⇦Chinese Physics Letters, 2016, Vol. 33, No. 11, Article code 115201 On Time-Fractional Cylindrical Nonlinear Equation * H. G. Abdelwahed1**, E. K. ElShewy2, A. A. Mahmoud2 Affiliations 1College of Science and Humanitarian Studies, Physics Department, Prince Sattam Bin Abdul Aziz University, Kingdom of Saudi Arabia 2Theoretical Physics Research Group, Physics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Received 16 June 2016 *Supported by the Deanship of Scientific Research at Prince Sattam Bin Abdulaziz University under Grant No 2016/01/6239.
**Corresponding author. Email: hgomaa_eg@mans.edu.eg; hgomaa_eg@yahoo.com Citation Text: Abdelwahed H G, ElShewy E K and Mahmoud A A 2016 Chin. Phys. Lett. 33 115201 Abstract Properties of cylindrical electron acoustic solitons are studied in vortex plasmas. The modified cylindrical Korteweg–de Vries (KdV) equation is acquired and converted into the time fractional cylindrical modified KdV equation by Agrawal's analysis. Via the Adomian decomposition method, a cylindrical soliton solution to the equation is obtained. The cylindrical time fractional effect on the wave properties is investigated. Further, the increase of the fractional order of time $\alpha$ and hot to trapped electrons temperature $\beta$ are minimized in both solitary width and amplitude. These influences on the structures of the soliton may be an alternative to the use of higher order perturbation analysis. DOI:10.1088/0256-307X/33/11/115201 PACS:52.30.-q, 05.45.Df, 52.35.Fp, 52.27.Lw © 2016 Chinese Physics Society Article Text Space observations confirmed the electron acoustic (EA) propagation in the magnetosphere.[1-7] According to fundamentals of kinetic theory, the BGK modes investigated the electron trapping and phase-space holes.[8-10] Lu et al.[9] performed electrostatic particle simulations for generation of electrostatic solitary mechanism in the auroral region of the Earth. They recorded that, for large EA amplitude, part of the beam and hot electrons were trapped. Moreover, Wu et al. discussed the effects of perpendicular thermal velocities and electron hole evolutions on transverse instability by simulation in weak magnetized plasmas.[11] They obtained that the transverse instability resorted to stability with perpendicular thermal velocity. El-Taibany et al. derived the modified form fifth-order nonlinear equation to explicate the soliton noises spotted by the spacecraft in Earth's magnetosphere.[12] More specifically, a comparative investigation on excited electrostatic waves in electron beam-plasmas have been discussed.[13] It was reported that the EA wave is one of the unstable modes excited in such a plasma. Sabry et al. examined the nonextensive effectiveness on spherical and cylindrical electron acoustic wave packets.[14] They found that the prolonged spherical envelope soliton is affected by the nonextensive parameter rather than the cylindrical one. The investigation of wave propagation in nonplanar geometry is a realistic modality in laboratories and space plasma. Accordingly, many investigations debated plasma properties in a nonplanar geometry.[15-17] In the last few years, research works have been investigating nonplanar electron-acoustic solitons.[18-21] Pakzad introduced nonextensive hot electrons to discuss cylindrical electron acoustic solitons using numerical simulations.[19] It was noted that cylindrical solitary amplitude is less than the spherical ones. Shuchy et al. derived the Gardner equation for critical cylindrical and spherical electron-acoustic solitons in nonthermal plasmas.[20] Spherical properties of electron-acoustic solitons in superthermal hot electrons have been reported.[21] In physics interactions in plasmas, hot electrons having different high-energy tails caused by trapping in a wave potential will not follow the equilibrium Boltzmann velocity distribution.[22-25] In other words, nonlinear fractional equations and its applications received one of the master attentions in fluid mechanics flow, viscoelasticity, optics and physics of space plasmas.[26-29] A lot of research has discussed the studies on plasma physics by using the fractional nonlinear evolution equations.[29-33] Guo et al. reported the wave progress in ion pair plasma via the Gardner equation with a time fraction term. They used the method of variational iteration to investigate the nonthermal electrons effect on the produced wave.[34] Later, Demiray et al. for the first time studied the cylindrical electron-acoustic mode using the Shamel vortex case via analytical and spectral scheme.[35] In this Letter, the cylindrical electron-acoustic model with vortex hot electrons is studied. The modified cylindrical Korteweg–de Vries (mcKdV) equation is derived and Agrawal's analysis[36] is applied to obtain the time-fractional mcKdV (TFmcKdV) equation solved by means of the Adomian-decomposition method.[37,38] Our fluid model has three unmagnetized homogenous collisionless plasmas with a pressure term having ions in a stationary state, two electrons, one is a mobile cold electron fluid and the other is trapped vortex-like distributed hot electrons. The system and all normalization of physical quantities in small but finite amplitude are given in[35] $$\begin{align} &\frac{\partial}{\partial t}n_{\rm c} (r,t)+\frac{\partial}{\partial r}[{n_{\rm c} (r,t)u_{\rm c} (r,t)}]\\ &+\frac{1}{r}({n_{\rm c} (r,t)^u_{\rm c} (r,t)})=0,~~ \tag {1a}\\ &\frac{\partial}{\partial t}u_{\rm c} (r,t)+u_{\rm c} (r,t)\frac{\partial}{\partial r}u_{\rm c} (r,t)-\delta \frac{\partial}{\partial r}\phi (r,t)=0,~~ \tag {1b}\\ &\frac{\partial ^2}{\partial r^2}\phi (r,t)+\frac{1}{r}\frac{\partial}{\partial r}\phi (r,t)\\ =\,&\frac{1}{\delta}n_{\rm c} (r,t)+n_{\rm h} (r,t)-(1+\frac{1}{\delta}),~~ \tag {1c} \end{align} $$ where $n_{\rm c,h}(r, t)$ is the density of cold (trapped hot) electron, $\phi (r, t)$ is the electric potential, $u_{\rm c} (r,t)$ is the cold electron fluid velocity, and $\delta =n_{\rm h0} /n_{\rm c0}$ is the electron density ratio, and $\theta =T_{\rm h}/T_{\rm c} \gg 1$ denotes the ratio of the electron temperature ratio. The normalizations are defined as $$\begin{align} &\phi (r, t)\to k_{\rm B} T_{\rm h}/e,\\ &n_{\rm c,h} \to n_{\rm c0}~~{\rm and}~~n_{\rm h0},\\ &u_{\rm c} (r,t)\to \sqrt {{k_{\rm B} T_{\rm h}}/{(m\delta)}},\\ &t \to \omega _{\pi}^{-1}, (\omega _{\pi} =\sqrt {{4\pi n_{\rm c0} e^2}/m}),\\ &r\to \lambda _{\rm D} =\sqrt {{k_{\rm B} T_{\rm h}}/(4\pi n_{\rm h0} e^2)}, \end{align} $$ where $e$ is the electron charge, $m$ is the electron mass, $\lambda _{\rm D}$ is the Debye length of hot electron, $\omega _{\pi}^{-1}$ is the inverse of plasma frequency, and $k_{\rm B}$ is the Boltzmann constant. The hot density of electrons $n_{\rm h} (r,t)$ can be given by a trapping-vortex-like distribution[17-19] $$\begin{alignat}{1} n_{\rm h} (r,t)=\,&1+\phi (r,t)-\frac{4}{3}b\phi ^{3/2}(r,t)\\ &+\frac{1}{2}\phi ^2(r,t)+\ldots,~~ \tag {2a}\\ b=\,&(1-\beta)/\pi ^{1/2},~~ \beta ={T_{\rm h}}/{T_{\rm ht}},~~ \tag {2b} \end{alignat} $$ where $b$ is a parameter including the temperature of resonant (both free with temperature $T_{\rm h}$ and trapped with temperature $T_{\rm ht}$) hot electrons. The third term $-\frac{4}{3}b\phi ^{3/2}$ denotes the contribution of electrons resonant to electron density,[24] $$ \tau =\varepsilon ^{3/2}t, ~~\xi =\varepsilon ^{1/2}(r-\lambda t),~~ \tag {3} $$ where $\lambda$ and $\varepsilon$ are the normalized phase velocity and a smallness perturbation amplitude. We expand all physical quantities in Eq. (1) in $\varepsilon $, $$\begin{align} n_{\rm c} (\xi,\tau)=\,&1+\varepsilon ^2n_1 (\xi,\tau)+\varepsilon ^3n_2 (\xi,\tau)\\ &+\varepsilon ^4n_3 (\xi,\tau)+\ldots,~~ \tag {4a}\\ u_{\rm c} (\xi,\tau)=\,&\varepsilon ^2u_1 (\xi,\tau)+\varepsilon ^3u_2 (\xi,\tau)\\ &+\varepsilon ^4u_3 (\xi,\tau)+\ldots,~~ \tag {4b}\\ \phi (\xi,\tau)=\,&\varepsilon ^2\phi _1 (\xi,\tau)+\varepsilon ^3\phi _2 (\xi,\tau)\\ &+\varepsilon ^4\phi _3 (\xi,\tau)+\ldots.~~ \tag {4c} \end{align} $$ Substituting Eqs (2)-(4) into Eq. (1) and eliminating the second order perturbation quantities $n_2 (\xi,\tau)$, $u_2 (\xi,\tau)$ and $\phi _2 (\xi,\tau)$, we can obtain the mcKdV equation as follows:[33] $$\begin{align} &\frac{\partial \phi _1 (\xi,\tau)}{\partial \tau}+\frac{\phi _1 (\xi,\tau)}{2\tau} +\frac{({1-\beta})}{\sqrt \pi}\phi _1 ^{\frac{1}{2}}({\xi,\tau})\frac{\partial \phi _1 (\xi,\tau)}{\partial \xi}\\ &+\frac{1}{2}\frac{\partial ^3\phi _1 (\xi,\tau)}{\partial \xi ^3}=0.~~ \tag {5} \end{align} $$ For simplicity $\phi _1 ({\xi,\tau})={\it \Phi} ({\xi,\tau})$, the mcKdV equation in (1+1)-dimension reads $$\begin{align} &\frac{\partial {\it \Phi} ({\xi,\tau})}{\partial \tau}+\frac{{\it \Phi} ({\xi,\tau})} {2\tau}+\frac{({1-\beta})}{\sqrt \pi}{\it \Phi} ^{\frac{1}{2}}({\xi,\tau})\frac{\partial {\it \Phi} ({\xi,\tau})}{\partial \xi}\\ &+\frac{1}{2}\frac{\partial ^3{\it \Phi} ({\xi,\tau})}{\partial \xi ^3}=0.~~ \tag {6} \end{align} $$ The TFmcKdV equation can be subedited in the following way. Using a potential function ${\it \Psi}(\xi, \tau)$ where ${\it \Phi} (\xi,\tau)={\it \Psi}_\xi (\xi, \tau)$ denotes potential equation (6) as $$\begin{align} &\frac{\partial ^2{\it \Psi}({\xi,\tau})}{\partial \xi \partial \tau}+\frac{{\it \Phi} ({\xi,\tau})}{2\tau}+\frac{2}{3}\frac{({1-\beta})}{\sqrt \pi}\frac{\partial}{\partial \xi} \Big({\frac{\partial {\it \Psi}({\xi,\tau})}{\partial \xi}}\Big)^{\frac{3}{2}}\\ &+\frac{1}{2}\frac{\partial ^4{\it \Psi}({\xi,\tau})}{\partial \xi ^4}=0.~~ \tag {7} \end{align} $$ Using the semi-inverse method, the functional of potential Eq. (7) reads $$\begin{align} J({\it \Psi})=\,&\mathop \smallint \nolimits_R d\tau {\it \Psi}(\xi, \tau)\Big[c_1 \frac{\partial ^2{\it \Psi} ({\xi,\tau})}{\partial \xi \partial \tau}+c_2 \frac{{\it \Phi} ({\xi,\tau})}{2\tau}\\ &+c_3 \frac{2}{3}\frac{({1-\beta})}{\sqrt \pi}\frac{\partial}{\partial \xi}\Big({\frac{\partial {\it \Psi} ({\xi,\tau})}{\partial \xi}}\Big)^{\frac{3}{2}}\\ &+c_4 \frac{1}{2}\frac{\partial ^4{\it \Psi}({\xi,\tau})}{\partial \xi ^4}\Big],~~ \tag {8} \end{align} $$ where $c_1$, $c_2$, $c_3$ and $c_4$ are constants to be determined. Integrating Eq. (8) by parts where, and taking the variation with respect to ${\it \Psi}(\xi, \tau)$ after integration, ${\it \Psi}_\xi |_R ={\it \Psi}_\tau |_T =0$, assuming that ${\it \Phi} (\xi, \tau)$ is a fixed, one can obtain $$ c_1 =\frac{1}{2},~c_2 =1,~c_3 =\frac{2}{5},~c_4 =\frac{1}{2}.~~ \tag {9} $$ Lagrangian of Eq. (7) reads $$\begin{alignat}{1} \!\!\!\!\!\!\!\!&L({{\it \Psi},{\it \Psi}_\tau,{\it \Psi}_\xi,{\it \Psi}_{\xi \xi}})\\ \!\!\!\!\!\!\!\!=\,&-\frac{1}{2}{\it \Psi} _\tau ({\xi,\tau}){\it \Psi}_\xi ({\xi,\tau})+\frac{{\it \Phi} ({\xi,\tau})} {2\tau}{\it \Psi}({\xi,\tau})\\ \!\!\!\!\!\!\!\!&-\frac{4}{15}\frac{({1-\beta})}{\sqrt \pi}\Big({\frac{\partial {\it \Psi}({\xi,\tau})}{\partial \xi}}\Big)^{\frac{5}{2}} +\frac{1}{4}\Big({\frac{\partial ^2{\it \Psi}({\xi,\tau})}{\partial \xi ^2}}\Big)^2,~~ \tag {10} \end{alignat} $$ where ${\it \Psi}_\tau ({\xi,\tau})=\frac{\partial {\it \Psi}({\xi,\tau})}{\partial \tau}$, ${\it \Psi}_\xi ({\xi,\tau})=\frac{\partial {\it \Psi}({\xi,\tau})}{\partial \xi}$, and ${\it \Psi}_{\xi \xi} ({\xi,\tau})=\frac{\partial ^2{\it \Psi}({\xi,\tau})}{\partial \xi ^2}$. Similarly, Lagrangian in the time-fractional range can be given as $$\begin{align} &F({{\it \Psi},{}_0D_\tau ^\alpha {\it \Psi},{\it \Psi}_\xi,{\it \Psi}_{\xi \xi}})\\ =\,&-\frac{1}{2}{}_0D_\tau ^\alpha {\it \Psi}({\xi,\tau}){\it \Psi}_\xi ({\xi,\tau})+\frac{{\it \Phi} ({\xi,\tau})} {2\tau ^\alpha}{\it \Psi}({\xi,\tau})\\ &-\frac{4}{15}\frac{({1-\beta})}{\sqrt \pi}\Big({\frac{\partial {\it \Psi}({\xi,\tau})}{\partial \xi}}\Big)^{\frac{5}{2}} +\frac{1}{4}\Big({\frac{\partial ^2{\it \Psi}({\xi,\tau})}{\partial \xi ^2}}\Big)^2,\\ &0 < \alpha \leq 1,~~ \tag {11} \end{align} $$ where $_0D_t^\alpha {\it \Psi}(\xi, \tau)$ is the left Riemann–Liouville fractional derivative definition[37] $$\begin{align} _a D_t^\alpha f(t)=\,&\frac{1}{{\it \Gamma} (k-\alpha)}\frac{d^k}{dt^k}[\mathop \smallint \nolimits_a^t f(\tau)],\\ &k-1\leq \alpha \leq k, ~t\in [a,b].~~ \tag {12} \end{align} $$ The time-fractional potential functional form is given as $$\begin{align} J({\it \Psi})=\mathop \smallint \nolimits_R d\;\tau F({\it \Psi},{}_0D_\tau ^\alpha {\it \Psi}, {\it \Psi}_\xi,{\it \Psi}_{\xi \xi}).~~ \tag {13} \end{align} $$ Following Agrawal's method,[22] taking variation of Eq. (12) to ${\it \Psi}(\xi, \tau)$ gives $$\begin{align} &\delta J({\it \Psi})=\mathop \smallint \nolimits_R d\tau \Big[\Big(\frac{\partial F}{\partial _0 D_\tau ^\alpha {\it \Psi}}\Big) \delta _0 D_\tau ^\alpha {\it \Psi}+\Big(\frac{\partial F}{\partial {\it \Psi}_\xi}\Big)\delta {\it \Psi}_\xi \\ &+\Big(\frac{\partial F}{\partial {\it \Psi}_{\xi \xi}}\Big)\delta {\it \Psi}_{\xi \xi} +\Big(\frac{\partial F} {\partial {\it \Psi}}\Big)\delta {\it \Psi}\Big],~~ \tag {14} \end{align} $$ $$\begin{align} &\int_a^b dtf(t){}_aD_t^\alpha g(t)=\int_a^b {dt\,g(t){}_tD_b^\alpha f(t)}, f(t),\\ &g(t)\in[a,b].~~ \tag {15} \end{align} $$ The right Riemann–Liouville fractional derivative ${}_tD_b^\alpha$ is defined by $$\begin{align} _t D_b^\alpha f(t)=\,&\frac{(-1)^k}{{\it \Gamma} (k-\alpha)}\frac{d^k}{dt^k}[\mathop \smallint \nolimits_t^b f(\tau)], \\ &k-1\leq \alpha \leq k,~ t\in [a,b].~~ \tag {16} \end{align} $$ By integrating $$\begin{align} \delta J({\it \Psi})=\,&\mathop \smallint \nolimits_R d\tau \Big[_\tau D_{T_0} ^\alpha \Big(\frac{\partial F} {\partial _0 D_\tau ^\alpha {\it \Psi}}\Big)-\frac{\partial}{\partial \xi}\Big(\frac{\partial F}{\partial {\it \Psi}_\xi}\Big)\\ &+\frac{\partial ^2}{\partial \xi ^2}\Big(\frac{\partial F}{\partial {\it \Psi}_{\xi \xi}}\Big) +\Big(\frac{\partial F}{\partial {\it \Psi}}\Big)\Big]\delta {\it \Psi},~~ \tag {17} \end{align} $$ the optimizing functional $J({\it \Psi})$, i.e., $\delta J({\it \Psi})=0$, gives the Euler–Lagrange equation for the time-fractional of the potential equation in the form $$\begin{align} &_\tau D_{T_0}^\alpha \Big({\frac{\partial F}{\partial _0 D_\tau ^\alpha {\it \Psi}}}\Big) -\frac{\partial}{\partial \xi}\Big({\frac{\partial F}{\partial {\it \Psi}_\xi}}\Big)\\ &+\frac{\partial ^2}{\partial \xi ^2}\Big({\frac{\partial F}{\partial {\it \Psi}_{\xi \xi}}}\Big) +\Big({\frac{\partial F}{\partial {\it \Psi}}}\Big)=0.~~ \tag {18} \end{align} $$ Using Eq. (11), we obtain $$\begin{alignat}{1} \!\!\!\!\!\!\!\!&-\frac{1}{2}[{_\tau D_{T_0}^\alpha {\it \Psi}_\xi ({\xi,\tau})}]+\frac{1}{2}[ {_\tau D_{T_0}^\alpha {\it \Psi}_\xi ({\xi,\tau})}]+\frac{{\it \Phi} ({\xi,\tau})}{2\tau ^\alpha}\\ \!\!\!\!\!\!\!\!&+\frac{2}{3}\frac{({1-\beta})}{\sqrt \pi}\frac{\partial}{\partial \xi}\Big({\frac{\partial {\it \Psi} ({\xi,\tau})}{\partial \xi}}\Big)^{\frac{3}{2}}+\frac{1}{2}\frac{\partial ^4{\it \Psi}({\xi,\tau})}{\partial \xi ^4}=0.~~ \tag {19} \end{alignat} $$ Substituting for the potential function ${\it \Psi}_\xi (\xi,\tau)={\it \Phi} (\xi,\tau)$, the TFmcKdV equation is $$\begin{alignat}{1} \!\!\!\!\!\!\!\!&-\frac{1}{2}[{_\tau D_{T_0}^\alpha {\it \Psi}_\xi ({\xi,\tau})}]+\frac{1}{2}[ {_\tau D_{T_0}^\alpha {\it \Psi}_\xi ({\xi,\tau})}]+\frac{{\it \Phi} ({\xi,\tau})}{2\tau ^\alpha}\\ \!\!\!\!\!\!\!\!&+\frac{({1-\beta})}{\sqrt \pi}{\it \Phi} ^{\frac{1}{2}}({\xi,\tau})\frac{\partial {\it \Phi} ({\xi,\tau})} {\partial \xi}+\frac{1}{2}\frac{\partial ^3{\it \Phi} ({\xi,\tau})}{\partial \xi ^3}=0,~~ \tag {20} \end{alignat} $$ where $_0 D_\tau ^\alpha {\it \Phi} (\xi, \tau)$ and $_\tau D_{T_0}^\alpha {\it \Phi} (\xi, \tau)$ are the left and right Rieman–Liouville fractional derivatives and defined by Eqs. (12) and (16). The TFmcKdV Eq. (20) can be rewritten as $$\begin{align} &{}_0^{R} D_\tau ^\alpha {\it \Phi} (\xi, \tau)+\frac{{\it \Phi} ({\xi,\tau})}{2\tau ^\alpha}+\frac{({1-\beta})} {\sqrt \pi}{\it \Phi} ^{\frac{1}{2}}({\xi,\tau})\frac{\partial {\it \Phi} ({\xi,\tau})}{\partial \xi}\\ &+\frac{1}{2}\frac{\partial ^3{\it \Phi} ({\xi,\tau})}{\partial \xi ^3}=0,~ 0\leq \alpha \leq 1,~t\in [{0,T_0}].~~ \tag {21} \end{align} $$ The Riesz fractional derivative ${}_0^{R} D_\tau ^\alpha$ is given by[39] $$\begin{align} &{}_0^{R} D_\tau ^\alpha {\it \Psi}({\xi,\tau})=-\frac{1}{2}[{_\tau D_{T_0}^\alpha {\it \Psi}({\xi,\tau})}]+\frac{1} {2}[{_\tau D_{T_0}^\alpha {\it \Psi}({\xi,\tau})}]\\ =\,&\frac{1}{2}\frac{1}{{\it \Gamma} (k-\alpha)}\frac{d^k}{dt^k}[{\int_a^t {dt| {\tau -t} |^{k-\alpha -1}{\it \Psi}({\xi,t})}}],\\ &k-1\leq \alpha \leq k,~~t\in [a,b].~~ \tag {22} \end{align} $$ The Adomian decomposition technique decomposes the solution to a nonlinear equation ${\it \Phi} (\xi,\tau)$ into infinite series as $$\begin{align} {\it \Phi} (\xi,\tau)=\sum\limits_{n=0}^\infty {{\it \Phi} _n} (\xi,\tau),~~ \tag {23a} \end{align} $$ and the nonlinear term $F({\it \Phi} (\xi,\tau))$ is represented by the Adomian series $$\begin{align} F({\it \Phi} (\xi,\tau))=\sum\limits_{n=0}^\infty {A_n},~~ \tag {23b} \end{align} $$ where $A_n$ are the Adomian polynomials given by $$\begin{alignat}{1} A_0 =\,&F({\it \Phi} _0 (\xi,\tau)),~~ \tag {24a}\\ A_1 =\,&{\it \Phi} _1 \frac{\partial F({\it \Phi} _0)}{\partial {\it \Phi} _0},~~ \tag {24b}\\ A_2 =\,&{\it \Phi} _2 \frac{\partial F({\it \Phi} _0)}{\partial {\it \Phi} _0}+\frac{1}{2}{\it \Phi} _1^2 \frac{\partial ^2F({\it \Phi} _0)}{\partial {\it \Phi} _0^2},~~ \tag {24c} \end{alignat} $$ and other polynomials can be generated in the same manner. Applying the operator ${}_0^{R} D_\tau ^{-\alpha}$ on both sides of Eq. (16) yields $$\begin{align} {\it \Phi} (\xi,\tau)=\,&{\it \Phi} (\xi,0)-{}_0^{R} D_\tau ^{-\alpha} \Big[\frac{{\it \Phi} (\xi,\tau)}{2\tau}\\ &+A{\it \Phi}^{1/2}(\xi,\tau)\frac{\partial {\it \Phi} (\xi,\tau)}{\partial \xi}+B\frac{\partial ^3{\it \Phi} (\xi,\tau)}{\partial \xi ^3}\Big], \\ A=\,&\frac{({1-\beta})}{\sqrt \pi}, ~~B=\frac{1}{2},~~ \tag {25} \end{align} $$ where ${}_0^{R} D_\tau ^{-\alpha}$ is the Riemann–Liouville fractional integral, which is defined by $$\begin{alignat}{1} \!\!\!\!\!\!\!\!\!\!{}_0^{R} D_\tau ^{-\alpha} f(t)=\frac{1}{{\it \Gamma} (\alpha)}\int_0^t {\frac{f(\tau)}{(t-\tau)^{(1-\alpha)}}} d\tau,~0\leq \alpha < 1.~~ \tag {26} \end{alignat} $$ Substituting Eq. (18) into Eq. (20) leads to $$\begin{alignat}{1} \!\!\!\!\!\!\!\!\sum\limits_{n=0}^\infty {{\it \Phi} _n (\xi,\tau)} =\,&{\it \Phi} (\xi,0)-{}_0^{R} D_\tau ^{-\alpha} \Big[\frac{1}{2\tau}\sum\limits_{n=0}^\infty {{\it \Phi} _n (\xi,\tau)} \\ \!\!\!\!\!\!\!\!&+A\sum\limits_{n=0}^\infty {A_n} +B\frac{\partial ^3\sum\limits_{n=0}^\infty {{\it \Phi} _n (\xi,\tau)}} {\partial \xi ^3}\Big].~~ \tag {27} \end{alignat} $$ The components ${\it \Phi} _n (\xi,\tau)$ of the solution ${\it \Phi} (\xi,\tau)$ can be computed by applying the following recursive relation $$\begin{alignat}{1} {\it \Phi} _0 (\xi,\tau)=\,&{\it \Phi} (\xi,0),~~ \tag {28a}\\ {\it \Phi} _{k+1} (\xi,\tau)=\,&-{}_0^{R} D_\tau ^{-\alpha} \Big[\frac{1}{2\tau}{\it \Phi} _k (\xi,\tau)+AA_k\\ &+B\frac{\partial ^3{\it \Phi} _k (\xi,\tau)}{\partial \xi ^3}\Big],~k\geq 1.~~ \tag {28b} \end{alignat} $$ The initial condition will be taken as $$\begin{align} {\it \Phi} (\xi,0)=\phi _m \sec h^4(c\xi),~~ \tag {29} \end{align} $$ where $\phi _m$ and $c$ are constants, and their values depend on the physical parameters of the system $\phi _m =(\frac{15v}{8A})^2$ and $c=\sqrt {\frac{v}{16B}}$, with $\nu$ being the traveling velocity. The zero order of ADM solution can be taken from the initial value of state variable, which is taken as the solution to the regular equation at time equal to zero, in this case as $$\begin{align} {\it \Phi} _0 (\xi,\tau)={\it \Phi} ({\xi,0})=\phi _m {\rm sech}^4(c\xi).~~ \tag {30} \end{align} $$ Substituting with this zero order in the recursive relation Eq. (23b) leads to the first recursive, in the same manner, higher recursive order can be given by using the Maple package. Substituting from these different recursive orders into Eq. (23a) represents the solution of the TFMCKdV equation.
cpl-33-11-115201-fig1.png
Fig. 1. The electrostatic potential ${\it \Phi}(\xi,\tau)$ against the position $\xi$ at $\tau=5$, $\nu=0.1$, and $\beta=-0.5$ for different values of the fractional parameter $\alpha$.
cpl-33-11-115201-fig2.png
Fig. 2. The electrostatic potential ${\it \Phi}(\xi,\tau)$ against the position $\xi$ at $\tau=5$, $\nu=0.1$, and the fractional parameter $\alpha=0.3$ for different values of $\beta$.
cpl-33-11-115201-fig3.png
Fig. 3. The electrostatic potential ${\it \Phi}(\xi,\tau)$ against the position $\xi$ at $\beta=-0.5$, $\nu=0.15$, and the fractional parameter $\alpha=0.3$ for different values of $\tau$.
cpl-33-11-115201-fig4.png
Fig. 4. The electrostatic potential ${\it \Phi}(\xi,\tau)$ against the position $\xi$ at $\beta=-0.5$, $\tau=5$, and the fractional parameter $\alpha=0.3$ for different values of $\nu$.
Three components of cylindrical collisionless unmagnetized plasma having stationary state ions, fluid of cold electrons and vortex-like distributed hot electrons have been inspected. The mcKdV equation is gained via RPT. The TFmcKdV equation has been derived by employing a variational method represented by Agrawal.[27] The derived TFmcKdV equation is solved via ADM.[37,38] The fourth order of ADM approximation is used. Numerical calculations are used to study the importance of fractional order of time $\alpha$ and other parameters of plasma in solitary wave features. Clearly, the effect of $\alpha$ on the soliton profile is shown in Fig. 1. It is noted that $\alpha$ reduces soliton width and amplitude. This means that the increase of $\alpha$ increases the Riesz fractional derivative order that affects the higher recursive order solutions and reduces the soliton amplitude and width. Effects of hot-to-trapped-electron temperature $\beta$ and time $\tau$ on the solitary properties are shown in Figs. 2 and 3. It is illustrated that $\beta$ decreases both the amplitude and width while $\tau$ increases both width and amplitude. The reason for $\beta$ effect is that the parameter $\beta$ which refers to shore of resonant hot electrons to electron density affects directly the nonlinear coefficient that controls the solitary wave amplitude. Finally, the speed $v$ will increase the soliton amplitude and distort the soliton width. The above calculations show that the fractional time order in cylindrical coordinates can be used to modify the solitary shape and to verify the experimental data for the definite set of plasma parameters. Further, the fractional order of the mcKdV equation modulates the shape of the wave progress in plasmas without needing to study higher orders of the mcKdV equation.[12,40] References Generation of broadband electrostatic noise by electron acoustic solitonsGeneration of electron-acoustic waves in the magnetosphereNonlinear electron-acoustic solitary waves in a relativistic electron-beam plasma system with non-thermal electronsModulation of electron-acoustic waves in a plasma with kappa distributionElectron acoustic waves in a magnetized plasma with kappa distributed ionsA Schamel equation for ion acoustic waves in superthermal plasmasHigher-order corrections to broadband electrostatic shock noise in auroral zoneExact Nonlinear Plasma OscillationsGeneration mechanism of electrostatic solitary structures in the Earth's auroral regionPerpendicular electric field in two-dimensional electron phase-holes: A parameter studyEffects of Perpendicular Thermal Velocities on the Transverse Instability in Electron Phase Space HolesHigher-order nonlinearity of electron-acoustic solitary waves with vortex-like electron distribution and electron beamElectrostatic waves in an electron-beam plasma systemPropagation of cylindrical and spherical electron-acoustic solitary wave packets in unmagnetized plasmaThree-dimensional cylindrical Kadomtsev–Petviashvili equation in a dusty electronegative plasmaEffects of nonthermal distribution of electrons and polarity of net dust-charge number density on nonplanar dust-ion-acoustic solitary wavesThree dimensional dust-acoustic solitary waves in an electron depleted dusty plasma with two-superthermal ion-temperaturePlanar electron-acoustic solitary waves and double layers in a two-electron-temperature plasma with nonthermal ionsCylindrical and spherical electron acoustic solitary waves with nonextensive hot electronsCylindrical and spherical electron-acoustic Gardner solitons and double layers in a two-electron-temperature plasma with nonthermal ionsSpherical electron acoustic solitary waves in plasma with suprathermal electronsA modified Korteweg-de Vries equation for ion acoustic wavess due to resonant electronsElectron-acoustic solitary waves via vortex electron distributionHigher-order solution of an electron acoustic solitary waves via vortex electron distributionElectron-acoustic solitary waves and double layers in a relativistic electron-beam plasma systemFractional conservation laws in optimal control theoryA general finite element formulation for fractional variational problemsInvariant analysis and exact solutions of nonlinear time fractional Sharma–Tasso–Olver equation by Lie group analysisTime-fractional study of electron acoustic solitary waves in plasma of cold electron and two isothermal ionsIon-acoustic waves in unmagnetized collisionless weakly relativistic plasma of warm-ion and isothermal-electron using time-fractional KdV equationSpace—time fractional KdV—Burgers equation for dust acoustic shock waves in dusty plasma with non-thermal ionsIon-acoustic waves in plasma of warm ions and isothermal electrons using time-fractional KdV equationInvestigation of non-isothermal electron effects on the dust acoustic waves in four components dusty plasmaTime-fractional Gardner equation for ion-acoustic waves in negative-ion-beam plasma with negative ions and nonthermal nonextensive electronsA note on the cylindrical solitary waves in an electron-acoustic plasma with vortex electron distributionFractional variational calculus in terms of Riesz fractional derivativesDimensions of fractals in the largeEffect of higher-order nonlinearity to nonlinear electron-acoustic solitary waves in an unmagnetized collisionless plasma
[1] Dubouloz N, Pottelette R, Malingre M and Treumann R A 1991 Geophys. Res. Lett. 18 155
[2] Singh S V and Lakhina G S 2001 Planet. Space Sci. 49 107
[3] Elwakil S A, Zahran M A and El-Shewy E K 2007 Phys. Scr. 75 803
[4] Demiray H 2016 Phys. Plasmas 23 032109
[5] Devanandhan S, Singh S V, Lakhina G S and Bharuthram R 2012 Phys. Plasmas 19 082314
[6] Williams G, Verheest F, Hellberg M A, Anowar M G M and Kourakis I 2014 Phys. Plasmas 21 092103
[7] Abdelwahed H G 2015 Phys. Plasmas 22 092102
[8] Bernstein I B, Greene J M and Kruskal M D 1957 Phys. Rev. 108 546
[9] Lu Q M, Wang D Y and Wang S 2005 J. Geophys. Res. 110 A03223
[10] Lu Q M, Lembege B, Tao J B and Wang S 2008 J. Geophys. Res. 113 A11219
[11] Wu M Y, Wu H, Lu Q M and Xue B S 2010 Chin. Phys. Lett. 27 095201
[12] El-Taibany W F and Moslem W M 2005 Phys. Plasmas 12 032307
[13] Lu, Q M, Wang S and Xiankang D 2005 Phys. Plasmas 12 072903
[14] Sabry R and Omran M A 2013 Astrophys. Space Sci. 344 455
[15] Moslem W M, Abdelsalam U M, Sabry R, El-Shamy E F and El-Labany S K 2010 J. Plasma Phys. 76 453
[16] Mamun A A and Shukla P K 2009 Phys. Rev. E 80 037401
[17] Borhanian J and Shahmansouri M 2013 Phys. Plasmas 20 013707
[18] Mannan A and Mamun A A 2012 Astrophys. Space Sci. 340 109
[19] Pakzad H R 2011 Phys. Plasmas 18 082105
[20] Shuchy S T, Mannan A and Mamun A A 2012 JETP Lett. 95 282
[21] Borhanian J and Shahmansouri M 2012 Astrophys. Space Sci. 342 401
[22] Schamel H 1973 J. Plasma Phys. 9 377
[23] Mamun A A and Shukla P K 2002 J. Geophys. Res. 107 1135
[24] Elwakil S A, El-Shewy E K and Zahran, M A 2004 Chaos Solitons Fractals 22 13
[25] Sahu B and Roychoudhury R 2004 J. Plasma Phys. 11 1947
[26] Frederico G S F and Torres D F M 2008 Nonlinear Dyn. 53 215
[27] Agrawal O P 2008 J. Math. Anal. Appl. 337 1
[28] Wang G W and Xu T Z 2014 Nonlinear Dyn. 76 571
[29] El-Wakil S A, Abulwafa E M, El-Shewy E K and Mahmoud A A 2012 Plasma Phys. 78 641
[30] El-Wakil S A, Abulwafa E M, El-Shewy E K and Mahmoud A A 2012 Adv. Space Res. 49 1721
[31] El-Shewy E K, Mahmoud A A, Tawfik A M, Abulwafa E M and Elgarayhi A 2014 Chin. Phys. B 23 070505
[32] El-Wakil S A, Abulwafa E M, El-Shewy E K and Mahmoud A A 2011 Chin. Phys. B 20 040508
[33] Nazari-Golshan A, Nourazar S S, Parvin P and Ghafoori-Fard H 2014 Astrophys. Space Sci. 349 205
[34] Guo, L M and Zhang Z 2015 Phys. Plasmas 22 052306
[35] Demiray H and Bayındır C 2015 Phys. Plasmas 22 092105
[36] Agrawal O P 2007 J. Phys. A 40 6287
[37] Momian S and Odibat Z 2007 Chaos Solitons Fractals 31 5 1248
[38] Ray S S and Bera R K 2005 Appl. Math. Comput. 167 561
[39]Miller S K and Ross B 1993 An Introduction to the Fractional Calculus and Fractional Differential Equation (New York: John Wiley & Sons)
[40] El-Shewy E K 2005 Chaos Solitons Fractals 26 1073