\phi _k^l}{2}\!=\!\frac{\sqrt{b^2c^2-1}}{b}\sinh \frac{x_k^{l+1}\!-\!x_k^l-2bc\!+\!2\chi _1}{2}, \sinh \chi _1\!{=}\!\frac{1}{\sqrt{b^2c^2-1}}, \end{aligned}$$

(320)

$$\begin{aligned}&{a}\sin \frac{\phi _k^l+\phi _{k+1}^l}{2}={c}\sin \frac{u_{k+1}^l-u_k^l}{2}, \end{aligned}$$

(321)

$$\begin{aligned}&{a}\cos \frac{\phi _k^l+\phi _{k+1}^l}{2}\!=\!\sqrt{c^2\!-\!a^2}\sinh \frac{x_{k\!+\!1}^l\!-\!x_k^l\!-\!2ac^{-1}\!+\!2\chi _2}{2},\ \sinh \chi _2\!{=}\!\frac{a}{\sqrt{c^2\!-\!a^2}}. \end{aligned}$$

(322)

By making a shift $k\rightarrow k+1$ in (319) first, then adding and subtracting with (319), we obtain

$$\begin{aligned}&\sin \frac{\phi _{k+1}^{l+1}-\phi _{k+1}^l+\phi _k^{l+1}-\phi _k^l}{4}\cos \frac{\phi _{k+1}^{l+1}-\phi _{k+1}^l-\phi _k^{l+1}+\phi _k^l}{4}\nonumber \\&\quad =bc\sin \frac{u_{k+1}^{l+1}+u_{k+1}^l+u_k^{l+1}+u_k^l}{4} \cos \frac{u_{k+1}^{l+1}+u_{k+1}^l-u_k^{l+1}-u_k^l}{4}, \end{aligned}$$

(323)

$$\begin{aligned}&\cos \frac{\phi _{k+1}^{l+1}-\phi _{k+1}^l+\phi _k^{l+1}-\phi _k^l}{4}\sin \frac{\phi _{k+1}^{l+1}-\phi _{k+1}^l-\phi _k^{l+1}+\phi _k^l}{4}\nonumber \\&\quad =bc\cos \frac{u_{k+1}^{l+1}+u_{k+1}^l+u_k^{l+1}+u_k^l}{4} \sin \frac{u_{k+1}^{l+1}+u_{k+1}^l-u_k^{l+1}-u_k^l}{4}. \end{aligned}$$

(324)

Similarly, from (320)–(322), one can obtain

$$\begin{aligned}&\sqrt{b^2c^2-1}\sinh \frac{x_{k+1}^{l+1}-x_{k+1}^l+x_k^{l+1}-x_k^l-4bc+4\chi _1}{4}\cosh \frac{x_{k+1}^{l+1}-x_{k+1}^l-x_k^{l+1}+x_k^l}{4}\nonumber \\&=\cos \frac{\phi _{k+1}^{l+1}-\phi _{k+1}^l+\phi _k^{l+1}-\phi _k^l}{4}\cos \frac{\phi _{k+1}^{l+1}-\phi _{k+1}^l-\phi _k^{l+1}+\phi _k^l}{4}, \end{aligned}$$

(325)

$$\begin{aligned}&\sqrt{b^2c^2-1}\cosh \frac{x_{k+1}^{l+1}-x_{k+1}^l+x_k^{l+1}-x_k^l-4bc+4\chi _1}{4}\sinh \frac{x_{k+1}^{l+1}-x_{k+1}^l-x_k^{l+1}+x_k^l}{4}\nonumber \\&\quad =-\sin \frac{\phi _{k+1}^{l+1}-\phi _{k+1}^l+\phi _k^{l+1}-\phi _k^l}{4}\sin \frac{\phi _{k+1}^{l+1}-\phi _{k+1}^l-\phi _k^{l+1}+\phi _k^l}{4}, \end{aligned}$$

(326)

$$\begin{aligned}&a\sin \frac{\phi _{k+1}^{l+1}+\phi _{k+1}^l+\phi _k^{l+1} +\phi _k^l}{4}\cos \frac{\phi _{k+1}^{l+1}-\phi _{k+1}^l+\phi _k^{l+1}-\phi _k^l}{4}\nonumber \\&\quad =c\sin \frac{u_{k+1}^{l+1}+u_{k+1}^l-u_k^{l+1}-u_k^l}{4}\cos \frac{u_{k+1}^{l+1}-u_{k+1}^l-u_k^{l+1}+u_k^l}{4} , \end{aligned}$$

(327)

$$\begin{aligned}&a\cos \frac{\phi _{k+1}^{l+1}+\phi _{k+1}^l+\phi _k^{l+1}+\phi _k^l}{4}\sin \frac{\phi _{k+1}^{l+1}-\phi _{k+1}^l+\phi _k^{l+1}-\phi _k^l}{4}\nonumber \\&\quad =c\cos \frac{u_{k+1}^{l+1}+u_{k+1}^l-u_k^{l+1}-u_k^l}{4}\sin \frac{u_{k+1}^{l+1}-u_{k+1}^l-u_k^{l+1}+u_k^l}{4} , \end{aligned}$$

(328)

and

$$\begin{aligned}&\sqrt{c^2-a^2}\sinh \frac{x_{k+1}^{l+1}-x_k^{l+1}+x_{k+1}^l-x_k^l-4ac^{-1}+4\chi _2}{4}\nonumber \\&\qquad \times \cosh \frac{x_{k+1}^{l+1}-x_{k+1}^l-x_k^{l+1}+x_k^l}{4}\nonumber \\&\quad =a\cos \frac{\phi _{k+1}^{l+1}+\phi _{k+1}^l+\phi _k^{l+1}+\phi _k^l}{4}\cos \frac{\phi _{k+1}^{l+1}-\phi _{k+1}^l+\phi _k^{l+1}-\phi _k^l}{4}, \end{aligned}$$

(329)

$$\begin{aligned}&\sqrt{c^2-a^2}\cosh \frac{x_{k+1}^{l+1}-x_k^{l+1}+x_{k+1}^l-x_k^l-4ac^{-1}+4\chi _2}{4}\nonumber \\&\qquad \times \sinh \frac{x_{k+1}^{l+1}-x_{k+1}^l-x_k^{l+1}+x_k^l}{4}\nonumber \\&\quad =-a\sin \frac{\phi _{k+1}^{l+1}+\phi _{k+1}^l+\phi _k^{l+1}+\phi _k^l}{4}\sin \frac{\phi _{k+1}^{l+1}-\phi _{k+1}^l+\phi _k^{l+1}-\phi _k^l}{4}. \end{aligned}$$

(330)

Thus, Eqs. (323) and (328) give the equation

$$\begin{aligned}&\frac{1}{a b} \sin \frac{u_{k+1}^{l+1}-u_{k+1}^l-u_k^{l+1}+u_k^l}{4}\cos \frac{\phi _{k+1}^{l+1}-\phi _{k+1}^l-\phi _k^{l+1}+\phi _k^l}{4}\nonumber \\&\quad =\sin \frac{u_{k+1}^{l+1}+u_{k+1}^l+u_k^{l+1}+u_k^l}{4}\cos \frac{\phi _{k+1}^{l+1}+\phi _{k+1}^l+\phi _k^{l+1}+\phi _k^l}{4}. \end{aligned}$$

(331)

Equations (325) and (329) lead to

$$\begin{aligned}&a\sqrt{b^2c^2-1}\sinh \frac{x_{k+1}^{l+1}-x_{k+1}^l+x_k^{l+1}-x_k^l-4bc+4\chi _1}{4}\cos \frac{\phi _{k+1}^{l+1}+\phi _{k+1}^l+\phi _k^{l+1}+\phi _k^l}{4}\nonumber \\&\quad =\sqrt{c^2-a^2}\sinh \frac{x_{k+1}^{l+1}-x_k^{l+1}+x_{k+1}^l-x_k^l-4ac^{-1}+4\chi _2}{4}\nonumber \\&\qquad \times \cos \frac{\phi _{k+1}^{l+1}-\phi _{k+1}^l-\phi _k^{l+1}+\phi _k^l}{4}. \end{aligned}$$

(332)

The substitution of (332) into (331) leads to

$$\begin{aligned} \frac{1}{b}\sin \frac{u_{k+1}^{l+1}-u_{k+1}^l-u_k^{l+1}+u_k^l}{4}=\Delta _k^l \sin \frac{u_{k+1}^{l+1}+u_{k+1}^l+u_k^{l+1}+u_k^l}{4}, \end{aligned}$$

(333)

with

$$\begin{aligned} \Delta _k^l=\frac{\sqrt{c^2-a^2}\sinh \frac{x_{k+1}^{l+1}-x_k^{l+1}+x_{k+1}^l-x_k^l-4ac^{-1}+4\chi _2}{4}}{\sqrt{b^2c^2-1}\sinh \frac{x_{k+1}^{l+1}-x_{k+1}^l+x_k^{l+1}-x_k^l-4bc+4\chi _1}{4}}. \end{aligned}$$

(334)

On the other hand, by multiplying (323) and (324), (325) and (326), we obtain, respectively

$$\begin{aligned}&b^2c^2\sin \frac{u_{k+1}^{l+1}+u_{k+1}^l+u_k^{l+1}+u_k^l}{2} \sin \frac{u_{k+1}^{l+1}+u_{k+1}^l-u_k^{l+1}-u_k^l}{2}\nonumber \\&\quad =\sin \frac{\phi _{k+1}^{l+1}-\phi _{k+1}^l+\phi _k^{l+1}-\phi _k^l}{2}\sin \frac{\phi _{k+1}^{l+1}-\phi _{k+1}^l-\phi _k^{l+1}+\phi _k^l}{2}, \end{aligned}$$

(335)

$$\begin{aligned}&({b^2c^2{-}1})\sinh \frac{x_{k+1}^{l+1}{-}x_{k+1}^l{+}x_k^{l+1}{-}x_k^l-4bc+4\chi _1}{2}\sinh \frac{x_{k+1}^{l+1}-x_{k+1}^l-x_k^{l+1}+x_k^l}{2}\nonumber \\&\quad =-\sin \frac{\phi _{k+1}^{l+1}-\phi _{k+1}^l+\phi _k^{l+1}-\phi _k^l}{2}\sin \frac{\phi _{k+1}^{l+1}-\phi _{k+1}^l-\phi _k^{l+1}+\phi _k^l}{2}, \end{aligned}$$

(336)

which leads to exactly (3) by eliminating the right side of the equations. Meanwhile, from (319)–(322), we have

$$\begin{aligned}&J_k^l=\frac{{b^2c^2-1}}{b^2}\sinh ^2\frac{x_k^{l+1}-x_k^l-2bc+2\chi _1}{2}+c^{2}\sin ^2\frac{u_k^l+u_k^{l+1}}{2}=\frac{1}{b^2}, \end{aligned}$$

(337)

$$\begin{aligned}&I_k^l=({c^2-a^2})\sinh ^2\frac{x_{k+1}^l-x_k^l-2ac^{-1}+2\chi _2}{2}+c^{2}\sin ^2\frac{u_{k+1}^l-u_k^l}{2}=a^2. \end{aligned}$$

(338)

Here $a^2$ and $\frac{1}{b^2}$ are constants and thus $J_k^l$ (337) and $I_k^l$ (338) actually give the conserved quantities. The proof is completed. $\square$

B Proof of Theorem 1.2

Proof

We can rewrite (144) and (147)–(149) as

$$\begin{aligned} \frac{1}{b}\sinh \frac{\varphi _k^{l+1}-\varphi _{k}^l}{2}&=\lambda \sin \frac{u_k^{l+1}+u_{k}^l}{2}, \end{aligned}$$

(339)

$$\begin{aligned} \frac{1}{b }\cosh \frac{\varphi _k^{l\!+\!1}\!-\!\varphi _k^l}{2}&\!=\!\frac{\sqrt{b^2\lambda ^2\!+\!1}}{b}\sin \frac{\tilde{x}_k^{l+1}\!-\!\tilde{x}_k^l\!+\!2b\lambda \!+\!2\omega _1}{2},\ \sin \omega _1\!{=}\!\frac{1}{\sqrt{b^2\lambda ^2\!+\!1}}, \end{aligned}$$

(340)

$$\begin{aligned} a\sinh \frac{\varphi _k^l+\varphi _{k+1}^l}{2}&=d\sin \frac{u_{k+1}^l-u_k^l}{2}, \end{aligned}$$

(341)

$$\begin{aligned} a\cosh \frac{\varphi _k^l\!+\!\varphi _{k+1}^l}{2}&\!=\!\sqrt{a^2\!+\!\lambda ^2}\sin \frac{\tilde{x}_{k\!+\!1}^l-\tilde{x}_k^l\!-\!2a\lambda ^{-1}\!+\!2\omega _2}{2},\ \sin \omega _2\!{=}\!\frac{a}{\sqrt{a^2\!+\!\lambda ^2}}. \end{aligned}$$

(342)

By making a shift of $k\rightarrow k+1$ in (339), then adding and subtracting it and (339), one obtains

$$\begin{aligned}&\sinh \frac{\varphi _{k+1}^{l+1}-\varphi _{k+1}^l+\varphi _k^{l+1}-\varphi _{k}^l}{4}\cosh \frac{\varphi _{k+1}^{l+1}-\varphi _{k+1}^l-\varphi _k^{l+1}+\varphi _{k}^l}{4}\nonumber \\&\quad =bd\sin \frac{u_{k+1}^{l+1}+u_{k+1}^l+u_k^{l+1}+u_{k}^l}{4} \cos \frac{u_{k+1}^{l+1}+u_{k+1}^l-u_k^{l+1}-u_{k}^l}{4}, \end{aligned}$$

(343)

$$\begin{aligned}&\cosh \frac{\varphi _{k+1}^{l+1}-\varphi _{k+1}^l+\varphi _k^{l+1}-\varphi _{k}^l}{4}\sinh \frac{\varphi _{k+1}^{l+1}-\varphi _{k+1}^l-\varphi _k^{l+1}+\varphi _{k}^l}{4}\nonumber \\&\quad =bd\cos \frac{u_{k+1}^{l+1}+u_{k+1}^l+u_k^{l+1}+u_{k}^l}{4} \sin \frac{u_{k+1}^{l+1}+u_{k+1}^l-u_k^{l+1}-u_{k}^l}{4}. \end{aligned}$$

(344)

Similarly, from Eqs. (340)–(342), we arrive at

$$\begin{aligned}&\sqrt{1+b^2\lambda ^2}\sin \frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^l+\tilde{x}_k^{l+1}-\tilde{x}_k^l+4b\lambda +4\omega _1}{4}\cos \frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^l{-}\tilde{x}_k^{l+1}{+}\tilde{x}_k^l}{4}\nonumber \\&\quad =\cosh \frac{\varphi _{k+1}^{l+1}-\varphi _{k+1}^l+\varphi _k^{l+1}-\varphi _k^l}{4}\cosh \frac{\varphi _{k+1}^{l+1}-\varphi _{k+1}^l-\varphi _k^{l+1}+\varphi _k^l}{4}, \end{aligned}$$

(345)

$$\begin{aligned}&\sqrt{1+b^2\lambda ^2}\cos \frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^l+\tilde{x}_k^{l+1}-\tilde{x}_k^l+4b\lambda +4\omega _1}{4}\sin \frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^l{-}\tilde{x}_k^{l+1}{+}\tilde{x}_k^l}{4}\nonumber \\&\quad =\sinh \frac{\varphi _{k+1}^{l+1}-\varphi _{k+1}^l+\varphi _k^{l+1}-\varphi _k^l}{4}\sinh \frac{\varphi _{k+1}^{l+1}-\varphi _{k+1}^l-\varphi _k^{l+1}+\varphi _k^l}{4}, \end{aligned}$$

(346)

$$\begin{aligned}&a\sinh \frac{\varphi _{k+1}^{l+1}+\varphi _{k+1}^l+\varphi _k^{l+1}+\varphi _k^l}{4}\cosh \frac{\varphi _{k+1}^{l+1}-\varphi _{k+1}^l+\varphi _k^{l+1}-\varphi _k^l}{4}\nonumber \\&\quad =d\sin \frac{u_{k+1}^{l+1}+u_{k+1}^l-u_k^{l+1}-u_{k}^l}{4}\cos \frac{u_{k+1}^{l+1}-u_{k+1}^l-u_k^{l+1}+u_{k}^l}{4}, \end{aligned}$$

(347)

$$\begin{aligned}&a\cosh \frac{\varphi _{k+1}^{l+1}+\varphi _{k+1}^l+\varphi _k^{l+1}+\varphi _k^l}{4}\sinh \frac{\varphi _{k+1}^{l+1}-\varphi _{k+1}^l+\varphi _k^{l+1}-\varphi _k^l}{4}\nonumber \\&\quad =d\cos \frac{u_{k+1}^{l+1}+u_{k+1}^l-u_k^{l+1}-u_{k}^l}{4}\sin \frac{u_{k+1}^{l+1}-u_{k+1}^l-u_k^{l+1}+u_{k}^l}{4}, \end{aligned}$$

(348)

and

$$\begin{aligned}&\sqrt{a^2+\lambda ^2}\sin \frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_k^{l+1}+\tilde{x}_{k+1}^l-\tilde{x}_k^l-4a\lambda ^{-1}+4\omega _2}{4}\cos \frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^l-\tilde{x}_k^{l+1}+\tilde{x}_k^l}{4}\nonumber \\&\quad =a\cosh \frac{\varphi _{k+1}^{l+1}+\varphi _{k+1}^l+\varphi _k^{l+1}+\varphi _k^l}{4}\cosh \frac{\varphi _{k+1}^{l+1}-\varphi _{k+1}^l+\varphi _k^{l+1}-\varphi _k^l}{4}, \end{aligned}$$

(349)

$$\begin{aligned}&\sqrt{a^2+\lambda ^2}\cos \frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_k^{l+1}+\tilde{x}_{k+1}^l-\tilde{x}_k^l-4a\lambda ^{-1}+4\omega _2}{4}\sin \frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^l-\tilde{x}_k^{l+1}+\tilde{x}_k^l}{4}\nonumber \\&\quad =a\sinh \frac{\varphi _{k+1}^{l+1}+\varphi _{k+1}^l+\varphi _k^{l+1}+\varphi _k^l}{4}\sinh \frac{\varphi _{k+1}^{l+1}-\varphi _{k+1}^l+\varphi _k^{l+1}-\varphi _k^l}{4}, \end{aligned}$$

(350)

respectively. Note that Eqs. (343) and (348) give

$$\begin{aligned}&\frac{1}{ab} \sin \frac{u_{k+1}^{l+1}-u_{k+1}^l-u_k^{l+1}+u_k^l}{4}\cosh \frac{\varphi _{k+1}^{l+1}-\varphi _{k+1}^l-\varphi _k^{l+1}+\varphi _k^l}{4}\nonumber \\&\quad =\sin \frac{u_{k+1}^{l+1}+u_{k+1}^l+u_k^{l+1}+u_k^l}{4}\cosh \frac{\varphi _{k+1}^{l+1}+\varphi _{k+1}^l+\varphi _k^{l+1}+\varphi _k^l}{4} . \end{aligned}$$

(351)

Equations (345) and (349) lead to

$$\begin{aligned}&a\sqrt{1+b^2\lambda ^2}\sin \frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^l+\tilde{x}_k^{l+1}-\tilde{x}_k^l+4b\lambda +4\omega _1}{4}\nonumber \\&\qquad \times \cosh \frac{\varphi _{k+1}^{l+1}+\varphi _{k+1}^l+\varphi _k^{l+1}+\varphi _k^l}{4}\nonumber \\&\quad =\sqrt{a^2+\lambda ^2}\sin \frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_k^{l+1}+\tilde{x}_{k+1}^l-\tilde{x}_k^l-4a\lambda ^{-1}+4\omega _2}{4}\nonumber \\&\qquad \times \cosh \frac{\varphi _{k+1}^{l+1}-\varphi _{k+1}^l-\varphi _k^{l+1}+\varphi _k^l}{4}, \end{aligned}$$

(352)

Substituting (352) into (351), we have

$$\begin{aligned} \frac{1}{b} \sin \frac{u_{k+1}^{l+1}-u_{k+1}^l-u_k^{l+1}+u_k^l}{4} =\tilde{\Delta }_k^l\sin \frac{u_{k+1}^{l+1}+u_{k+1}^l+u_k^{l+1}+u_k^l}{4}, \end{aligned}$$

(353)

with

$$\begin{aligned} \tilde{\Delta }_k^l=\frac{\sqrt{a^2+\lambda ^2}\sin \frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_k^{l+1}+\tilde{x}_{k+1}^l-\tilde{x}_k^l-4a\lambda ^{-1}+4\omega _2}{4}}{\sqrt{1+b^2\lambda ^2}\sin \frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^l+\tilde{x}_k^{l+1}-\tilde{x}_k^l+4b\lambda +4\omega _1}{4}}. \end{aligned}$$

(354)

Subsequently, by multiplying Eqs. (343) and (344), (345) and (346), we obtain

$$\begin{aligned}&b^2\lambda ^2\sin \frac{u_{k+1}^{l+1}+u_{k+1}^l+u_k^{l+1}+u_{k}^l}{2} \sin \frac{u_{k+1}^{l+1}+u_{k+1}^l-u_k^{l+1}-u_{k}^l}{2}\nonumber \\&\quad =\sinh \frac{\varphi _{k+1}^{l+1}-\varphi _{k+1}^l+\varphi _k^{l+1}-\varphi _{k}^l}{2}\sinh \frac{\varphi _{k+1}^{l+1}-\varphi _{k+1}^l-\varphi _k^{l+1}+\varphi _{k}^l}{2}, \end{aligned}$$

(355)

$$\begin{aligned}&({1+b^2\lambda ^2})\sin \frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^l+\tilde{x}_k^{l+1}-\tilde{x}_k^l+4b\lambda +4\omega _1}{2}\sin \frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^l-\tilde{x}_k^{l+1}{+}\tilde{x}_k^l}{2}\nonumber \\&\quad =\sinh \frac{\varphi _{k+1}^{l+1}-\varphi _{k+1}^l+\varphi _k^{l+1}-\varphi _k^l}{2}\sinh \frac{\varphi _{k+1}^{l+1}-\varphi _{k+1}^l-\varphi _k^{l+1}+\varphi _k^l}{2}, \end{aligned}$$

(356)

which can be recast to

$$\begin{aligned}&({1+b^2\lambda ^2})\sin \frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^l+\tilde{x}_k^{l+1}-\tilde{x}_k^l+4b\lambda +4\omega _1}{2}\sin \frac{\tilde{x}_{k+1}^{l+1}-\tilde{x}_{k+1}^l-\tilde{x}_k^{l+1}{+}\tilde{x}_k^l}{2}\nonumber \\&=b^2\lambda ^2\sin \frac{u_{k+1}^{l+1}+u_{k+1}^l+u_k^{l+1}+u_{k}^l}{2} \sin \frac{u_{k+1}^{l+1}+u_{k+1}^l-u_k^{l+1}-u_{k}^l}{2}. \end{aligned}$$

(357)

Then, we obtain the fully discrete gsG equation with $\nu =1$. In addition, from (339)–(342), we know

$$\begin{aligned} \tilde{J}_k^l=&\left( \frac{1}{b}\cos \frac{\tilde{x}_k^{l+1}-\tilde{x}_k^l+2b\lambda }{2}+d\sin \frac{\tilde{x}_k^{l+1}-\tilde{x}_k^l+2b\lambda }{2}\right) ^2\nonumber \\&-\lambda ^2\sin ^2\frac{u_k^{l+1}+u_{k}^l}{2}=\frac{1}{b^2}, \end{aligned}$$

(358)

$$\begin{aligned} \tilde{I}_k^l=&\left( a\cos \frac{\tilde{x}_{k+1}^l-\tilde{x}_k^l-2a\lambda ^{-1}}{2}+d\sin \frac{\tilde{x}_{k+1}^l-\tilde{x}_k^l-2a\lambda ^{-1}}{2}\right) ^2\nonumber \\&-\lambda ^2\sin ^2\frac{u_{k+1}^l-u_k^l}{2}={a^2}. \end{aligned}$$

(359)

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