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Journals >Acta Physica Sinica >Volume 68 >Issue 21 >Page 210201-1 > Article

Acta Physica Sinica
Vol. 68, Issue 21, 210201-1 (2019)

Fractional order model and Lump solution in dusty plasma

Jun-Chao Sun¹, Zong-Guo Zhang², Huan-He Dong¹, and Hong-Wei Yang^1、*

Author Affiliations

¹College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, China

²School of Mathematics Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China

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DOI: 10.7498/aps.68.20191045 Cite this Article

Jun-Chao Sun, Zong-Guo Zhang, Huan-He Dong, Hong-Wei Yang. Fractional order model and Lump solution in dusty plasma[J]. Acta Physica Sinica, 2019, 68(21): 210201-1 Copy Citation Text

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Abstract

In recent years, the dust plasma research plays an important role in the field of space, industry, and laboratory. In this paper, starting from the control equations of the double temperature dust plasma, we derive the (2+1)-dimensional Kadomtsev-Petviashvili (KP) equation to describe the double temperature dust plasma sound waves by using the multi-scale analysis, and reduce it by using the perturbation method. Then by using the semi inverse method and fractional variational principle, the (2+1)-dimensional KP equation is introduced into the time-space fractional KP equation (TFS-KP). The fractional KP equation has potential applications in describing physical phenomena in practical problems. Furthermore, based on the symmetrical analysis method, by which lie discussed the time fractional KP (TF-KP) equation of the conservation law, the dual temperature dust plasma acoustic conserves quantity. Finally, based on the bilinear method, the lump solution of fractional KP equation is obtained. The existence of this solution indicates the rogue waves existing in double temperature dusty plasma. The influence of fractional order on rogue wave is also analyzed.

Keywords

bilinear method dust plasma fractional Kadomtsev-Petviashvili equation Lump solutions

$ n_{{\rm il}0}+n_{{\rm ih} 0} = \sum\limits_{j = 1}^{N} Z_{{\rm d}0_{j}}n_{{\rm d}0_{j}}+n_{{\rm e}0}, $

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$ \left\{\begin{aligned}& \frac{\partial n_{{\rm d}_{j}}}{\partial t}+\nabla\cdot(n_{{\rm d}_{j}}{{u}_{{\rm d}_{j}}}) = 0,\\ &\frac{\partial {{u}_{{\rm d}_{j}}}}{\partial t}+({{u}_{{\rm d}_{j}}}\cdot\nabla){{u}_{{\rm d}_{j}}} = \frac{Z_{{\rm d}_{j}}}{m_{{\rm d}_{j}}}\nabla\varPhi,\\ & \frac{\partial ^{2}\varPhi}{\partial x^{2}}+\frac{\partial ^{2}\varPhi}{\partial y^{2}} = \sum\limits_{j = 1}^{N} Z_{{\rm d}_{j}}n_{{\rm d}_{j}}+n_{\rm e}-n_{{\rm il}}-n_{{\rm ih}}, \end{aligned} \right. $

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$ \left\{ \begin{aligned}& \frac{\partial n_{{\rm d}}}{\partial t}+\frac{\partial (n_{{\rm d}}u_{{\rm d}})}{\partial x}+\frac{\partial (n_{{\rm d}}v_{{\rm d}})}{\partial y} = 0,\\ & \frac{\partial u_{{\rm d}}}{\partial t}+u_{{\rm d}}\frac{\partial u_{{\rm d}}}{\partial x}+v_{{\rm d}}\frac{\partial u_{{\rm d}}}{\partial y} = \frac{\partial \phi}{\partial x},\\ & \frac{\partial v_{{\rm d}}}{\partial t}+u_{{\rm d}}\frac{\partial v_{{\rm d}}}{\partial x}+v_{{\rm d}}\frac{\partial v_{{\rm d}}}{\partial y} = \frac{\partial \phi}{\partial y},\\ & \frac{\partial ^{2}\phi}{\partial x^{2}}+\frac{\partial ^{2}\phi}{\partial y^{2}} = n_{{\rm d}}-1+C_{1}\phi+C_{2}\phi^{2}+C_{3}\phi^{3}, \end{aligned} \right. $

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$ \begin{aligned} & C_{1} = \frac{\gamma}{\mu-1}\left[ {C_{\rm e} \left( {H_{\rm e}-\frac{1}{2}} \right)+C_{\rm i}\left( {H_{\rm i}-\frac{1}{2}} \right)} \right],\\ & C_{2} = \frac{\gamma}{2(1-\mu)}\left[ {\mu C_{\rm e}^{2}\left( {H_{\rm e}^{2}-\frac{1}{4}} \right)+C_{{\rm i}}^{2}\left( {H_{{\rm i}}^{2}-\frac{1}{4}} \right)} \right],\\ & C_{3} = \frac{\gamma}{6(\mu-1)}\left[\mu C_{\rm e}^{3}\left( {H_{\rm e}^{2}-\frac{1}{4}} \right)\left( {H_{\rm e}-\frac{3}{2}} \right)\right.\\ &\quad\quad \left.+C_{{\rm i}}^{3}\left( {H_{{\rm i}}^{2}-\frac{1}{4}} \right)\left( {H_{\rm i}+\frac{3}{2}} \right)\right], \end{aligned} $

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$ \xi = \varepsilon(x-v_{0}t), ~\eta = \varepsilon^{2} y,~ \tau = \varepsilon^{3}t, $

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$\begin{split} & \frac{\partial}{\partial t} = \varepsilon^{3}\frac{\partial}{\partial \tau}-\varepsilon v_{0}\frac{\partial}{\partial \xi},\\ & \frac{\partial}{\partial x} = \varepsilon\frac{\partial}{\partial \xi},\; \; \frac{\partial}{\partial y} = \varepsilon^{2}\frac{\partial}{\partial \eta}. \end{split}$

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$ \left\{ \begin{aligned} & n_{{\rm d}} = 1+\varepsilon^{2}n_{{\rm d}_{1}}+\varepsilon^{4}n_{{\rm d}_{2}}+\cdots,\\ & u_{{\rm d}} = \varepsilon^{2}u_{{\rm d}_{1}}+\varepsilon^{4}u_{{\rm d}_{2}}+\cdots,\\ & v_{{\rm d}} = \varepsilon^{3}v_{{\rm d}_{1}}+\varepsilon^{5}v_{{\rm d}_{2}}+\cdots,\\ & \phi = \varepsilon^{2} \phi_{1}+\varepsilon^{4}\phi_{2}+\cdots. \end{aligned} \right. $

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$ \left\{\begin{aligned} & \varepsilon^{3}\frac{\partial n_{{\rm d}}}{\partial \tau}-\varepsilon v_{0}\frac{\partial n_{{\rm d}}}{\partial \xi}+\varepsilon\frac{\partial (n_{{\rm d}}u_{{\rm d}})}{\partial \xi}+\varepsilon^{2}\frac{\partial (n_{{\rm d}}v_{{\rm d}})}{\partial \eta} = 0,\\ & \varepsilon^{3}\frac{\partial u_{{\rm d}}}{\partial \tau}-\varepsilon v_{0}\frac{\partial u_{{\rm d}}}{\partial \xi}+\varepsilon u_{{\rm d}}\frac{\partial u_{{\rm d}}}{\partial \xi}+\varepsilon ^{2}v_{{\rm d}}\frac{\partial u_{{\rm d}}}{\partial \eta} = \varepsilon\frac{\partial \phi}{\partial \xi},\\ & \varepsilon^{3}\frac{\partial v_{{\rm d}}}{\partial \tau}-\varepsilon v_{0}\frac{\partial v_{{\rm d}}}{\partial \xi}+\varepsilon u_{{\rm d}}\frac{\partial v_{{\rm d}}}{\partial \xi}+\varepsilon ^{2}v_{{\rm d}}\frac{\partial v_{{\rm d}}}{\partial \eta} = \varepsilon^{2}\frac{\partial \phi}{\partial \eta},\\ & \varepsilon^{2}\frac{\partial ^{2}\phi}{\partial \xi^{2}}+\varepsilon^{4}\frac{\partial ^{2}\phi}{\partial \eta^{2}} = n_{{\rm d}}-1+C_{1}\phi+C_{2}\phi^{2}+C_{3}\phi^{3}. \end{aligned}\right. $

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$ \varepsilon^{3}: \left\{\begin{aligned} &-v_{0}\frac{\partial n_{{\rm d}_{1}}}{\partial \xi}+\frac{\partial u_{{\rm d}_{1}}}{\partial \xi} = 0,\\ & -v_{0}\frac{\partial u_{{\rm d}_{1}}}{\partial \xi} = \frac{\partial \phi_{1}}{\partial \xi}, \end{aligned} \right. ~~~~~~~ $

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$ \varepsilon^{4}: \left\{ \begin{aligned} & -v_{0}\frac{\partial v_{{\rm d}_{1}}}{\partial \xi} = \frac{\partial \phi_{1}}{\partial \eta},\\ & \frac{\partial ^{2}\phi_{1}}{\partial \xi^{2}} = n_{{\rm d}_{2}}+C_{1}\phi_{2}+C_{2}\phi_{1}^{2}, \end{aligned} \right. $

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$ \varepsilon^{5}: \left\{ \begin{aligned} & \frac{\partial n_{{\rm d}_{1}}}{\partial \tau}-v_{0}\frac{\partial n_{{\rm d}_{2}}}{\partial \xi}+\frac{\partial u_{{\rm d}_{2}}}{\partial \xi}+\frac{\partial u_{{\rm d}_{1}}n_{{\rm d}_{1}}}{\partial \xi}+\frac{\partial v_{{\rm d}_{1}}}{\partial \eta} = 0,\\ & \frac{\partial u_{{\rm d}_{1}}}{\partial \tau}-v_{0}\frac{\partial u_{{\rm d}_{2}}}{\partial \xi}+u_{{\rm d}_{1}}\frac{\partial u_{{\rm d}_{1}}}{\partial \xi} = \frac{\partial \phi_{2}}{\partial \xi}. \end{aligned} \right. $

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$ \left\{\begin{aligned} & n_{{\rm d}_{1}} = \frac{u_{{\rm d}_{1}}}{v_{0}},\\ & u_{{\rm d}_{1}} = -\frac{\phi_{1}}{v_{0}},\\ & -v_{0}\frac{\partial v_{{\rm d}_{1}}}{\partial \xi} = \frac{\partial \phi_{1}}{\partial \eta},\\ & \frac{\partial ^{2}\phi_{1}}{\partial \xi^{2}} = n_{{\rm d}_{2}}+C_{1}\phi_{2}+C_{2}\phi_{1}^{2}. \end{aligned} \right. $

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$ \frac{\partial}{\partial \xi}\left( {\frac{\partial \phi_{1}}{\partial \tau}+b_{1}\phi_{1}\frac{\partial \phi_{1}}{\partial \xi}+b_{2}\frac{\partial ^{3}\phi_{1}}{\partial ^{3}\xi}} \right)+b_{3}\frac{\partial ^{2}\phi_{1}}{\partial ^{2}\eta} = 0, $

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$ \left\{\begin{aligned}& b_{1} = -\frac{3}{2v_{0}}-v_{0}^{2}C_{2},\\ & b_{2} = \frac{v_{0}^{3}}{2}, \; b_{3} = \frac{v_{0}}{2}.\\ \end{aligned} \right. $

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$D_t^\omega f = \left\{ {\begin{aligned} &{\frac{{{\partial ^n}f}}{{\partial {t^n}}},} \quad\quad\quad\quad\quad\quad\quad\quad {\omega = n,\;n \in N,}\\ &{\frac{1}{{{\varGamma(n\!-\!\omega)}(n \!- \!\omega )}}\frac{{{\partial ^n}}}{{\partial {T^n}}}\int_0^{{T_0}}\!\! f {{(T\! - \!\tau )}^{n - \omega - 1}}{\rm{d}}\tau ,}\\ &\quad\quad\quad \quad \quad\quad\qquad\quad\quad {n - 1 <\omega < n.} \end{aligned}} \right.$

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$ A_{\tau}+b_{1}AA_{\xi}+b_{2}A_{\xi\xi\xi}+b_{3}D^{-1}A_{\eta\eta} = 0,$

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$ B_{\xi\tau}+b_{1}B_{\xi}B_{\xi\xi}+b_{2}B_{\xi\xi\xi\xi}+b_{3}B_{\eta\eta} = 0,\\ $

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$ \begin{split} J(B) \!=\, & \iint\nolimits_{R}{\rm{d}}\xi {\rm{d}}\eta \int_T {\rm{d}} \tau [B({c_1}{B_{\xi \tau }} \!+\! {c_2}{b_1}{B_\xi }{B_{\xi \xi }}\\ & + {c_3}{b_2}{B_{\xi \xi \xi \xi }} + {c_4}{b_3}{B_{\eta \eta }})], \end{split}$

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$ \begin{split} J(B) =\, & \iint\nolimits_{R} {\rm{d}}\xi {\rm{d}}\eta \int_{T}{\rm{d}}\tau\left[ {-c_{1}B_{\xi}B_{\tau} \!-\! \frac{1}{2}c_{2}b_{1}B_{\xi}^{3}}\right.\\ & { +c_{3}b_{2}(B_{\xi\xi})^{2}+c_{4}b_{3}(B_{\eta})^{2}} \Big], \end{split}$

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$ \begin{split} & F(\xi,\eta,\tau,B,B_{\tau},B_{\xi},B_{\xi\xi},B_{\eta})\\ = \, & \frac{\partial F}{\partial B}-\frac{\partial}{\partial \tau}\left( {\frac{\partial F}{\partial B_{\tau}}} \right)-\frac{\partial}{\partial \xi}\left( {\frac{\partial F}{\partial B_{\xi}}} \right)\\ & -\frac{\partial}{\partial \eta}\left( {\frac{\partial F}{\partial B_{\eta}}} \right)+ \frac{\partial^{2}}{\partial \xi^{2}}\left( {\frac{\partial F}{\partial B_{\xi\xi}}} \right)\\ = \,& 2c_{1}B_{\xi\tau}+3c_{2}b_{1}B_{\xi}B_{\xi\xi}+2c_{3}b_{2}B_{\xi\xi\xi\xi}\\ & +2c_{4}b_{3}B_{\eta\eta} = 0,\end{split} $

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$ c_{1} = \frac{1}{2},\; \; c_{2} = \frac{1}{3},\; \; c_{3} = \frac{1}{2},\; \; c_{4} = \frac{1}{2}. $

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$\begin{split} L(B_{\tau},B_{\xi},B_{\eta},B_{\xi\xi}) & = -\frac{1}{2}B_{\tau}B_{\xi} -\frac{1}{6}b_{1}(B_{\xi})^{3} \\ & +\frac{1}{2}b_{2}(B_{\xi\xi})^{2} \!-\! \frac{1}{2}b_{3}(B_{\eta})^{2}, \end{split}$

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$ \begin{split} & F(D_\tau ^\omega B,D_\xi ^\alpha B,D_\eta ^\beta B,D_\xi ^{\alpha \alpha }B) \\=\, & - \frac{1}{2}D_\tau ^\omega BD_\xi ^\alpha B - \frac{1}{6}{a_1}{{(D_\xi ^\alpha B)}^3}\\ & + \frac{1}{2}{a_2}{{(D_\xi ^{\alpha \alpha }B)}^2} - \frac{1}{2}{a_3}{{(D_\eta ^\beta B)}^2}, \end{split}$

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$\begin{split} J_{F}(B) =\, & \iint\nolimits_{R} ({\rm{d}}\xi)^{\alpha} ({\rm{d}}\eta)^{\beta} \int_{T} ({\rm{d}}\tau)^{\omega} \\ &\times F(D_{\tau}^{\omega}B,D_{\xi}^{\alpha}B,D_{\eta}^{\beta}B, D_{\xi}^{\alpha\alpha}B). \end{split} $

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$ \begin{split} \delta {J_F}(B) =\, & \int_R {{{({\rm{d}}\xi )}^\alpha }} \int_R {{{({\rm{d}}\eta )}^\alpha }} \int_T {{{({\rm{d}}\tau )}^\omega }} \\ & \times\left[ {\left( {\frac{{\partial F}}{{\partial D_\tau ^\omega B}}} \right)\delta D_\tau ^\omega B + \left( {\frac{{\partial F}}{{\partial D_\xi ^\alpha B}}} \right)\delta D_\xi ^\alpha B} \right.\\ & \left. { + \left( {\frac{{\partial F}}{{\partial D_\xi ^{\alpha \alpha }B}}} \right)\delta D_\xi ^{\alpha \alpha } + \left( {\frac{{\partial F}}{{\partial D_\eta ^\beta B}}} \right)\delta D_\eta ^\beta B} \right], \end{split} $

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$ \int_{a}^{T}({\rm{d}}\tau)^{j}f(\tau) = j\int_{a}^{T}({\rm{d}}\tau)(T-\tau)^{j}f(\tau). $

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$\begin{split} & \int_{a}^{b}({\rm{d}}\tau)^{i}f(x)D_{x}^{i}g(x) =\varGamma(1+i)[g(x)f(x)|_{a}^{b}\\ & \quad\quad\quad\quad\quad\quad\quad\quad -\int_{a}^{b}({\rm{d}}x)^{i}g(x)D_{x}^{i}f(i)],\\ & f(x),g(x)\in[a,b]. \end{split} $

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$ \begin{split} \delta J_{F}(B) =\, &\, \int_{R}({\rm{d}}\xi)^{\alpha}\int_{R}({\rm{d}}\eta)^{\alpha}\int_{T}({\rm{d}}\tau)^{\omega} \\& \times\left[ {-D_{\tau}^{\omega}\left( {\frac{\partial F}{\partial D_{\tau}^{\omega}B}} \right)-D_{\xi}^{\alpha}\left( {\frac{\partial F}{\partial D_{\xi}^{\alpha}B}} \right) }\right.\\ &\left. {-D_{\eta}^{\beta}\left( {\frac{\partial F}{\partial D_{\eta}^{\beta}B}} \right)+D_{\xi}^{\alpha\alpha}\left( {\frac{\partial F}{\partial D_{\xi}^{\alpha\alpha}B}} \right)} \right].\end{split} $

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$\begin{split} & -D_{\tau}^{\omega}\left( {\frac{\partial F}{\partial D_{\tau}^{\omega}B}} \right)-D_{\xi}^{\alpha}\left( {\frac{\partial F}{\partial D_{\xi}^{\alpha}B}} \right) \\ & -D_{\eta}^{\beta}\left( {\frac{\partial F}{\partial D_{\eta}^{\beta}B}} \right)+D_{\xi}^{\alpha\alpha}\left( {\frac{\partial F}{\partial D_{\xi}^{\alpha\alpha}B}} \right) = 0. \end{split}$

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$ \begin{split} \, & D_{\tau}^{\omega}D_{\xi}^{\alpha}B +b_{1}(D_{\xi}^{\alpha}B)D_{\xi}^{\alpha\alpha}B+b_{2}D_{\xi}^{\alpha\alpha\alpha\alpha}B \\ &~~ +b_{3}D_{\eta}^{\beta\beta}B = 0. \end{split} $

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$ D_{\tau}^{\omega}A+b_{1}AD_{\xi}^{\alpha}A+b_{2}D_{\xi}^{\alpha\alpha\alpha}A+\int b_{3}D_{\eta}^{\beta\beta}A({\rm{d}}\xi)^{\alpha} = 0.$

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$D_{\xi}^{\alpha}(D_{\tau}^{\omega}A+b_{1}AD_{\xi}^{\alpha}A+b_{2}D_{\xi}^{\alpha\alpha\alpha}A) +b_{3}D_{\eta}^{\beta\beta}A = 0. $

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$ D_{\tau}^{\omega}A+b_{1}A\frac{\partial A}{\partial \xi}+b_{2}\frac{\partial ^{3}A}{\partial \xi^{3}} +b_{3}D^{-1}\left( {\frac{\partial ^{2}A}{\partial \eta^{2}}} \right) = 0. $

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$ \begin{split} D_{\tau}^{\gamma}A ={}& Q(\xi, \eta, \tau, A, A_{\xi}, A_{\xi\xi\xi}, D_{\tau}^{\omega}A, A_{\eta\eta}, \cdots), \\ & \omega \!>\! 0.\\ \end{split} $

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$ \left\{ \begin{aligned}& \overline{\xi} = \xi+\varepsilon X(\xi,\eta,\zeta,\tau,A)+O(\varepsilon ^{2}),\\ & \overline{\eta} = \eta+\varepsilon Y(\xi,\eta,\zeta,\tau,A)+O(\varepsilon ^{2}),\\ & \overline{\tau} = \tau+\varepsilon T(\xi,\eta,\zeta,\tau,A)+O(\varepsilon ^{2}),\\ & \overline{A} = A+\varepsilon \psi(\xi,\eta,\zeta,\tau,A)+O(\varepsilon ^{2}),\\ & D_{\tau}^{\omega}\overline{A}\rightarrow D_{\tau}^{\omega}A+\varepsilon \psi_{\omega}^{\tau}+O(\varepsilon ^{2}),\\ & \frac{\partial \overline{A}}{\partial \xi}\rightarrow \frac{\partial A}{\partial \xi}+\varepsilon \psi_{\xi}+O(\varepsilon ^{2}),\\ & \frac{\partial ^{3}\overline{A}}{\partial \xi^{3}}\rightarrow \frac{\partial ^{3}A}{\partial \xi^{3}}+\varepsilon \psi_{\xi\xi\xi}+O(\varepsilon ^{2}),\\ & \frac{\partial ^{2}\overline{A}}{\partial \eta^{2}}\rightarrow \frac{\partial ^{2}A}{\partial \eta^{2}}+\varepsilon \psi_{\eta\eta}+O(\varepsilon ^{2}), \end{aligned} \right. $

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$ \left\{\begin{aligned} \psi_{\omega}^{\tau} =\, & D_{\tau}^{\omega}(\psi)+X D_{\tau}^{\omega}(A_{\xi})-D_{\tau}^{\omega}(X A_{\xi})\\ &+Y D_{\tau}^{\omega}(A_{\eta})-D_{\tau}^{\omega}(Y A_{\eta}) \\ & +D_{\tau}^{\omega}(D_{\tau}(T)A)-D_{\tau}^{\gamma+1}(T A) \\ &+T D_{\tau}^{\gamma+1}(A),\\ \psi_{\xi} =\, & D_{\xi}(\psi)-A_{\xi}D_{\xi}(X)-A_{\eta}D_{\xi}(Y) \\ & -A_{\tau}D_{\xi}(T),\\ \psi_{\xi\xi\xi} =\, & D_{\xi}(\psi^{\xi\xi})-A_{\xi\xi\xi}D_{\xi}(X)-A_{\xi\xi\eta}D_{\xi}(Y) \\ & -A_{\xi\xi\tau}D_{\xi}(T),\\ \psi_{\eta\eta} =\, & D_{\eta}(\psi^{\eta})-A_{\xi\eta}D_{\eta}(X)-A_{\eta\eta}D_{\eta}(Y) \\ & -A_{\eta\tau}D_{\eta}(T), \end{aligned} \right. $

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$ \left\{\begin{aligned} {D_\tau} =\, & \frac{\partial}{{\partial \tau}} + {A_{\tau}}\frac{\partial}{{\partial \tau}} + {A_{\tau\tau}}\frac{\partial}{\partial {A_{\tau}}}+{A_{\xi\tau}}\frac{\partial}{{\partial {A_\xi}}}\\ &+{A_{\eta\tau}}\frac{\partial}{{\partial {A_\eta}}}+{A_{\zeta\tau}}\frac{\partial}{{\partial {A_\zeta}}} + \cdots,\\ {D_\xi} =\, & \frac{\partial}{{\partial \xi}} + {A_\xi}\frac{\partial}{{\partial A}} + {A_{\xi\xi}}\frac{\partial}{{\partial {A_\xi}}} + {A_{\tau\xi}}\frac{\partial}{{\partial {A_\tau}}} \\ &+ {A_{\eta\xi}}\frac{\partial}{{\partial {A_\eta}}}+ {A_{\zeta\xi}}\frac{\partial}{{\partial {A_\zeta}}} + \cdots, \\ {D_\eta} =\, & \frac{\partial}{{\partial \eta}} + {A_\eta}\frac{\partial}{{\partial A}} + {A_{\eta\eta}}\frac{\partial}{{\partial {A_\eta}}} + {A_{\tau\eta}}\frac{\partial}{{\partial {A_\tau}}} \\ &+ {A_{\xi\eta}}\frac{\partial}{{\partial {A_\xi}}}+ {A_{\zeta\eta}}\frac{\partial}{{\partial {A_\zeta}}} + \cdots. \end{aligned}\right. $

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$D_T^\omega (f(t)g(t)) = \sum\limits_{n = 0}^\infty \left( {\begin{aligned} \omega \\ n \end{aligned}} \right) D_t^{\omega - n}f(t)D_t^ng(t),\; \omega > 0,$

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$ \left( {\begin{aligned} \omega \\ n \end{aligned}} \right)= \frac{(-1)^{n-1}\omega\varGamma(n-\omega)} {\varGamma(1-\omega)\varGamma(n+1)},\\ $

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$ \begin{split} \psi_{\omega}^{\tau} \!=\! \,& D_{\tau}^{\omega}(\psi) \!-\! \omega D_{\tau}^{\omega}(T)\frac{\partial^{\omega}A}{\partial \tau^{\omega}} \!-\! \sum\limits_{n = 1}^{\infty}\left( {\begin{aligned} \omega \\ n \end{aligned}} \right)D_{\tau}^{n}(X)D_{\tau}^{\omega-n}A_{\xi}\\ & -\sum\limits_{n = 1}^{\infty}\left( {\begin{aligned} \omega \\ n \end{aligned}} \right)D_{\tau}^{n}(Y) D_{\tau}^{\omega-n}A_{\eta}\\ &-\sum\limits_{n = 1}^{\infty}\left( {\begin{aligned} \omega\;\;\; \\ {n+1} \end{aligned}} \right) D_{\tau}^{n+1}(T)D_{\tau}^{\omega-n}A.\\[-17pt] \end{split} $

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$ \frac{{\rm{d}}^{m}f(g(t))}{{\rm d}t^{m}} = \sum\limits_{k = 0}^{m}\sum\limits_{r = 0}^{k}\left( \begin{aligned} k \\ r\end{aligned} \right)\frac{1}{k!}[-g(t)^{k-r}]\frac{{\rm d}^{k}f(g(t))}{{\rm{d}}t^{k}},$

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$\begin{split} D_{\tau}^{\omega}\psi =\, & \frac{\partial^{\omega}\psi}{\partial \tau^{\omega}}+\psi_{A}\frac{\partial^{\omega}A}{\partial \tau^{\omega}}-A\frac{\partial^{\omega}\psi_{A}}{\partial \tau^{\omega}}\\ &+\sum\limits_{n = 1}^{\infty}\left( {\begin{aligned} \omega \\ n \end{aligned}} \right) \frac{\partial^{n}\psi_{A}}{\partial \tau^{n}}D_{\tau}^{\omega-n}A+Ra, \end{split}$

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$ \begin{split} \, & Ra = \sum\limits_{n = 2}^{\infty}\sum\limits_{m = 2}^{n}\sum\limits_{k = 2}^{m} \sum\limits_{r = 0}^{k-1}\left[\left( {\begin{aligned} \omega \\ n \end{aligned}} \right)\left( {\begin{aligned} n\\ m \end{aligned}} \right)\left( {\begin{aligned} k\\ r \end{aligned}} \right)\right.\frac{1}{k!}\\ & \left.\!\! \! \times\frac{\tau^{n-\omega}}{\varGamma(n \!+\! 1 \!-\! \omega)}(-A)^{r}\frac{\partial^{A}}{\partial \tau^{A}}(A)^{k-r} \frac{\partial^{n-m+k} \psi}{\partial \tau^{n-m}\partial A^{k}} \right]. \end{split}$

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$ \begin{split} \psi_{\omega}^{\tau} = \, & \frac{\partial^{\omega}\psi}{\partial \tau^{\omega}}+(\psi_{A}-\omega D_{\tau}(T))\frac{\partial^{\omega}A}{\partial \tau^{\omega}}-A\frac{\partial^{\omega}\psi_{A}}{\partial \tau^{\omega}}\\ &+\sum\limits_{n = 1}^{\infty}\left[ { \left( {\begin{aligned} \omega \\ n \end{aligned}} \right) \frac{\partial^{\omega}\psi_{A}}{\partial \tau^{\omega}}- \!\left( {\begin{aligned} \omega\;\;\; \\ {n+1} \end{aligned}} \right) D_{\tau}^{n+1}(T)} \right]\! D_{\tau}^{\omega-n}A \\ &-\sum\limits_{n = 1}^{\infty} \left( {\begin{aligned} \omega \\ n \end{aligned}} \right) [D_{\tau}^{n}(X)D_{\tau}^{\omega-n}(A_{\xi})\\ &+D_{\tau}^{n}(Y)D_{\tau}^{\omega-n}(A_{\eta})]+Ra. \\[-17pt]\end{split} $

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$ M = X\frac{\partial}{\partial \xi}+Y\frac{\partial}{\partial \eta}+T\frac{\partial}{\partial \tau}+\psi\frac{\partial}{\partial A}.\\ $

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$ \left\{ \begin{aligned} & Pr^{(n)}M(\varDelta)|_{\varDelta = 0} = 0,\qquad n = 1,2,3,\cdots,\\ & \varDelta = D_{\tau}^{\omega}A+a_{1}A\frac {\partial A}{\partial \xi}+a_{2}\frac {\partial ^{3}A}{\partial \xi^{3}}+a_{3}D^{-1} \left( {\frac {\partial ^{2}A}{\partial \eta^{2}}} \right). \end{aligned} \right. $

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$\begin{split} Pr^{(4)} M(\varDelta) =\, & T\frac{\partial^{\omega}}{\partial \tau^{\omega}}+X\frac{\partial}{\partial \xi}+Y\frac{\partial}{\partial \eta}+\psi\frac{\partial}{\partial A}\\ &+\psi_{\tau}^{\omega}\frac{\partial}{\partial D_{\tau}^{\gamma}A} +\psi_{\xi}\frac{\partial}{\partial A_{\xi}}\\ &+\psi_{\xi\xi\xi}\frac{\partial}{\partial A_{\xi\xi\xi}}+\psi_{\eta\eta}\frac{\partial}{\partial A_{\eta\eta}}. \end{split} $

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$ \psi_{\omega}^{\tau}+a_{1}\psi\frac{\partial}{\partial \xi}+a_{1}\psi_{\xi}A+a_{2}\psi_{\xi\xi\xi}+a_{3}D^{-1}(\psi_{\eta\eta}) = 0.\\ $

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$ \left\{ \begin{aligned} & \left(\begin{aligned} \omega \\ n \end{aligned} \right) \frac{\partial^{\omega}\psi_{A}}{\partial \tau^{\omega}}- \left( {\begin{aligned} \omega\quad \\ {n+1} \end{aligned}} \right) D_{\tau}^{n+1}(\tau) = 0,\\ & X_{A} = X_{\tau} = 0,\\ & Y_{A} = Y_{\tau} = Y_{\xi} = 0,\\& \psi_{A}-\omega T_{\tau} = 0,\\ &\psi_{A}-a_{1}X_{\xi} = 0,\\ & \psi_{A}-a_{1}Y_{\eta} = 0. \end{aligned} \right. $

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$ \left\{ \begin{aligned} & \psi = c_{1}A, ~~ X = \frac{c_{1}\xi}{a_{1}}+c_{2},\\ & Y = \frac{c_{1}\eta}{a_{3}}+c_{3},~~ T = \frac{c_{1}\tau}{\omega}+c_{5}. \end{aligned} \right. $

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$ \left\{ \begin{aligned} & M_{1} = \frac{\partial}{\partial \tau},~~ M_{2} = \frac{\partial}{\partial \xi},~~M_{3} = \frac{\partial}{\partial \eta},\\ & M_{5} = \frac{\xi}{a_{1}}\frac{\partial}{\partial \xi}+\frac{\eta}{a_{3}}\frac{\partial}{\partial \eta}+\frac{\tau}{\omega}\frac{\partial}{\partial \tau}-A\frac{\partial}{\partial A}. \end{aligned} \right. $

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$ D_{\tau}(C^{\tau})+D_{\xi}(C^{\xi})+D_{\eta}(C^{\eta}) = 0.\\ $

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$\begin{split} L =\, & s(\xi, \eta, \tau)\left[D_{\tau}^{\omega}A+a_{1}A\frac{\partial A}{\partial \xi}+a_{2}\frac{\partial ^{3}A}{\partial \xi^{3}}\right.\\ & \left.+a_{3}D^{-1}\left( {\frac{\partial ^{2}A}{\partial \eta^{2}}} \right)\right],\end{split}$

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$ \int_{R}\int_{R}\int_{T}L(\xi, \eta, \tau, D_{\tau}^{\omega}A, A_{\xi}, A_{\xi\xi\xi}, A_{\eta\eta}){\rm{d}}\xi {\rm{d}}\eta {\rm{d}}\tau,\\ $

(53)

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$ \begin{split} \frac{\delta}{\delta A} =\, &\frac{\partial}{\partial A}+(D_{\tau}^{\omega})^{*}\frac{\partial}{\partial D_{\tau}^{\omega}A}+D_{\xi}\frac{\partial}{\partial A_{\xi}}\\ &-D^{3}_{\xi}\frac{\partial}{\partial A_{\xi\xi\xi}}-D^{2}_{\eta}\frac{\partial}{\partial A_{\eta\eta}}. \end{split} $

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$ (D_{\tau}^{\omega})^{*} = (-1)^{n}I_{p}^{n-\omega}(D_{\tau}^{n}) = {}_{\tau}^{C}D_{p}^{\omega}, $

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$ F^{*} = \frac{\delta L}{\delta A} = 0. $

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$ \begin{split} F^{*} = \, & a_{1}A_{\xi}+(D_{\tau}^{\omega})^{*}s-a_{1}D_{\xi}(As)\\ &-a_{2}D_{\xi}^{3}s-a_{3}D^{-1}(D_{\eta}^{2}s). \end{split}$

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$ W = \psi-T A_{\tau}-X A_{\xi}-Y A_{\eta}. $

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$ \left\{\begin{aligned} & W_{1} = -A_{\xi},\\ & W_{2} = -A_{\eta},\\ & W_{4} = -A_{\tau},\\ & W_{5} = -A-\frac{\xi}{a_{1}}A_{\xi}-\frac{\eta}{a_{3}}A_{\eta} -\frac{\tau}{\omega}A_{\tau}. \end{aligned} \right. $

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$ \begin{aligned} C^{\tau} =\, & T L+\sum\limits_{k = 0}^{n-1}(-1)^{k}_{0}D_{T}^{\omega-1-k}(W_{i})D_{\tau}^{k}\frac{\partial L}{\partial(_{0}D_{\tau}^{\omega}A)} \\ & -(-1)^{n}J\left( {W_{i},D_{\tau}^{n}\frac{\partial L}{\partial(_{0}D_{\tau}^{\omega}A)}} \right).\end{aligned} $

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$ J(a,b) \!=\! \frac{1}{\varGamma(n \!-\! \omega)}\int_{0}\int_{\tau} \frac{f(T,\xi,\eta,\zeta)g(\mu,\xi,\eta,\zeta)}{(\mu-T)^{\omega+1-n}} {\rm{d}}\mu {\rm{d}}T. $

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$ \begin{split} C^{i} =\, & \rho^{i}L+W_{\gamma}\left[\frac{\partial L}{\partial A_{i}}-D_{i}\left( {\frac{\partial L}{\partial A_{ij}}} \right)\right.\\ & \left.+D_{i}D_{k}\left( {\frac{\partial L}{\partial A_{ijk}}} \right)-\cdots \right]\\ &+D_{j}(W_{\gamma})\left( {\frac{\partial L}{\partial A_{ij}}-D_{k}\frac{\partial L}{\partial A_{ijk}}+\cdots} \right)\\ & +D_{j}D_{k}(W_{\gamma})\left( {\frac{\partial L}{\partial A_{ijk}}-\cdots} \right)+\cdots, \end{split} $

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$ \left\{\begin{aligned} C^{\tau} \, &= TL+_{0}D_{T}^{\omega-1}(W_{5})\frac{\partial L}{\partial _{0}D_{\tau}^{\gamma}A} \!+\! J\Big(\!{W_{5},D_{\tau}\frac{\partial L}{\partial D_{\tau}^{\gamma}A}}\!\Big), \\ &= s_{0}D_{\tau}^{\omega-1}(W_{5})+J(W_{5},s_{\tau}), \\ C^{1} \, & = XL+W_{5}\left[ {\frac{\partial L}{\partial A_{\xi}}+D_{\xi}D_{\xi}\Big( {\frac{\partial L}{\partial A_{\xi\xi\xi}}} \Big)} \right]\\ &\;\;\;+D_{\xi}(W_{5})\left[ {-D_{\xi}\Big( {\frac{\partial L}{\partial A_{\xi\xi\xi}}} \Big)} \right],\\ C^{2} \, & = YL\!+\!W_{5}\left[ {-D_{\eta}\Big( {\frac{\partial L}{\partial A_{\eta\eta}}} \Big)} \right]\!+\!D_{\eta}(W_{5})\Big( \!{\frac{\partial L}{\partial A_{\eta\eta}}}\! \Big). \end{aligned} \right. $

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$ \left\{ \begin{aligned} C^{\tau} =\,& {}s_{0}D_{\tau}^{\omega-1}\Big( {-A-\frac{\xi}{a_{1}}A_{\xi}\!-\!\frac{\eta} {a_{3}}A_{\eta}\!-\!\frac{\zeta}{a_{3}}A_{\zeta}-\frac{\tau}{\omega}A_{\tau}} \Big)\\ &+J\!\left[ {\Big( {\!-\!A\!-\!\frac{\xi}{a_{1}}A_{\xi}\!-\!\frac{\eta}{a_{3}}A_{\eta}\!-\!\frac{\zeta}{a_{3}} A_{\zeta}\!-\!\frac{\tau}{\omega}A_{\tau}} \Big),s_{\tau}} \right],\\ C^{1} =\,& XL+ \Big( {-A-\frac{\xi}{a_{1}}A_{\xi}-\frac{\eta}{a_{3}} A_{\eta}-\frac{\zeta}{a_{3}}A_{\zeta} -\frac{\tau}{\omega}A_{\tau}} \Big)\\ & \times\left[ {\frac{\partial L}{\partial A_{\xi}}+D_{\xi}D_{\xi}\Big( {\frac{\partial L}{\partial A_{\xi\xi\xi}}} \Big)} \right]\\ &+D_{\xi}\Big( {-A-\frac{\xi}{a_{1}}A_{\xi}-\frac{\eta}{a_{3}}A_{\eta} -\frac{\zeta}{a_{3}}A_{\zeta}-\frac{\tau}{\omega}A_{\tau}} \Big)\\ & \times \left[ {-D_{\xi}\Big( {\frac{\partial L}{\partial A_{\xi\xi\xi}}} \Big)} \right],\\ C^{2} =\, & YL+ \Big( {-A-\frac{\xi}{a_{1}}A_{\xi}-\frac{\eta}{a_{3}}A_{\eta}- \frac{\zeta}{a_{3}}A_{\zeta}-\frac{\tau}{\omega}A_{\tau}} \Big) \\ & \times\left[ {-D_{\eta}\Big( {\frac{\partial L}{\partial A_{\eta\eta}}} \Big)} \right]\\ &+D_{\eta}\Big( {-A-\frac{\xi}{a_{1}}A_{\xi}-\frac{\eta}{a_{3}}A_{\eta} -\frac{\zeta}{a_{3}}A_{\zeta}-\frac{\tau}{\omega}A_{\tau}} \Big)\\ & \times\Big( {\frac{\partial L}{\partial A_{\eta\eta}}} \Big). \end{aligned} \right. $

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$\begin{split} D_{x}^{n}(a,b)\, &\equiv \left( {\frac{\partial}{\partial x}-\frac{\partial}{\partial y}} \right)^{n}a(x)b(y)\mid_{y = x} \\ &= \frac{\partial^{n}}{\partial y^{n}}a(x+y)b(x-y)\mid_{y = 0},\end{split}$

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$\begin{split}& D_{t}^{m}D_{x}^{n}(a,b)\\ \equiv &\frac{\partial^{m}}{\partial s^{m}}\frac{\partial^{n}}{\partial y^{n}} a(t\!+\!s,x\!+\!y)b(t\!-\!s,x\!-\!y)\mid_{s = 0,y = 0}. \end{split}$

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$ \left\{\begin{aligned} & D_{x}^{2}(f\cdot f) = 2f_{xx}f-2(f_{x})^{2},\\ & D_{x}D_{t}(f\cdot f) = 2f_{xt}f-2f_{x}f_{t},\\ & D_{x}^{4}(f\cdot f) = 2f_{xxxx}f-8f_{xxx}f_{x}+6(f_{xx})^{2}. \end{aligned}\right. $

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$ \left.\begin{aligned}& T \!=\! \frac{p_{1}\tau^{\omega}}{\varGamma(1+\omega)},~ X \!=\! \frac{p_{2}\xi^{\alpha}}{\varGamma(1+\alpha)},~ Y \!=\! \frac{p_{3}\eta^{\beta}}{\varGamma(1+\beta)}, \end{aligned}\right. $

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$ \left.\begin{aligned} & \frac{\partial^{\omega} A}{\partial \tau^{\omega}} = p_{1}\frac{\partial A}{\partial T},~ \frac{\partial^{\alpha} A}{\partial \xi^{\alpha}} = p_{2}\frac{\partial A}{\partial X},~ \frac{\partial^{\beta} A}{\partial \eta^{\beta}} = p_{3}\frac{\partial A}{\partial Y}, \end{aligned}\right. $

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$ \frac{\partial}{\partial \xi}\left( {\frac{\partial A}{\partial T}+b_{1}A\frac{\partial A}{\partial X}+b_{2}\frac{\partial ^{3}A}{\partial ^{3}X}} \right)+b_{3}\frac{\partial ^{2}A}{\partial ^{2}Y} = 0. $

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$ A = R(\ln f)_{XX}, $

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$ R = {12b_{2}}/{b_{1}}, $

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$ (D_{X}D_{T}+b_{2}{D_{X}^{4}}+b_{3}D_{Y}^{2})(f\cdot f) = 0, $

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$ \begin{split}&(D_{X}D_{T}+b_{2}{D_{X}^{4}}+b_{3}D_{Y}^{2})(f\cdot f)\\ =\, & 2f_{XT}f-2f_{X}f_{T}+b_{2}(2f_{XXXX}f-8f_{XXX}f_{X}\\ & +6f_{XX}^{2})+b_{3}(2f_{YY}f-2f_{Y}^{2}),\\[-15pt] \end{split} $

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$\begin{split} & f = m^{2}+n^{2}+a_{9}, m = a_{1}X+a_{2}Y+a_{3}T+a_{4}, \\ & n = a_{5}X+a_{6}Y+a_{7}T+a_{8}. \\[-15pt]\end{split}$

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$ \left\{\begin{aligned} & a_{3} = -\frac{b_{3}\left(a_{1}a_{2}^{2}-a_{1}a_{6}^{2}+2a_{2}a_{5}a_{6}\right)}{a_{1}^{2}+a_{5}^{2}},\\ & a_{7} = -\frac{b_{3}\left(a_{5}a_{6}^{2}-a_{5}a_{2}^{2}+2a_{1}a_{2}a_{6}\right)}{a_{1}^{2}+a_{5}^{2}},\\ & a_{9} = -\frac{3b_{2}\left(a_{1}^{2}+a_{5}^{2}\right)^{3}}{b_{3}\left(a_{1}a_{6}-a_{2}a_{5}\right)^{2}}, \end{aligned} \right. $

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$ a_{1}a_{5}\neq0,~~ a_{1}a_{6}-a_{2}a_{5}\neq0. $

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$ \begin{split} A =\, &\frac{12b_{2}}{b_{1}}\left\{ {\frac{2a_{1}^{2}+2a_{5}^{2}}{(a_{1}X+a_{2}Y+a_{3}T+a_{4})^{2}+(a_{5}X+a_{6}Y+a_{7}T+a_{8})^{2}+a_{9}}}\right.\\ &\left. { -\frac{[2a_{1}(a_{1}X+a_{2}Y+a_{3}T+a_{4})+2a_{5}(a_{5}X+a_{6}Y+a_{7}T+a_{8})]^{2}}{[(a_{1}X+a_{2}Y+a_{3}T+a_{4})^{2}+(a_{5}X+a_{6}Y+a_{7}T+a_{8})^{2}+a_{9}]^{2}}} \right\}, \end{split} $

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$ \left.\begin{aligned} &T \!=\! \frac{p_{1}\tau^{\omega}}{\varGamma(1\!+\!\omega)},~X \!=\! \frac{p_{2}\xi^{\alpha}}{\varGamma(1\!+\!\alpha)},~ Y \!=\! \frac{p_{3}\eta^{\beta}}{\varGamma(1\!+\!\beta)}. \end{aligned}\right. $

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Jun-Chao Sun, Zong-Guo Zhang, Huan-He Dong, Hong-Wei Yang. Fractional order model and Lump solution in dusty plasma[J]. Acta Physica Sinica, 2019, 68(21): 210201-1

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