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Journals >Acta Physica Sinica >Volume 69 >Issue 1 >Page 010203-1 > Article

Acta Physica Sinica
Vol. 69, Issue 1, 010203-1 (2020)

The Boussinesq equation: Lax pair, Bäcklund transformation, symmetry group transformation and consistent Riccati expansion solvability

Ping Liu^1、*, Heng-Rui Xu², and Jian-Rong Yang³

Author Affiliations

¹School of Electronic and Information Engineering, University of Electronic Science and Technology of China, Zhongshan Institute, Zhongshan 528402, China

²School of Physics, University of Electronic Science and Technology of China, Chengdu 610054, China

³School of Physics and Electronic Information, Shangrao Normal University, Shangrao 334001, China

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DOI: 10.7498/aps.69.20191316 Cite this Article

Ping Liu, Heng-Rui Xu, Jian-Rong Yang. The Boussinesq equation: Lax pair, Bäcklund transformation, symmetry group transformation and consistent Riccati expansion solvability[J]. Acta Physica Sinica, 2020, 69(1): 010203-1 Copy Citation Text

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Abstract

The Boussinesq equation is a very important equation in fluid mechanics and some other disciplines. A Lax pair of the Boussinesq equation is proposed. With the help of the truncated Painlevé expansion, auto-B?cklund transformation of the Boussinesq equation and B?cklund transformation between the Boussinesq equation and the Schwarzian Boussinesq equation are demonstrated. Nonlocal symmetries of the Boussinesq equation are discussed. One-parameter subgroup invariant solutions and one-parameter group transformations are obtained. The consistent Riccati expansion solvability of the Boussinesq equation is proved and some interaction structures between soliton-cnoidal waves are obtained by consistent Riccati expansion.

Keywords

B?cklund transformation Boussinesq equation consistent Riccati expansion lax pair

$ u_{tt}+\alpha u_{xx}+\beta (u^2)_{xx}+\gamma u_{xxxx} = 0, $

(1)

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$ u_{tt}- u_{xx}- (u^2)_{xx}- u_{xxxx} = 0. $

(2)

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$ u_{tt}- u_{xx}- (u^2)_{xx}+ u_{xxxx} = 0. $

(3)

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$ u_{tt}- u_{xx}- (u^2)_{xx}- u_{xxtt} = 0. $

(4)

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$ v_{\tau\tau}+ (v^2)_{\chi\chi}+\frac{1}{3}v_{\chi\chi\chi\chi} = 0, $

(5)

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$ \psi_{\chi\chi\chi} = -\frac{3}{2} v\psi_\chi-\frac{3}{4}\psi v_{\chi}-\frac{3}{4}\psi{\int v_{\tau}\, {\rm d }\chi}+\lambda \psi \tag{6a},$

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$ \psi_{\tau} = \psi_{\chi\chi}+v \psi \tag{6b}.$

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$ \begin{split} & u(x,{\mkern 1mu} t) = {\left( {\frac{{3 \gamma }}{{{\beta ^2}}}} \right)^{\frac{1}{3}}}v\left( {\chi ,\tau } \right) - \frac{\alpha }{{2 \beta }},\\ &\chi = \pm \frac{{{\beta ^{\left( {\frac{1}{6}} \right)}} {3^{\left( {\frac{2}{3}} \right)}} x}}{{3 {\gamma ^{\frac{1}{3}}}}},\\ & \tau = \pm \frac{{{3^{\left( {\frac{5}{6}} \right)}} {\beta ^{\left( {\frac{1}{3}} \right)}} t}}{{3 {\gamma ^{\left( {\frac{1}{6}} \right)}}}}.\end{split}$

(7)

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$\begin{split} \phi_{xxx} =\, & - { \frac { \sqrt{3\, \gamma } \beta\phi }{12\,\gamma ^{2}}} \, { \int } u_t\, {\rm d} x \! - { \frac { (2\,u\,\beta + \alpha ) }{4\,\gamma }} \phi_x \\ & - { \frac {\phi \,\beta \,}{ 4\,\gamma }} u_x \pm { \frac { \,\phi \,\lambda \,\sqrt{\beta }}{3\,\gamma }},\end{split} \tag{8a} $

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$ \phi_t = \sqrt{3\,\gamma} \phi_{xx} + { \frac {\sqrt{3} \,\phi \,(2\,\beta \,u + \alpha )\,}{6\,\sqrt{ \gamma }}},\tag{8b} $

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$ u = u_0+\frac{u_1}{f}+\frac{u_2}{f^2}, $

(9)

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$ {u_2} = - { \frac {6\,\gamma \, {f_{x}}^{2}}{\beta }}. $

(10)

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$ {u_1} = { \frac {6\,\gamma \,{f _{{xx}}}}{\beta }}. $

(11)

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$ {u_0} = { \frac {3}{2}} \, { \frac {\gamma \,{f_{{xx}}}^{2}}{\beta {f_{x}}^{2} }} - { \frac {2\,\gamma \,{f_{{xxx}} }}{\beta{f_{x}}}} - { \frac {1}{2}} \, { \frac {{f_{t}}^{2}}{\beta{f_{x}}^{2} }} - { \frac {\alpha }{2\,\beta }}\ . $

(12)

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$ \gamma \left( { \frac {{f_{{xxx}}}}{{f_{x} }}} - { \frac {3}{2}} \,{ \frac {{f_{ {xx}}}^{2}}{{f_{x}}^{2}}} \right)_x + \left({ \frac {{f_{t}}}{ {f_{x}}}} \right)_t + { \frac {{f_{t}}\,}{{f _{x}}}} \left({ \frac {{f_{t}}}{{f_{x}}}} \right)_x = 0, $

(13)

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$ u = {u_0} = { \frac {3}{2}} \, { \frac {\gamma \,{f_{{xx}}}^{2}}{\beta {f_{x}}^{2} }} - { \frac {2\,\gamma \,{f_{{xxx}} }}{\beta{f_{x}}}} - { \frac {1}{2}} \, { \frac {{f_{t}}^{2}}{\beta{f_{x}}^{2} }} - { \frac {\alpha }{2\,\beta }} $

(14)

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$\begin{split} u = \, & u_0+\frac{u_1}{f}+\frac{u_2}{f^2} = { \frac {3}{2}} \, { \frac {\gamma \,{f_{{xx}}}^{2}}{\beta {f_{x}}^{2} }} - { \frac {2\,\gamma \,{f_{{xxx}} }}{\beta{f_{x}}}}\\ & - { \frac {1}{2}} \, { \frac {{f_{t}}^{2}}{\beta{f_{x}}^{2} }} - { \frac {\alpha }{2\,\beta }}+\frac{{ 6\,\gamma \,{f _{{xx}}} }}{f\beta}-\frac{ { 6\,\gamma \, {f_{x}}^{2} }}{\beta f^2}\end{split}$

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$ u_{tt}+\alpha u_{xx}+\beta (u^2)_{xx}+\gamma u_{xxxx} = 0, \tag{16a}$

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$ \gamma \left( { \frac {{f_{{xxx}}}}{{f_{x} }}} - { \frac {3}{2}} \,{ \frac {{f_{ {xx}}}^{2}}{{f_{x}}^{2}}} \right)_x + \left({ \frac {{f_{t}}}{ {f_{x}}}} \right)_t + { \frac {{f_{t}}\,}{{f _{x}}}} \left({ \frac {{f_{t}}}{{f_{x}}}} \right)_x = 0, \tag{16b}$

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$ u = { \frac {3}{2}} \, { \frac {\gamma \,{f_{{xx}}}^{2}}{\beta {f_{x}}^{2} }} - { \frac {2\,\gamma \,{f_{{xxx}} }}{\beta{f_{x}}}} - { \frac {1}{2}} \, { \frac {{f_{t}}^{2}}{\beta{f_{x}}^{2} }} - { \frac {\alpha }{2\,\beta }}, \tag{16c}$

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$ f_x = g, \tag{16d}$

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$ f_{xx} = h . \tag{16e}$

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$ \sigma \!=\! \left(\!\!\! \begin{array}{cccc} \sigma^u \\ \sigma^f\\ \sigma^g\\ \sigma^h \end{array}\!\!\!\right) \!=\! \left(\!\!\!\begin{array}{cccc} { \dfrac {6\,\gamma \,{f _{{xx}}}}{\beta }} \\ -f^2\\ -2ff_x\\ -2{f_x}^2-2ff_{xx} \end{array}\!\!\!\right) \!=\! \left(\!\!\!\begin{array}{cccc} { \dfrac {6\,\gamma \,h}{\beta }} \\ -f^2\\ -2fg\\ -2g^2-2fh \end{array}\!\!\right)\ . $

(17)

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$ \underline{V} = C_1 \underline{ V_1} +C_2 \underline{V_2}+C_3 \underline{V_3}+C_4\underline{ V_4}+C_5 \underline{V_5}+C_{6} \underline{V_{6}}\,, $

(18)

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$\begin{split} & \underline{V_1} = \frac{x}{2}\partial_x + t\partial_ t-u\partial_u - { \frac {\alpha }{2 \,\beta }} \partial_u -\frac{g}{2}\partial_g-h\,\partial_h, \\ & \underline{ V_2} = \partial_x, \ \ \ \ \underline{V_3} = \partial_t, \\ & \underline{V_4} = -h\partial_u +{ \frac {\beta f^{2}}{6\,\gamma }} \partial_f +{ \frac {g\,\beta \,f}{3\,\gamma }}\partial_g +{ \frac {\beta (f\,h+g^2)}{3\,\gamma }}\partial_h,\qquad \qquad \\ & \underline{V_5} = -f\partial_f-g \partial_g-h \partial_h, \ \ \ \ \ \underline{V_{6}} = -\partial_f . \\[-10pt]\end{split}$

(19)

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$g_{\varepsilon}(\underline{V_1}) : \{x,t,u,f,g,h\} \longrightarrow \left\{x\,{\rm e}^{ \frac {\varepsilon }{2} },t\,{\rm e}^{\varepsilon },{\rm e}^{ - \varepsilon }\,u + { \frac { \alpha }{2\,\beta }}({\rm e}^{ - \varepsilon }-1),f,g\,{\rm e}^{ - \frac {\varepsilon }{2} }, h\,{\rm e}^{ - \varepsilon } \right\}, \tag{20a}$

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$ g_{\varepsilon}(\underline{V_2}) : \{x,t,u,f,g,h\} \longrightarrow \{x+\varepsilon, \ t, \ u, \ f, \ g, \ h\}, \tag{20b}$

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$ g_{\varepsilon}(\underline{V_3}) : \{x,t,u,f,g,h\} \longrightarrow \{x, \ t+\varepsilon, \ u, \ f, \ g, \ h\}, \tag{20c}$

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$ \begin{array}{l} g_{\varepsilon}(\underline{V_4}) : \{x,t,u,f,g,h\} \longrightarrow \left\{ \begin{array}{ll} x,\ t, \ { u- { \dfrac {6\,\gamma \,\varepsilon ^{2}\,\beta \,g^{2}}{(\varepsilon \,\beta \,f - 6\,\gamma )^{2}}} + { \dfrac {6\, \gamma \,\varepsilon \,h}{\varepsilon \,\beta \,f - 6\,\gamma }}} , \ { \dfrac {6\, \gamma \,f}{6\,\gamma-\varepsilon \,\beta \,f }},\\ { { \dfrac { 36\,g\,\gamma ^{2}}{(\varepsilon \,\beta \,f - 6\,\gamma )^{2}}}},\ \ { { \dfrac {36\,\gamma ^{2 }\,( - 2\,g^{2}\,\varepsilon \,\beta + f\,h\,\varepsilon \,\beta - 6\,h\,\gamma )}{(\varepsilon \,\beta \,f - 6\,\gamma )^{3}}}} \end{array} \right\}, \end{array} \tag{20d}$

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$ g_{\varepsilon}(\underline{V_5}) : \{x,t,u,f,g,h\} \longrightarrow \{x, \ t, \ u, \ {\rm e}^{( - \varepsilon )}f, \ {\rm e}^{( - \varepsilon )} g, \ {\rm e}^{( - \varepsilon )}h\}, \tag{20e}$

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$ g_{\varepsilon}(\underline{V_6}) : \{x,t,u,f,g,h\} \longrightarrow \{x, \ t, \ u, \ f- \varepsilon, \ g, \ h\}. \tag{20f} $

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$ \left\{ \begin{array}{ll} { \overline{u}_1 = {\rm e}^{ - \varepsilon }\,u(x {\rm e}^{ -\frac {\varepsilon }{2} },t {\rm e}^{-\varepsilon }) + { \dfrac { \alpha }{2\,\beta }}({\rm e}^{ - \varepsilon }-1)}, \qquad { \overline{f}_1 = f(x {\rm e}^{ -\frac {\varepsilon }{2} },t {\rm e}^{-\varepsilon })}, \\ { \overline{g}_1 = g(x {\rm e}^{ -\frac {\varepsilon }{2} },t {\rm e}^{-\varepsilon })\,{\rm e}^{ - \frac {\varepsilon }{2} }}, \qquad \qquad \qquad \ \qquad { \overline{h}_1 = h(x {\rm e}^{ -\frac {\varepsilon }{2} },t {\rm e}^{-\varepsilon })\,{\rm e}^{ - \varepsilon }} \end{array}\right \}, \tag{21a}$

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$ \left\{ \overline{u}_2 = u(x-\varepsilon, t), \ \ \ \overline{f}_2 = f(x-\varepsilon, t),\ \ \ \overline{g}_2 = g(x-\varepsilon, t), \ \ \ \overline{h}_2 = h (x-\varepsilon, t)\right\}, \tag{21b}$

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$ \{ \overline{u}_3 = u(x, t-\varepsilon), \ \overline{f}_3 = f(x, t-\varepsilon),\ \ \ \ \ \overline{g}_3 = g(x, t-\varepsilon), \ \ \ \overline{h}_3 = h(x, t-\varepsilon) \}, \tag{21c}$

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$ \left \{\begin{array}{ll} { \overline{u}_4 = u(x,t)- { \dfrac {6\,\gamma \,\varepsilon ^{2}\,\beta \,g(x,t)^{2}}{(\varepsilon \,\beta \,f(x,t) - 6\,\gamma )^{2}}} + { \dfrac {6\, \gamma \,\varepsilon \,h(x,t)}{\varepsilon \,\beta \,f(x,t) - 6\,\gamma }}}, \ { \overline{f}_4 = { \dfrac {6\, \gamma \,f(x,t)}{6\,\gamma-\varepsilon \,\beta \,f(x,t) }}}, \\ { \overline{g}_4 = { \dfrac { 36 \gamma^{2}g(x,t)\,}{(\varepsilon \beta f(x,t) - 6 \gamma )^{2}}}},\ { \overline{h}_4 = { \dfrac {36 \gamma ^{2 } ( \varepsilon \beta f(x,t) h(x,t) - 2 \varepsilon \beta g(x,t)^{2}- 6 \gamma h(x,t) )}{(\varepsilon \,\beta \,f(x,t) - 6\,\gamma )^{3}}}} \end{array} \!\!\!\right\}, \tag{21d}$

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$ \{ \overline{u}_5 = u(x,t), \ \ \ \overline{f}_5 = {\rm e}^{( - \varepsilon )}f (x,t),\ \ \ \overline{g}_5 = {\rm e}^{( - \varepsilon )}\,g(x,t),\ \ \ \overline{h}_5 = {\rm e}^{( - \varepsilon )}h (x,t)\}, \tag{21e}$

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$ \{\overline{u}_6 = u(x,t), \ \ \ \overline{f}_6 = f(x,t)- \varepsilon,\ \ \ \ \overline{g}_6 = g(x,t), \qquad \ \ \ \overline{h}_6 = h(x,t) \}. \tag{21f}$

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$ \xi = { \frac {{C_{3}} + {C_{1}}\,t}{{C_{1}}\,(2\,{C _{2}} + {C_{1}}\,x)^{2}}}. $

(22)

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$ \begin{split}u= \,&{ \frac { {U}(\xi )}{(2\,{C_{2}} + {C_{1}}\,x)^{2}}} - { \frac {(4\,{C_{2}} + {C_{1}}\,x)\, \alpha\,{C_{1}}\,x}{2\,\beta( 2\, {C_{2}} + {C_{1}}\,x)^{2} }} \\ & - { \frac {6\,\gamma\,{C_{4}}\, {H}(\xi ) \,{\rm e}^{( - 2\, {F}(\xi )\,{C_{1}})}\, }{(2\,{C_{2}} + {C_{1}}\,x)^{2}\,\delta }}\\ & \times {\rm tanh}\!\left[{ \frac {\delta \, {\rm ln}(2\,{C _{2}} + {C_{1}}\,x ) + \delta {F}(\xi )\,{C_{1}} }{3\,\gamma\,{C_{1}}}}\right ] \\ & +{ \frac {6\,\beta\,\gamma\,{C_{4}}^{2}\, {G}(\xi )^{2}\,{\rm e} ^{( - 2\, {F}(\xi )\,{C_{1}})} }{\delta ^{2}\,(2\,{C_{2}} + {C_{1}}\,x)^{2}\,}}\\ & \times {\rm sech}\!\left[{ \frac {\delta \, {\rm ln}(2\,{C _{2}} + {C_{1}}\,x ) + \delta {F}(\xi )\,{C_{1}} }{3\,\gamma\,{C_{1}}}}\right ]^{2}, \end{split}\tag{23a}$

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$ f \!=\! { \frac {3\,\gamma\,{C_{5}} }{C_4\,\beta}}-\frac{\delta}{C_4\,\beta} {\rm tanh} \!\left[{ \frac {\delta \, {\rm ln}(2\,{C _{2}} \!+\! {C_{1}}\,x ) \!+\! \delta {F}(\xi )\,{C_{1}} }{3\,\gamma\,{C_{1}}}}\right ],\tag{23b}$

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$\begin{split} g =\, &{ \frac { {G}(\xi )\,{\rm e}^{( - {F}(\xi ) \,{C_{1}})}}{\,(2\,{C_{2} } + {C_{1}}\,x)}} \\ & \times{\rm sech}\!\left[{ \frac {\delta \, {\rm ln}(2\,{C _{2}} + {C_{1}}\,x ) + \delta {F}(\xi )\,{C_{1}} }{3\,\gamma\,{C_{1}}}}\right ]^{2},\end{split}\tag{23c}$

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$ \begin{split} h = & { \frac { {H}(\xi )\,{\rm e}^{( - 2\, {F}( \xi )\,{C_{1}})}}{(2\,{C_{2}} + {C_{1}}\,x)^{2}\,}} \\ & \times{\rm sech}\!\left[{ \frac {\delta \, {\rm ln}(2\,{C _{2}} + {C_{1}}\,x ) + \delta {F}(\xi )\,{C_{1}} }{3\,\gamma\,{C_{1}}}}\right ]^{2} \\ &+{ \frac {2\,\beta\,{C_{4}}\, {G}(\xi )^{2}\, \,{\rm e}^{( - 2\, {F}(\xi )\,{C _{1}})}}{(2\,{C_{2}} + {C_{1}}\,x)^{2}\,\delta }} \\ & \times\frac{{\rm sinh}\!\left[{ \dfrac {\delta \, {\rm ln}(2\,{C _{2}} + {C_{1}}\,x ) + \delta {F}(\xi )\,{C_{1}} }{3\,\gamma\,{C_{1}}}}\right ]}{{\rm cosh}\!\left[{ \dfrac {\delta \, {\rm ln}(2\,{C _{2}} + {C_{1}}\,x ) + \delta {F}(\xi )\,{C_{1}} }{3\,\gamma\,{C_{1}}}}\right ]^3} , \end{split}\tag{23d}$

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$ u = - { \frac {4\,x\, \alpha \,{C_{1}}\,{C_{2}} + x^{2}\,\alpha \,{C_{1}}^{2} - 2\, {U}(\xi )\,\beta }{2\,\beta \,(2\,{C_{2}} + {C_{1}}\,x)^{2}} }\,, \tag{24a}$

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$ f = (2\,{C_{2}} + {C_{1}}\,x)^{\left( - \frac {2\,{C_{5}}}{{C_{1}}}\right)}\, {F}(\xi ) - { \frac {{C_{6}}}{{C_{5}}}}\,,\tag{24b}$

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$ g = (2\,{C_{2}} + {C_{1}}\,x)^{\left( - 1 - \frac {2\,{C_{5}}}{{C_{1}}}\right) }\, {G}(\xi )\,, \tag{24c}$

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$ h = (2\,{C_{2}} + {C_{1}}\,x)^{\left( - 2 - \frac {2\,{C_{5}}}{{C_{1}}}\right ) }\, {H}(\xi )\, . \tag{24d}$

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$ \xi = { \frac {{C_{3}} + {C_{1}}\,t}{{C_{1}}\,(2\,{C _{2}} + {C_{1}}\,x)^{2}}}\ . $

(25)

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$ g = - 2\,(2\,{C_{2}} + {C_{1}}\,x)^{( - 1 - \frac {2\,{C_{5}}}{{C_{1}}})}\,({C_{1}}{F_{\xi }}\,\xi \, + C_{5} {F} ), $

(26)

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$\begin{split} h =\,& 2\,(2\,{C_{2}} + {C_{1}}\,x)^{( - 2 - \frac {2 \,{C_{5}}}{{C_{1}}})} (2 {C_{1}}^{2} {F_{\xi \xi }} \xi ^{2}+ 3 {C_{1}}^{2} { F_{\xi }}\,\xi \\ & + {C_{1}} {C_{5}}{F} \, + 4\,{C_{1}} {F_{\xi }} {C_{5}} \xi + 2\,{C_{5}}^{2} {F} ),\\[-12pt]\end{split}$

(27)

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$ \xi = \frac{C_2 t-C_3x}{C_2}, $

(28)

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$ u = {U}(\xi)+{ \frac {3\,\gamma\, C_{4} {\rm e} ^{ \frac {\delta \,{F}(\xi) }{ 3\,{C_{2}}\,\gamma } } {H}(\xi) }{\delta \,\left[{\rm e}^{ \frac {\delta \,(x + {F}(\xi) )}{3\,{C_{2} }\,\gamma } } + 1\right] }} - { \frac {24\,\beta\,\gamma\,{C_{4}}^{2} {G}(\xi)^{2} }{\delta ^{2} \left[{\rm e}^{ \frac {\delta \, (x + {F}(\xi) )}{3\,{C_{2} }\,\gamma } } + 1\right]^{2}}}, \tag{29a}$

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$ f = - { \frac {\delta }{ \beta \,{C_{4}}}} {\rm tanh}\left[ { \frac {\delta \,(x + F (\xi) )}{6\,\gamma\,{C_{2}} }} \right ] + { \frac {3\,\gamma \,{C_{5}}}{ \beta \,{C_{4}}}}, \tag{29b}$

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$ g = - { \frac {2\, {G}(\xi) }{ {\rm cosh}\left[{ \dfrac {\delta \,(x + {F}(\xi) )}{3\,{C_{2}}\,\gamma }} \right] + 1} }, \tag{29c}$

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$ h = { \frac { {\rm e}^{ \frac {\delta \, (2\,{F}(\xi) +x)}{3\,{C_{2}}\,\gamma } } {H}(\xi) }{\left[{\rm e}^{ \frac {\delta \,(x + {F}(\xi) )}{3\,{C_{2} }\,\gamma } } + 1\right]^{2}}} - { \frac {16\,\beta\,{C_{4}}\, G(\xi)^{2}\,{\rm e}^{ \frac {\delta \,(x+F(\xi)) }{3\,{C_{2}}\,\gamma } } }{ \delta \,\left[{\rm e}^{ \frac {\delta \,(x + {F}(\xi) )}{3\,{C_{2} }\,\gamma } } + 1\right]^{3}}} .\tag{29d}$

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$ \begin{split}& 27 {C_{2}}^{2}\,{C_{3}}^{5} \gamma ^{2} {F_{\xi \xi }}^{3} + 4 {F_{\xi \xi }} {C_{3}}^{4}\,{C_{2}} \delta ^{2}\,{F_{ \xi }}^{3}\\ + & ( - 6 {C_{3}}^{3} {C_{2}}^{2} \delta ^{2} {F_{ \xi \xi }} + 9\,{C_{3}}^{5}\,{F_{\xi \xi \xi \xi }}\,{C_{2}}^{2}\,\gamma ^{2})\,{F_{\xi }}^{2}\\ + & \{[ 2\,{C_{2}}^{3}\,( 2\,\delta ^{2}\,{C_{3}}^{2}-9\,\gamma \,{C_{2}}^{4})\\ - & 36\,{C_{2}}^{2}\,{C_{3}}^{5}\,\gamma ^{2}\,{F_{\xi \xi \xi }} ]\,{F_{\xi \xi }} - 18\,{C_{2}}^{3}\,{C_{3 }}^{4}\,{F_{\xi \xi \xi \xi }}\,\gamma ^{2}\}\,{F_{\xi }} \\ - & {C_{3}}^{5} {F_{\xi \xi }}\,\delta ^{2}\,{F_{\xi }}^{4} \\ + & \left[36 \gamma ^{2} {C_{3}}^{4} {C_{2}}^{3} {F_{\xi \xi \xi }} + { \frac {{C_{2}}^{4}\,(9\, \gamma {C_{2}}^{4} - \delta ^{2} {C_{3}}^{2})}{{C_{3}}}}\right] \\ \times & {F _{\xi \xi }} + 9\gamma ^{2} {C_{2}}^{4} {F_{\xi \xi \xi \xi }} {C _{3}}^{3} = 0 . \\[-10pt] \end{split} $

(30)

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$ \begin{split} u =& - { \frac {1}{6}} \, { \frac {({C_{2}} - {F_{\xi }}\,{C_{3}})^{2}\,\delta ^{2}\,}{\beta \,\gamma \,{C_{2}}^{4}}}{\rm tanh}\left[ { \frac {\delta \,(x + {F} )}{6\,{C_{2}}\, \gamma }}\right ]^{2} \\ &+ { \frac {\delta \,{C_{3}}^{2}\,{F_{\xi \xi }} }{{C _{2}}^{3}\,\beta }} {\rm tanh}\left[ { \frac {\delta \,(x + {F} )}{6\,{C_{2}}\, \gamma }}\right ] \\ &- \frac {1}{18 \gamma \beta{C_{2}}^{4}({C_{2}} - {F _{\xi }}\,{C_{3}})^{2} } \{ 3\,{C_{2}}^{2}\,(3\,\gamma \,{C_{2}}^{4} \\ & + 3\,{C_{2}}^{2} \,\alpha \,\gamma \,{C_{3}}^{2} - 4\,\delta ^{2}\,{C_{3}}^{2})\,{ F_{\xi }}^{2}\\ & + [36\,{F_{\xi \xi \xi }}\,{C_{3}}^{4}\,{C_{2}}^{ 2}\,\gamma ^{2} - 2\,{C_{2}}^{3}\,{C_{3}}\,(9\,{C_{2}}^{2}\, \alpha \,\gamma\\ &- 4\,\delta ^{2})]\,{F_{\xi }} - 27\gamma ^{2} {F_{\xi \xi }}^{2}\,{C_{3}}^{4}\,{C_{2}}^{2} \\ &- 2 \delta ^{2} {F_{\xi }}^{4} {C_{3}}^{4} - 36 {F_{\xi \xi \xi }} {C_{3}}^{3} {C_{2}}^{3 }\,\gamma ^{2} \\ &+ {C_{2}}^{4}\,(9\alpha \gamma {C_{2}}^{2} - 2 \delta ^2)+ 8\,\delta ^{2}\,{C_{2}}\,{F_{\xi }}^{3}\,{C_{3}} ^{3} \}. \end{split}$

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$ u = {U} (\xi ), \tag{32a} $

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$ f = {\rm e}^{\left( - \frac {{C_{5}}\,x}{{C_{2}}}\right)}\, {F}(\xi ) - { \frac {{C_{6}}}{{C_{5}}}}, \tag{32b}$

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$ g = G(\xi )\,{\rm e}^{\left( - \frac {{C_{5}}\,x}{{C_{2}}}\right)}, \tag{32c}$

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$ h = {H}(\xi )\,{\rm e}^{\left( - \frac {{C_{5}}\,x}{{C_{2}}}\right)}, \tag{32d}$

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$ \xi = \frac{C_2 t-C_3x}{C_2}. $

(33)

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$ \begin{split} u = & - { \frac {1}{2}}\frac{1}{({F_{\xi }} { C_{3}} + {F} {C_{5}})^{2} {C_{2}}^{2} \beta} \left\{ [4\,\gamma {F_{\xi \xi \xi }}\,{C_{3}}^{4}\right. \\ &+ 2\,{C_{3}} {C_{5}} {F} (2\,{C_{5}}^{2}\,\gamma + \alpha \,{C_{2}}^{2})] {F_{\xi }} - 3\,\gamma \,{F_{\xi \xi }}^{2}\,{C_{3}}^{4} \\ & +{C_{2}}^{2} (\alpha \,{C_{3}}^{ 2} + {C_{2}}^{2}) {F_{\xi }}^{2} + 6\, \gamma \,{F}{F_{\xi \xi }}\,{C_{3}}^{2}{C_{5 }}^{2}\,\\ & \left.+ 4\,\gamma \,{F_{\xi \xi \xi }} {C_{3}}^{3} {F} {C_{5}} + {C_{5}}^{2} {F} ^{2} ({C_{5}}^{2}\,\gamma + \alpha {C_{2}}^{2})\right\} . \end{split} $

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$ R_w = a_0+a_1R(w)+a_2 R(w)^2, $

(35)

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$ R(w) = -\frac{\sqrt{\theta}}{2\, a_2}\tanh \left(\frac{\sqrt{\theta}\, w}{2}\right)+\frac{ a_1}{2\, a_2}, $

(36)

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$ \theta\equiv {a_1}^2-4a_0\, a_2. $

(37)

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$ \begin{split} & P(x, t, v) = 0,\ \ \ P = \{P_1, P_2,..., P_m\}, \\ & x = \{x_1, x_2,..., x_n\}, \ \ \ v = \{v_1, v_2,..., v_m\}, \end{split} $

(38)

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$ v_i = \sum\limits_{j = 0}^{J_i} v_{i,j}R^j(w), $

(39)

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$ P_{j,i}(x,t,v_{l,k},w) = 0. $

(40)

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$ u = u_3+u_4 R(w)+u_5 {R(w)}^2,$

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$\begin{split} {u_{3}} = & -\, { \frac { \gamma \,({a_{1}}^{2} + 8\,{a_{2}}\, {a_0})\,{w_{x}}^{2}}{ 2\, \beta }} - { \frac { \alpha + 6\,\gamma \,{w_{{xx}}}\,{a_{1}}}{2\,\beta }} \\ & - { \frac {2\,\gamma \,{w_{{xxx}}}}{{w_{x}}\, \beta }} - { \frac { {w_{t}}^{2} - 3\,\gamma \,{w_{{xx}}}^{2}}{2\,\beta \,{w_{x}} ^{2}}}, \\[-18pt]\end{split} \tag{42a}$

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$ {u_{4}} = - { \frac {6\,{a_{2}}\,\gamma \,({w_{x}}^{2 }\,{a_{1}} + {w_{{xx}}})}{\beta }}, \tag{42b}$

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$ {u_{5}} = - { \frac {6\,\gamma \,{w_{x}}^{2}\,{a_{2}} ^{2}}{\beta }},\tag{42c}$

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$\begin{split} {w_{t}}^{2} {w_{ {xx}}}\, & - \gamma (4 {a _{2}} {a_0}\! -\! {a_{1}}^{2}) {w_{ {xx}}} {w_{x}}^{4} \!-\! ( \gamma {w_{ {xxxx}}}\! +\! {w_{ {tt}}}) {w_{x}} ^{2}\\ &+ 4 \gamma {w_{x}} {w_{ {xx}}} {w_{ {xxx}}} - 3 \gamma {w_{ {xx}}}^{3} = 0. \\[-10pt]\end{split} $

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$ \begin{split} u = & u_3+u_4 R(w)+u_5 {R(w)}^2 \\ = & - \,{ \frac { \gamma \,({a_{1}}^{2} + 8\,{a_{2}}\, {a_0})\,{w_{x}}^{2}}{ 2\,\beta }} - { \frac { \alpha + 6\,\gamma \,{w_{{xx}}}\,{a_{1}}}{2\,\beta }} \\ & - { \frac {2\,\gamma \,{w_{{xxx}}}}{{w_{x}}\, \beta }} - { \frac { {w_{t}}^{2} - 3\,\gamma \,{w_{{xx}}}^{2}}{2\,\beta \,{w_{x}} ^{2}}} \\ & - { \frac {6\,{a_{2}}\,\gamma \,({w_{x}}^{2 }\,{a_{1}} + {w_{{xx}}})}{\beta }} R(w) \\ & - { \frac {6\,\gamma \,{w_{x}}^{2}\,{a_{2}} ^{2}}{\beta }} R(w)^2, \end{split} $

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$ \begin{split} u = & - { \frac {3}{2}} \, { \frac {{w_{x}}^{2}\,\gamma \,\,\theta }{ \beta }} {\rm tanh}\left( { \frac {w\,\sqrt{\theta }}{2}} \right)^{2}\\ & + { \frac {3\,\gamma \,{w_{{xx}}}\, \, \sqrt{\theta }}{\beta }}{\rm tanh}\left({ \frac {w\,\sqrt{\theta }}{2}} \right) - { \frac {\alpha }{2\,\beta }} \\ & - { \frac {\gamma \,(4\,{a _{2}}\,{a_{0}} - {a_{1}}^{2})\,{w_{x}}^{2}}{\beta }} - { \frac {2\,\gamma \,{w_{ {xxx}}}}{{w_{x}}\, \beta }}\\ & - { \frac {1}{2}} \,{ \frac { {w_{t}}^{2} - 3\,\gamma \,{w_{ {xx}}}^{2}}{\beta \,{w_{x}} ^{2}}} . \end{split} $

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$ w = {k_{1}}\,x + \omega_1\,t + {a_{3}}\,E_{\pi}( {\rm sn}({k_{2}}\,x + \omega_2\,t, \,m), \,n, \,m), $

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$ \left\{\begin{aligned} & {a_{0}} = { \frac {n\,{ a_{1}}^{2}\,{a_{3}}^{2} - 4\,m^{2} + 4\,m^{2}\,n - 4\,n^{2} + 4\, n}{4\, {a_{3}}^{2}\,{a_{2}}\,n}}, \\ & \gamma = { \frac {3\,[{k_{1}}^{2}\,{a _{4}}\,n^{2} - {k_{1}}\,{a_{4}}^{2}\,(1 + m^{2})\,n + m^{2}\,{a_{ 4}}^{3}]\,{k_{1}}\,{\omega _{2}}^{2}\,n\,{a_{3}}^{2}}{{k_{2}}^{2} \,[{k_{1}}^{2}\,({k_{1}} + 3\,{a_{4}})\,n^{2} - 2\,{k_{1}}\,{a_{4 }}\,({k_{1}} + {a_{4}})\,(1 + m^{2})\,n + m^{2}\,({a_{4}} + 3\,{k _{1}})\,{a_{4}}^{2}]^{2}}},\\ & {\omega _{1}} = - { \frac {{\omega _{ 2}} {k_{1}} [{k_{1}}^{2} (3 {a_{4}} - {k_{1}}) n^{2} - 2 {k _{1}} {a_{4}} (2 {a_{4}} - {k_{1}}) (1 + m^{2}) n + m^{2} ( 5 {a_{4}} - 3 {k_{1}}) {a_{4}}^{2}]}{{k_{2}} [{k_{1}}^{2}\,({ k_{1}} + 3 {a_{4}})\,n^{2} - 2 {k_{1}} {a_{4}} ( {k_{1}} + {a_{4}} ) (1 + m^{2})\,n + m^{2} ({a_{4}} + 3 \,{k_{1}})\,{a_{4}}^{2}]}},\end{aligned} \right. $

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$ \left\{\begin{aligned} & {\omega _{1}} = - { \frac {1}{2}} \,{ \frac {{\omega _{2}}\,{k_{1}}}{{k_{2}}}}, \\ & \gamma = - { \frac {3}{16}} \, { \frac {{\omega _{2}}^{2}\,{a_{3}}^{2}\,(3\,{a_{4}} - 2\,{k_{1}}\,n)}{{k_{2}}^{2}\,{a_{4}}\,({k_{1}}\,n - {a_{4}})^{ 2}}}, \\ & m = \pm { \frac { \sqrt{( 2\,{k_{1}}\,n- 3\,{a _{4}})\,{a_{4}}\,{k_{1}}\,n\,({k_{1}}\,n - 2\,{a _{4}})}}{{a_{4}}\,( 3\,{a_{4}}- 2\,{k_{1}}\,n)}},\\ &{a_{0}} = { \frac {1}{4}} \, { \frac {{a_{1}}^{2}}{{a_{2}}}} - { \frac {(n - 1)\,({k_{1}}^{2}\,n + 2\,{k_{1}}\,n\,{a_{3}}\,{k_{2}} - {k_{1}}^{2} - 4\,{k_{1}}\,{a_{3}}\,{k_{2}} - 3\,{a_{3}}^{2}\,{ k_{2}}^{2})}{{a_{2}}\,( 2\,{k_{1}}\,n - 3\,{a_{4}} )\,{a_{3}}^{2}\,{a_{4}}}}, \end{aligned} \right. $

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$ u = { \frac {3}{2}} \, { \frac {(a_{4}-{k_{1}}\,n\,S^{2} )^{2} \gamma \theta T^{2}}{(n\,S^{2} - 1)^{2} \beta }} + { \frac {6 \gamma {a_{3}} {k_{2}}^{2} n S C {D} \sqrt{\theta }\,T}{(n S^{2} - 1)^{2} \beta }} - \frac{ [ a_5 S^{8} + a_6 S^{6} + a_7 S^{4} +a_8 S^{2} + a_9 ]}{2[({k_{1}} n S^{2} -{a_{4}} )^{2} (n S^{2} - 1)^{2} \beta ]}, $

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$ \begin{split} T\equiv \, &{\rm tanh}\left\{{ \frac {1}{2}} \,\sqrt{\theta }\,\left [{ k_{1}}\,x + {\omega _{1}}\,t + {a_{3}}\,E_{\pi}( {\rm sn}({k_{2}}\,x + \omega_2\,t, \,m), \,n, \,m)\right ]\right\}, \\ S\equiv\, &{\rm sn}({k_{2}}\,x + \omega_2\,t, \,m),\ C\equiv{\rm cn}({k_{2}}\,x + \omega_2\,t, \,m), \ D\equiv{\rm dn}({k_{2}}\,x + \omega_2\,t, \,m), \\ a_5 = \, & - n^{3} (2 n \gamma \theta {k_{1}}^{4} - \alpha {k_{1}}^{2} n - {\omega _{1}}^{2} n + 8 \gamma {a_{3}} {k_{2}}^{3} {k_{ 1}} m^{2}), \\ a_6 =\, & 2 n^3[ 8 \gamma a_{3}{k_2}^{3}k_{1} ( m^2 + 1)+ 4 \gamma {a_{4}} \theta {k_{1}}^{3}- \alpha {k_{1}}^{2} - \alpha {k_{1}} {a_{4}} - 2 {\omega _{1}}^{2} - \omega _{1} {a_{3}} \omega _2] - 4 n^{2}m^{2} {a_{3}}\gamma {k_{2}}^{3}( a_4 +3 {k_{1}} ), \\ a_7 = \, & 24 n\gamma {a_{3}}{k_{2}}^{3}( {a_{4}}\,m^{2} - n^2 {k_{1}}) + n^2( {\omega _{ 2}}^{2} {a_{3}}^{2} - 4\,\gamma \,{a_{3}}^{2} {k_{2}}^{4} - 12 \gamma {k_{1}}^{2} {a_{4}}^{2} {a_{1}}^{2} \\ & + 6 {\omega _{1}}^{2} + 48 \gamma {a_{2} } {k_{1}}^{2} {a_{4}}^{2} {a_{0}} + 6\,{\omega _{1}} {a_{3 }} {\omega _{2}} - 4 \gamma {a_{3}}^{2} {k_{2}}^{4} m^{2} + 6 \alpha {k_{1}} {a_{4}} + \alpha {a_{3}}^{2} {k_{2}}^{2} ), \\ a_8 =\, & 4 \gamma n^2 {a_{3}} {k_{2}}^{3} ({k_{1}}+3 {a_{4}}) +8n \gamma{k_{1}} {a_{4}}^{3} \theta - 16\,n\gamma {a_{3}} {k _{2}}^{3}a_{4}( m^{2} + 1) \\ & -2n( {a_{3}}^{2} {\omega _{2}}^{2} + \alpha {k_{1}} {a_{4}} + 2 {\omega _{1}}^{2} + 3 { \omega _{1}} {a_{3}} {\omega _{2}} + {a_{4}}^{2} \alpha), \\ a_9 = & {a_{4}}^{2} \alpha +( {a_{3}} {\omega _{2}} + {\omega _{1}})^{2} + 8 {a_{3}} \gamma {k_{2}}^{3} n {a_{4}} - 2 \gamma {a_{4}} ^{4} \theta, \end{split} $

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Ping Liu, Heng-Rui Xu, Jian-Rong Yang. The Boussinesq equation: Lax pair, Bäcklund transformation, symmetry group transformation and consistent Riccati expansion solvability[J]. Acta Physica Sinica, 2020, 69(1): 010203-1

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