尘埃等离子体中的分数阶模型及其Lump解

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4.TF-KP方程的守恒律

守恒律对于研究等离子体的物理现象具有重要的意义. 守恒定律是一个数学化的表述, 它表明在一个物理系统的演化过程中, 某一物理量的总量保持不变. 为了进一步研究分数阶模型的性质, 接下来对(2+1)维TF-KP方程的对称性和守恒定律进行了研究.
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4.1.李群分析

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4.1.李群分析

(2+1)维TSF-KP方程可以写作如下形式:

$ 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. $

将(32)式转化为分数阶偏微分方程形式如下:

$ \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} $

假设在以下形式的单参数李群点变换下, (33)式是不变的

$ \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. $

其中$ \varepsilon \ll1 $

是参数; X, Y, T, $ \psi $

是子函数; $ \psi_{\omega}^{\tau} $

, $ \psi_{\xi} $

, $ \psi_{\xi\xi\xi} $

, $ \psi_{\eta\eta} $

为子函数的延拓, 其定义如下:

$ \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. $

这里的$ D_{\tau}^{\omega} $

是总分数阶导数算子. $ D_{\tau} $

, $ D_{\xi} $

, $ D_{\eta} $

分别是$ \tau $

, $ \xi $

, $ \eta $

的总导数算子, 其定义如下:

$ \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. $

接下来应用广义莱布尼茨规则^[34]:

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

其中

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

可以得到

$ \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} $

根据复合函数的链式法则:

$ \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}},$

当$ f(t)=1 $

时, 可以得到

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

其中

$ \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}$

因此, (39)式可写成如下形式:

$ \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} $

通过李对称理论, 得到无穷小生成元M

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

在微元变换下, 系统(33)的不变性导致了以下不变性条件:

$ \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. $

根据(43)式和(45)式, 可以得到

$\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} $

利用系统(33)的二次延拓, 可以得到以下不变性准则:

$ \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.\\ $

将(43)式代入(47)式并设各单项系数为零, 可以得到如下方程组:

$ \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. $

通过求解(48)式, 可以得到

$ \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. $

因此, (32)式一系列点对称的李代数可以写成:

$ \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. $

4.2.守恒律

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4.2.守恒律

4.1节中得到了一系列点对称的李代数. 本节将基于李对称分析方法, 研究TF-KP方程的守恒律, (32)式的守恒定律满足如下方程:

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

其中$ C^{\tau} $

, $ C^{\xi} $

, $ C^{\eta} $

是守恒向量. (32)式的形式拉格朗日方程如下所示:

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

其中$ s(\xi, \eta, \tau) $

是一个新的因变量. 根据形式拉格朗日方程, 将作用积分定义为

$ \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,\\ $

欧拉-拉格朗日算子定义为

$ \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} $

其中$ (D_{\tau}^{\omega})^{*} $

是$ D_{\tau}^{\omega} $

的伴随运算子, 定义如下:

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

其中$ I_{p}^{n-\omega} $

是右分数积分算子, $ _{\tau}^{C}D_{p}^{\omega} $

是右Caputo分数微分算子. 因此, 可以得到(32)式的伴随方程为欧拉-拉格朗日方程

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

(56)式可以写成:

$ \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}$

李特征函数W如下所示:

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

由(50)式可得

$ \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. $

根据Riemann-Liouville分数阶导数, 定义保守向量的分量为

$ \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} $

这里$ J(\cdot) $

定义如下:

$ 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. $

$ C^{i}(i=1, 2, 3) $

定义如下:

$ \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} $

其中$ \rho^{1}=X,\; \rho^{2}=Y $

.
以$ W_{5} $

为例, 根据上述定义计算式(32)的保守向量, 得到

$ \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. $

上述方程(63)可以表示为

$ \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. $

由此得到了$ W_{5} $