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Journals >Acta Physica Sinica >Volume 69 >Issue 1 >Page 010202-1 > Article

Acta Physica Sinica
Vol. 69, Issue 1, 010202-1 (2020)

Discrete integrable systems: Multidimensional consistency

Da-Jun Zhang^*

DOI: 10.7498/aps.69.20191647 Cite this Article

Da-Jun Zhang. Discrete integrable systems: Multidimensional consistency[J]. Acta Physica Sinica, 2020, 69(1): 010202-1 Copy Citation Text

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Abstract

In contrast to the well-established theory of differential equations, the theory of difference equations has not quite developed so far. The most recent advances in the theory of discrete integrable systems have brought a true revolution to the study of difference equations. Multidimensional consistency is a new concept appearing in the research of discrete integrable systems. This property, as an explanation to a type of discrete integrability, plays an important role in constructing the B?cklund transformations, Lax pairs and exact solutions for discrete integrable system. In the present paper, the multidimensional consistency and its applications in the research of discrete integrable systems are reviewed.

Keywords

discrete integrable systems multidimensional consistency

$ { {\varPhi}}_x ={ {M \varPhi}},\; \; { M} = \left(\!\!\!\begin{array}{cc}\eta & u \\ v & -\eta\end{array}\!\!\!\right),\; \; \; { {\varPhi}} = (\phi_1, \phi_2)^{\rm T}, $

(1)

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$ { {\varPhi}}_{n+j} = { {\varPhi}}(x+j\epsilon), $

(2)

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$ \begin{split} & { {\varPhi}}_{n+1} = \left(\!\!\!\begin{array}{cc} 1+\epsilon \eta & \epsilon u \\\epsilon v & 1- \epsilon\eta \!\!\!\end{array} \right) { {\varPhi}}_n,\\ &{ {\varPhi}}_n = (\phi_{1,n}, \phi_{2,n})^{\rm T}. \end{split}$

(3)

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$ \begin{split} &{ {\varPhi}}_{n+1} = \left(\!\!\!\begin{array}{cc} \lambda & Q_n \\ R_n & 1/\lambda \end{array}\!\!\!\right) { {\varPhi}}_n,\\&{ {\varPhi}}_n = (\phi_{1,n}, \phi_{2,n})^{\rm T}. \end{split}$

(4)

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$ (Q_n, R_n) = \epsilon (u,v),\; \; \; \lambda = {\rm e}^{\epsilon \eta} $

(5)

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$ H_{n+1}(x) = 2xH_n(x)-2nH_{n-1}(x). $

()

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$ {J_\alpha }(x) = {\left( {\frac{x}{2}} \right)^\alpha }\sum\limits_{k = 0}^\infty {\frac{{{{( - {x^2}/4)}^k}}}{{k!{\mkern 1mu} \Gamma (\alpha + k + 1)}}} $

()

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$ x^2 w''+xw'+(x^2-\alpha^2)w = 0, $

()

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$ x(w_{\alpha+1}+w_{\alpha-1})-2\alpha w_{\alpha} = 0, $

()

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$ f{''}(t) = 2f^3+tf-\alpha $

()

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$ f_{\alpha+1}(t) = -f_{\alpha}(t)-\dfrac{\alpha+\dfrac{1}{2}}{f^{'}_{\alpha}(t)-f^2_{\alpha}(t)-\dfrac{t}{2}}, $

()

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$ u_t+6uu_x+u_{xxx} = 0, $

(6)

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$ (\widetilde w+w)_x = 2\lambda-\frac{1}{2}(\widetilde w -w)^2, \tag{7a}$

()

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$ (\widetilde w-w)_t = \frac{1}{2}[(\widetilde w-w)^3]_x-6\lambda (\widetilde w-w)_{x}-(\widetilde w-w)_{xxx}\tag{7b} $

()

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$ w_t+3(w_x)^2+ w_{xxx} = 0, $

(8)

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$ (w_1+w)_x = 2\lambda_1-\frac{1}{2}(w_1 -w)^2,\tag{9a}$

()

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$ (w_2+w)_x = 2\lambda_2-\frac{1}{2}(w_2 -w)^2.\tag{9b}$

()

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$ (w_{12}+w_1)_x = 2\lambda_2-\frac{1}{2}(w_{12} -w_1)^2;\tag{10a}$

()

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$ (w_{21}+w_2)_x = 2\lambda_1-\frac{1}{2}(w_{21} -w_2)^2.\tag{10b}$

()

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$4(\lambda_1-\lambda_2) = (w_1-w_2)(w_{12}-w),$

(11)

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$ (w_{n+1,m}-w_{n,m+1})(w_{n,m}-w_{n+1,m+1}) = p-q, $

(12)

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$ u_t = 6u^2u_x -u_{xxx} $

(13)

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$ \theta_{xx}-\theta_{tt} = \sin \theta $

(14)

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$ \tan \frac{\theta_{12}-\theta}{4} = \frac{p+q}{p-q}\tan \frac{\theta_{1}-\theta_2}{4}, $

(15)

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$ p \sin \frac{\theta_{12}-\theta_1+\theta_2-\theta}{4} = q\sin \frac{\theta_{12}+\theta_{1}-\theta_2-\theta}{4}. $

(16)

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$ p(u_{12} u_2 - u_1 u) = q (u_{12} u_1 - u_2 u), $

(17)

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$ p \sin \frac{\varphi_{12}+\varphi-\varphi_1-\varphi_2}{4} = q\sin \frac{\varphi_{12}+\varphi+\varphi_{1}+\varphi_2}{4}, $

(18)

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$ p(u_{12} u - u_1 u_2) = q(u u_1u_2 u_{12} -1). $

(19)

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$ u_t = u_{xx}-2u^2v,\; \; v_t = -v_{xx}-2v^2u $

(20)

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$ (u_1-u_2)(uv_{12}+1)+(p-q)u = 0, \tag{21a}$

()

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$ (v_1-v_2)(uv_{12}+1)-(p-q)v_{12} = 0, \tag{21b}$

()

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$ \widetilde { {\varPhi}} = { T}{ {\varPhi}},\; { T} = { T}(\gamma,{ U},\widetilde { U}) = \left(\!\!\!\begin{array}{cc} 2(\eta-\gamma) + u \widetilde v & u \\ \widetilde v & 1 \end{array}\!\!\!\right), $

(22)

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$ { T}_x-\widetilde { M}{ T}+{ {TM}} = 0, $

(23)

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$ u_{x} = \widetilde u- u- u^2 \widetilde v,\; \; v_{x} = v - \underset{\sim }{\mathop{v}}\, + \underset{\sim }{\mathop{u}}\, v^2, $

(24)

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$ \widetilde { {\varPhi}} = { T}(p/{2},{ U},\widetilde { U}){ {\varPhi}},\; \; \widehat{{ {\varPhi}} } = { T}(q/{2},{ U},{\widehat{{ U} }}){ {\varPhi}}, $

(25)

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$\begin{split} & u = u_{n,m},\\ & \widetilde u = u_{n+1,m},\\ &{\widehat{u }} = u_{n,m+1}, \\ & {\widehat{\widetilde{u }}} = u_{n+1,m+1}, \end{split}$

(26)

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$ (u-{\widehat{\widetilde{u }}})(\widetilde u- {\widehat{u }}) = p-q. $

(27)

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$ (u-{\widetilde{\overline{u}}})(\widetilde u- {\overline{u}}) = p-r,\tag{28a}$

()

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$ (u-{\widehat{\overline{u}}})({\overline{u}}- {\widehat{u }}) = r-q,\tag{28b}$

()

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$ {\widehat{\widetilde{u }}} = \frac{p-q}{\widehat{u}-\widetilde{u}} + u,\tag{29a}$

()

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$ {\widehat{\overline{u}}} = \frac{r-q}{{\widehat{u }}-{\overline{u}}}+u,\tag{29b}$

()

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$ {\widetilde{\overline{u}}} = \frac{p-r}{{\overline{u}}-\widetilde u}+u,\tag{29c}$

()

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$ \widehat{\widetilde{\overline{u}}} = \frac{(q-p)\widetilde u{\widehat{u }}+(r-q){\widehat{u }}{\overline{u}}+(p-r){\overline{u}}\widetilde u}{(q-r)\widetilde u+(p-q){\overline{u}}+(r-p){\widehat{u }}}. $

()

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${ Q}(u,\widetilde u,{\widehat{u }},\widehat{\widetilde u};p,q) = 0, $

(30)

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$ \begin{split} & { Q}(u,\widetilde u,{\widehat{u }},\widehat{\widetilde u};p,q) = 0, \;\; { Q}({\overline{u}},\overline{\widetilde u},\overline{{\widehat{u }}},\overline{\widehat{\widetilde u}};p,q) = 0,\\ & { Q}(u,\widetilde u,{\overline{u}},\overline{\widetilde u};p,r) = 0, \;\; { Q}({\widehat{u }},\widehat{\widetilde u},\overline{{\widehat{u }}},\overline{\widehat{\widetilde u}};p,r) = 0, \\ & { Q}(u,{\widehat{u }},{\overline{u}},\overline{{\widehat{u }}};q,r) = 0, \;\;{ Q}(\widetilde u,\widehat{\widetilde u},\overline{\widetilde u},\overline{\widehat{\widetilde u}};q,r) = 0. \end{split} $

(31)

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$ \begin{split} \,& k_0 u\widetilde u{\widehat{u }}{\widehat{\widetilde{u }}} + k_1(u\widetilde u {\widehat{u }}+ u \widetilde u {\widehat{\widetilde{u }}} + u{\widehat{u }} {\widehat{\widetilde{u }}}+\widetilde u{\widehat{u }}{\widehat{\widetilde{u }}}) \\ & ~~ + k_2(\widetilde u{\widehat{u }}+u {\widehat{\widetilde{u }}}) + k_3(u\widetilde u+{\widehat{u }}{\widehat{\widetilde{u }}})+ k_4(u{\widehat{u }}+\widetilde u {\widehat{\widetilde{u }}})\\ & ~~+k_5(u+\widetilde u+{\widehat{u }}+{\widehat{\widetilde{u }}})+k_6 = 0,\\[-10pt]\end{split} $

(32)

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$ {\rm H}1: \ \ (u-\widehat{\widetilde{u}})(\widetilde{u}-\widehat{u})-p+q = 0,\tag{33a}$

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$ {\rm H}2: \ \ (u-\widehat{\widetilde{u}})(\widetilde{u}-\widehat{u})-(p-q)(u+\widetilde{u}+\widehat{u}+\widehat{\widetilde{u}}+p+q) = 0, \tag{33b}$

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$ {\rm H}3(\delta): \ \ p(u\widetilde{u}+\widehat{u}\widehat{\widetilde{u}})-q(u\widehat{u}+\widetilde{u}\widehat{\widetilde{u}})+\delta(p^2-q^2) = 0, \tag{33c}$

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$ {\rm A}1(\delta): \ \ p(u+\widehat{u})(\widetilde{u}+\widehat{\widetilde{u}})\!-\!q(u+\widetilde{u})(\widehat{u}+\widehat{\widetilde{u}})-\delta^2pq(p-q) \!=\! 0,\tag{33d}$

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$ \begin{split}{\rm A}2: \ \ p(1\, & -q^2)(u\widetilde{u}+\widehat{u}\widehat{\widetilde{u}})-q(1-p^2)(u\widehat{u}+\widetilde{u}\widehat{\widetilde{u}})\\ & -(p^2-q^2)(1+u\widetilde{u}\widehat{u}\widehat{\widetilde{u}}) = 0,\\[-10pt]\end{split}\tag{33e} $

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$ \begin{split}{\rm Q}1(\delta): \ \ p(u\, &-\widehat{u})(\widetilde{u}-\widehat{\widetilde{u}})-q(u-\widetilde{u})(\widehat{u}-\widehat{\widetilde{u}})\\ &+\delta^2pq(p-q) = 0,\end{split} \tag{33f}$

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$ \begin{split} {\rm Q}2: \ \ & p(u-\widehat{u})(\widetilde{u}-\widehat{\widetilde{u}})-q(u-\widetilde{u})(\widehat{u}-\widehat{\widetilde{u}}) \\ +\, &pq(p-q)(u+\widetilde{u}+\widehat{u}+\widehat{\widetilde{u}}-p^2+pq-q^2) = 0,\end{split} \tag{33g}$

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$ \begin{split}{\rm Q}3(\delta):\ \ \, & p(1-q^2)(u\widehat{u}+\widetilde{u}\widehat{\widetilde{u}})-q(1-p^2)(u\widetilde{u}+\widehat{u}\widehat{\widetilde{u}}) \\ -\, &(p^2\!-\!q^2)\left(\widetilde{u}\widehat{u}\!+\! u\widehat{\widetilde{u}}\!+\!\frac{\delta^2(1\! - \!p^2)(1\!-\!q^2)}{4pq}\right) \!=\! 0,\end{split}\tag{33h}$

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$ \begin{split}{\rm Q}4: \ \ &{\rm{sn}}(p)(u\widetilde{u}+\widehat{u}\widehat{\widetilde{u}})-{\rm{sn}}(q)(u\widehat{u}+\widetilde{u}\widehat{\widetilde{u}})\\-\,&{\rm{sn}}(p-q)(\widetilde{u}\widehat{u}+u\widehat{\widetilde{u}} ) \\ +\,&{\rm{sn}}(p){\rm{sn}}(q){\rm{sn}}(p-q)(1+k^2u\widetilde{u}\widehat{u}\widehat{\widetilde{u}}) = 0, \end{split} \tag{33i}$

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$ \begin{split} &\frac{P-Q}{x_{11}-x_{20}}-\frac{P-Q}{x_{02}-x_{11}}\\ =\, & (x_{00}-x_{21})(x_{10}-x_{01})\\ & -(x_{01}-x_{22})(x_{21}-x_{12}),\end{split}$

(34)

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$ {\widetilde{y}} - x{\widetilde{x}}+z = 0,\; \; {\widehat{y}} - x\widehat{x}+z = 0, \tag{35a}$

()

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$ y - x\widehat{\widetilde{x}}+\widehat{\widetilde{z}}-\frac{P-Q}{\widetilde{x}-\widehat{x}} = 0. \tag{35b}$

()

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$ [{\rm B}-2]:\; \; \widetilde{y} = x \widetilde{x}-z, \; \; \widehat{y} = x \widehat{x}-z, \tag{36a}$

()

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$ y = x \widehat{ \widetilde{x}}- \widehat{ \widetilde{z}} +b_0( \widehat{ \widetilde{x}}-x)+b_1+\frac{P-Q}{ \widetilde{x}- \widehat{x}},\tag{36b}$

()

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$ [{\rm A}-2]:\; \;\widetilde{y} = z \widetilde{x}-x, \; \; \widehat{y} = z \widehat{x}-x, \tag{37a}$

()

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$ y = x \widehat{ \widetilde{z}}-b_0 x+\frac{P \widetilde{x}-Q \widehat{x}}{ \widehat{z}- \widetilde{z}},\tag{37b}$

()

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$ [{\rm C}-3]:\; \; {\widetilde{y}} \,z = {\widetilde{x}} - x,\; \; {\widehat{y}} \,z = {\widehat{x}} - x, \tag{38a}$

()

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$ \widehat{\widetilde{z}}\,y = b_0\,x + b_1 + z \frac{P\,{\widetilde{y}}\,{\widehat{z}}-Q\,{\widehat{y}}\,{\widetilde{z}}}{{\widetilde{z}}-{\widehat{z}}},\tag{38b} $

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${\rm B}1(u,\widetilde u,{\widehat{u }},\widehat{\widetilde u};p,q) = p(u {\widehat{\widetilde{u }}} -\widetilde{u} \widehat{u})-q( u\widetilde u{\widehat{u }}{\widehat{\widetilde{u }}} -1) = 0, $

(39)

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$ {\rm B}2(u,\widetilde u,{\widehat{u }},\widehat{\widetilde u};p,q) = p({\widehat{u }} {\widehat{\widetilde{u }}} -u \widetilde{u})- q( \widetilde u \,{\widehat{\widetilde{u }}}- u{\widehat{u }}) = 0. $

(40)

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$ \begin{split} & {{\text{底}}: \;\;{\rm B}1(u,\widetilde u,{\widehat{u }},\widehat{\widetilde u};p,q) = 0,}\\&{{\text{顶}}:\;\;{\rm B}1({\overline{u}},{{\widetilde{\overline{u}}}}, {{\widehat{\overline{u}}}}, {\widehat{{\widetilde{\overline{u}}}}};p,q) = 0,}\\& {{\text{左}}:\;\; {\rm B}1(u,\widetilde u,{\overline{u}}, {{\widetilde{\overline{u}}}};p,r) = 0,}\\&{{\text{右}}: \;\;{\rm B}1({\widehat{u }},\widehat{\widetilde u},{{\widehat{\overline{u}}}}, {\widehat{{\widetilde{\overline{u}}}}};p,r) = 0,}\\& {{\text{后}}: \;\;{\rm B}2(u,{\widehat{u }},{\overline{u}}, {{\widehat{\overline{u}}}};q,r) = 0,}\\&{{\text{前}}: \;\;{\rm B}2(\widetilde u,\widehat{\widetilde u}, {{\widetilde{\overline{u}}}}, {\widehat{{\widetilde{\overline{u}}}}};q,r) = 0.} \end{split}$

(41)

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$ u\widehat{\widetilde{{\overline{u}}}}-\widetilde u{\widehat{\overline{u}}}+ {\widehat{u }}{\widetilde{\overline{u}}} -{\overline{u}} {\widehat{\widetilde{u }}} = 0, \tag{42a}$

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$ \frac{(\widetilde u-{\overline{u}})({\widehat{u }}-\widehat{\widetilde{{\overline{u}}}})}{({\overline{u}}-{\widehat{u }})(\widehat{\widetilde{{\overline{u}}}}-\widetilde u)} = \frac{(u-{\widetilde{\overline{u}}})({\widehat{\widetilde{u }}}-{\widehat{\overline{u}}})}{({\widetilde{\overline{u}}}-{\widehat{\widetilde{u }}})({\widehat{\overline{u}}}-u)} . \tag{42b}$

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$ {\rm i})\; \; \; \; \; {\widehat{\widetilde{u }}}{\overline{u}} - {\widehat{\overline{u}}} \widetilde u + {\widetilde{\overline{u}}} {\widehat{u }} = 0,\tag{43a}$

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$ {\rm ii})\; \; \; \; \frac{({\widehat{\widetilde{u }}}-{\widehat{u }})({\widehat{\overline{u}}}-{\overline{u}})({\widetilde{\overline{u}}}-\widetilde u)}{({\widehat{\widetilde{u }}} - \widetilde u)({\widehat{\overline{u}}}-{\widehat{u }})({\widetilde{\overline{u}}}-{\overline{u}})} = 1, \tag{43b}$

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$ {\rm iii})\; \; \; ({\widehat{\widetilde{u }}}-{\widehat{\overline{u}}}){\widehat{u }} + ({\widehat{\overline{u}}}- {\widetilde{\overline{u}}}){\overline{u}} + ({\widetilde{\overline{u}}}-{\widehat{\widetilde{u }}}) \widetilde u = 0, \tag{43c}$

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$ {\rm iv}) \; \; \; \frac{({\widehat{\widetilde{u }}}-{\widehat{\overline{u}}})}{{\widehat{u }}} + \frac{({\widehat{\overline{u}}}- {\widetilde{\overline{u}}})}{{\overline{u}}} + \frac{({\widetilde{\overline{u}}}-{\widehat{\widetilde{u }}})}{ \widetilde u} = 0, \tag{43d}$

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$ {\rm v}) \; \; \; \; \frac{({\widehat{\overline{u}}}-{\widetilde{\overline{u}}})}{{\overline{u}}} = {\widehat{\widetilde{u }}} \left( \frac{1}{\widetilde u} - \frac{1}{ {\widehat{u }}}\right). \tag{43e}$

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$ { Q}(u,\widetilde u,{\widehat{u }},\widehat{\widetilde u};p,q) = 0, \tag{44a}$

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$ { Q}({\overline{u}},{{\widetilde{\overline{u}}}},{\widehat{\overline{u}}},\widehat{\widetilde{{\overline{u}}}};p,q) = 0.\tag{44b}$

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$ { Q}(u,\widetilde u,{\overline{u}},{\widetilde{\overline{u}}};p,r) = 0,\tag{45a}$

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$ { Q}(u,{\overline{u}}, {\widehat{u }},{\widehat{\overline{u}}};r,q) = 0.\tag{45b}$

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$ p(u {\widetilde{\overline{u}}} -\widetilde{u} \overline{u})-r( u\widetilde u{\overline{u}}{\widetilde{\overline{u}}} -1) = 0, \tag{46a}$

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$ r({\widehat{u }} {\widehat{\overline{u}}} -u \overline{u})- q( {\overline{u}} \,{\widehat{\overline{u}}}- u{\widehat{u }}) = 0, \tag{46b}$

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${\overline{u}} = \frac{v}{\epsilon}+\frac{\delta^2}{4\epsilon^2}, $

(47)

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$ p(v-\widehat{v})(\widetilde{v}-\widehat{\widetilde{v}})-q(v-\widetilde{v})(\widehat{v}-\widehat{\widetilde{v}})+\delta^2 pq(p-q) = 0, $

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$ \delta(u-\widetilde u)(v-\widetilde v) = -p[\delta^2(u+\widetilde u)-2 v\widetilde v]+p^2\delta (v+\widetilde v+p\delta)],\tag{48a}$

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$ \delta(u-{\widehat{u }})(v-{\widehat{v}}) = -q[\delta^2(u+{\widehat{u }})-2 v{\widehat{v}}]+q^2\delta (v+{\widehat{v}}+q\delta)],\tag{48b}$

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$ {\widetilde{\overline{u}}} = \frac{p-r}{{\overline{u}}-\widetilde u}+u,\; \; {\widehat{\overline{u}}} = \frac{r-q}{{\widehat{u }}-{\overline{u}}}+u. $

(49)

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$\begin{split} & \frac{\widetilde{g}}{{\widetilde{f}}} = \frac{ug+(p-r-u\widetilde u)f}{g-\widetilde u f},\\ & \frac{\widehat{g}}{{\widehat{f}}} = \frac{ug+(q-r-u {\widehat{u }}) f}{g-{\widehat{u }} f}. \end{split}$

(50)

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$ \widetilde{{ {\varPhi}}} = \gamma_1 { M}{ {\varPhi}},\; \; \; \widehat{{\rm {\varPhi}}} = \gamma_2 { {N\varPhi}}, \tag{51a}$

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$ { M} = \left(\!\!\!\begin{array}{cc} u & p-r-u\widetilde u\\ 1 & -\widetilde u \end{array}\!\!\!\right),\; \; \; { N} = \left(\!\!\!\begin{array}{cc} u & q-r-u {\widehat{u }}\\ 1 & -{\widehat{u }} \end{array}\!\!\!\right),\tag{51b} $

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$ {\widehat{ M}} { N} = {\widetilde{ N}} { M}, $

(52)

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$ \widetilde{\widetilde{\phi}}_2 = (u-\widetilde{\widetilde{u}}) \widetilde{\phi}_{2}-(p-r)\phi_{2}, $

(53)

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$ \widetilde{{ {\varPhi}}} = \gamma_1 { M}{ {\varPhi}},\; \; \; \widehat{{ {\varPhi}}} = \gamma_2 { N}{ {\varPhi}}, \tag{54a}$

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$\begin{split} & { M} = \left(\begin{array}{cc} -\partial_{{\widehat{u }}}Q& -Q\\ \partial_{{\widehat{\widetilde{u }}}}\partial_{{\widehat{u }}}Q& \partial_{{\widehat{\widetilde{u }}}}Q \end{array} \right)_{{\widehat{u }} = {\widehat{\widetilde{u }}} = 0,\; q = r}, \\&{ N} = \left(\begin{array}{cc} -\partial_{\widetilde u}Q& -Q\\ \partial_{{\widehat{\widetilde{u }}}}\partial_{\widetilde u}Q& \partial_{{\widehat{\widetilde{u }}}}Q \end{array} \right)_{\widetilde u = {\widehat{\widetilde{u }}} = 0,\; p = r},\end{split}\tag{54b}$

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$ \widetilde { {\varPhi}} = \frac{-1}{\sqrt{u\widetilde u}}\left(\begin{array}{lr}-p \widetilde u & r\\ ru\widetilde u & -pu \end{array}\right){ {\varPhi}}, \tag{55a}$

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$ \widehat{{ {\varPhi}} } =\frac{1}{\sqrt{u{\widehat{u }}}}\left(\begin{array}{lr} -r u & q u {\widehat{u }}\\ q & -r {\widehat{u }} \end{array}\right){ {\varPhi}}. \tag{55b}$

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$ a^2 = -p,\; \; \; b^2 = -q, $

(56)

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$ u_0 = a n +b m +\lambda, $

(57)

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$ (u-\widetilde{\overline{u}})(\widetilde{u}-\overline{u}) = -a^2+k^2,\tag{58a}$

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$ (u-{\widehat{\overline{u}}})(\overline{u}-\widehat{u}) = -k^2+b^2. \tag{58b}$

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$ \overline{u} = \overline{u}_{0}+v, $

(59)

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$ \overline{u}_{0} = a n +b m +k+\lambda, $

(60)

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$ \widetilde{v} = \frac{Ev}{v+F},\; \; \; \widehat{v} = \frac{Gv}{v+H}, $

(61)

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$\begin{split} & E = -(a+k),\; \; F = -(a-k),\\ & G = -(b +k),\; \; H = -(b-k).\end{split}$

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$ \widetilde{ {\varPhi}} = \left(\begin{array}{cc} E & 0\\ 1 & F \end{array} \right){ {\varPhi}},\; \; \; \widehat{{ {\varPhi}}} = \left(\begin{array}{cc} G & 0\\ 1 & H \end{array} \right){ {\varPhi}}. $

(62)

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$ f_{n,m} = E^n f_{0,m},\; \; \; g_{n,m} = \frac{E^n-F^n}{-2k}f_{0,m}+F^n g_{0,m}, $

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$ f_{n,m} = G^m f_{n,0},\; \; \; g_{n,m} = \frac{G^m-H^m}{-2k}f_{n,0}+ H^m g_{n,0}. $

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$ {\varPhi}_{n,m} = \left(\begin{array}{cc} E^nG^m & 0\\ \dfrac{E^nG^m-F^nH^m}{-2k} & F^nH^m \end{array} \right){ {\varPhi}}_{0,0}. $

(63)

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$ \rho_{n,m}\! =\! \biggl(\frac{E}{F})^n \! \biggl(\frac{G}{H})^m \rho_{0,0} \!=\! \biggl(\frac{a +k}{a -k})^n \!\biggl(\frac{b + k}{b-k})^m \rho_{0,0}, $

(64)

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$ v_{n,m} = \frac{v_{0,0}\rho_{n,m}/\rho_{0,0} } {1+\dfrac{v_{0,0}}{2k} - \dfrac{v_{0,0}}{2k}\rho_{n,m}/\rho_{0,0}}. $

(65)

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$ v_{n,m} = \frac{-2k \rho_{n,m}}{1+\rho_{n,m}}. $

(66)

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$ u_{n,m}^{1SS} = \overline{u}_{0}+v_{n,m} = an + bm + \lambda +\frac{k(1- \rho_{n,m})}{1+\rho_{n,m}}. $

(67)

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$ { Q}(u,\widetilde u, u,\widetilde u;p,r) = 0,\tag{68a}$

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$ { Q}(u, u, {\widehat{u }},{\widehat{u }};r,q) = 0. \tag{68b}$

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$ (\widetilde u-u)^2 = r-p,\quad ({\widehat{u }}-u)^2 = r-q. $

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$ u = an+bm+\gamma,\; \; a^2 = r-p, \; \; b^2 = r-q, $

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$ { Q}(u,\widetilde u, T(u), T(\widetilde u);p,r) = 0, \tag{69a}$

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$ { Q}(u, T(u), {\widehat{u }},T({\widehat{u }});r,q) = 0. \tag{69b}$

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$\begin{split} & r(u-\widetilde u)^2 = p(c^2+\delta^2r(p-r)),\\ & r(u-{\widehat{u }})^2 = q(c^2+\delta^2r(q-r)). \end{split}$

(70)

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$\begin{split} & p = \frac{c^2/r-\delta^2 r}{a^2-\delta^2},\quad q = \frac{c^2/r-\delta^2 r}{b^2-\delta^2},\\ & \alpha: = pa,\quad \beta: = qb, \end{split}$

(71)

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$ u_0 = \alpha n+\beta m+\lambda, $

()

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$\begin{split} & 4pu\widetilde u +r(u-\widetilde u)^2 = \delta^2pr(p-r),\\ & 4q u{\widehat{u }}+ r(u-{\widehat{u }})^2 = \delta^2r(q-r). \end{split}$

(72)

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$ p = \dfrac{1}{2} r(1- \cosh(\alpha')) = -\dfrac{1}{4}r(1-\alpha)^2/\alpha, $

(73)

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$ q = \dfrac{1}{2} r(1- \cosh(\beta')) = -\dfrac{1}{4}r(1-\beta)^2/\beta, $

(74)

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$ u_0 = A \alpha^n \beta^m +B \alpha^{-n}\beta^{-m},\quad AB = \delta^2 r^2/16. $

(75)

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$ \psi =\prod\limits_{i}{{{(1-{{p}_{i}}k)}^{{{n}_{i}}}}} $

()

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Da-Jun Zhang. Discrete integrable systems: Multidimensional consistency[J]. Acta Physica Sinica, 2020, 69(1): 010202-1

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