1 Introduction
The signature of pair-ion (PI) plasma (PIP) has been observed by the Cassini spacecraft[1] in spaces, specially, in upper region of Titan's atmosphere, and is also generatedby Oohara et al.[2-4] in laboratory experiments.PI has also identified in cometary comae,[5-7]Earth's ionosphere,[8-12]plasma processing reactor,[13] neutral beam sources,[14] and successively has encouraged many researchers to understand the intrinsic contributionof the PI in recognizing the collective properties of the PIP by investigating ion-plasma waves,[3] intermediate frequency waves,[3] ion-acoustic (IA) solitary waves (IASWs),[5]IA shock waves (IASHWs),[6] IA Gardner solitons (IAGSs),[7] IA double layers (IADLs),[8] and IA rogue waves (IARWs),[10-12] etc.
Non-thermal particles are appeared in a complex system due to the presence of the external forcefield, and are governed by the Cairns distribution/non-thermal distribution,[15] and have also beenidentified by the Freja[16] and Viking[17] satellites in the magnetosphere and upper ionosphere of the auroral zone.Paul and Bandyopadhyay[18] investigated dust-ion-acoustic solitary waves in a four components plasma medium having non-thermal plasma species,and observed that the height of the positive super-solitons increases with increasing the value of non-thermality of the plasma species.Selim[19] studied IASWs in a three components PIP medium (PIPM) by considering non-thermal electrons, andfound that the height and thickness of the compressive solitons increase with increasing the value of electron's non-thermality.Singh and Lakhina[20] considered a three components plasma model having non-thermal electrons, and studied IA super-solitons,and found that the width of the super-solitons increases with an increase in the value of electrons non-thermality.
Electrostatic rogue waves (RWs) are considered to have appeared due tothe nonlinear property of the plasma medium as well as the modulational instability (MI) ofthe carrier waves, and are governed by the nonlinear Schrödinger equation (NLSE),and have also been identified in the plasma physics.[10-12, 21]El-Tantawy et al.[10] considered a two components plasma modelhaving inertial ions and non-thermal electrons, and studied the mechanism of the formation ofIARWs, and also found that behind the critical value of the non-thermal parameter, the nonlinearityand height of the IARWs increase with the non-thermal parameter. Abdelwahed et al.[11]considered a three components PIPM and observed the IARWs in presence of super-thermal electrons, and found that the height of the IARWsdecreases with the positive ion number density. Elwakil et al.[12] analyzed the MI of the IAWs in aPIPM having non-thermal electrons. Rahman et al.[21] studied dust-acoustic RWs in a non-thermal dusty plasmamedium, and found that the height of the RWs increases with the non-thermality of the plasma species.
Recently, Sabry et al.[5] studied IASWs in a three components PIPM having non-thermal electrons, andfound that the presence of negative ion enhances the amplitude but decreases the width of the solitary pulse.El-Labany et al.[8] reported IADLs in a three componentsPIPM in presence of non-thermal electrons. Jannat et al.[6] examined IASHWs in afour components PIPM having inertial PI as well as inertialess electrons and positrons.It is inevitable to investigate the MI of the IAWs and the formation of the IARWs in a fourcomponents PIPM by considering inertial double ions and inertialess iso-thermal positronsas well as non-thermal Cairns distributed electrons.
The outline of the paper is as follows: The governing equations describing our plasma model are presented in Sec. 2. The standard NLSE is derived in Sec. 3. The MI of the IAWs is examined in Sec. 4. The first-order and second-order RWs are discussed in Sec. 5. Finally, a brief conclusion is provided in Sec. 6.
2 Governing Equations
We consider a four components plasma model consisting of negative ions, positive ions, electrons, and positrons. Overall, the charge neutrality condition can be written as $Z_+ n_{+0}+n_{p0}=Z_-n_{-0}+n_{e0}$; where $n_{+0}$,$n_{p0}$, $n_{-0}$, and $n_{e0}$ are, respectively, the equilibrium number densities of positive ions, iso-thermal positrons,negative ions, and non-thermal Cairn's distributed electrons, and also $Z_+$ and $Z_-$ are, respectively, the charge state ofthe positive ions and negative ions. Now, the basic set of normalized equations can be written as
$\frac{\partial n_+}{\partial t}+\frac{\partial}{\partial x}(n_+u_+)=0\,,$
$\frac{\partial u_+}{\partial t}+u_+\frac{\partial u_+}{\partial x}=-\frac{\partial\phi}{\partial x}\,,$
$\frac{\partial n_-}{\partial t}+\frac{\partial}{\partial x}(n_-u_-)=0\,,$
$\frac{\partial u_-}{\partial t}+u_-\frac{\partial u_-}{\partial x}=\delta_1\frac{\partial\phi}{\partial x}\,,$
$\frac{\partial^2\phi}{\partial x^2}=\delta_2 n_e-\delta_3n_p+(1-\delta_2+\delta_3)n_-n_+\,,$
where $n_+$ and $n_-$ are the positive and negative ion number density normalizedby their equilibrium value $n_{+0}$ and $n_{-0}$, respectively; $u_-$ and $u_+$ are the negative and positive ion fluid speed normalized by wave speed $C_+=(Z_+k_BT_e/m_+)^{1/2}$(with $T_e$ being the temperature non-thermal electron, $m_+$ being the positive ion mass,and $k_B$ being the Boltzmann constant); $\phi$ is the electrostatic wave potentialnormalized by $k_BT_e/e$ (with $e$ being the magnitude of single electron charge); thetime and space variables are normalized by $\omega_{p+}^{-1}=(m_+/4\pi Z_+^2 e^2 n_{+0})^{1/2}$and $\lambda_{D+}=(k_BT_e/4\pi e^2 Z_+n_{+0})^{1/2}$, respectively. Other plasma parameters are defined as $\delta_1=Z_-m_+/Z_+m_-$,$\delta_2=n_{e0}/Z_+n_{+0}$, and $\delta_3=n_{p0}/Z_+n_{+0}$. Now, the expression for electron numberdensity, which is obeying non-thermal Cairn's distribution[15] is given by
$ n_e=(1-\beta\phi+\beta\phi^2)\exp(\phi)\,,$
where $\beta=4\alpha/(1+3\alpha)$, with $\alpha$ being the parameter determining the fast particles present in our plasma
model. The positron number density which is iso-thermally distributed is given by
$n_p=\exp(-\delta_4\phi)\,,$
where $\delta_4=T_e/T_p$ and $T_e>T_p$. Now, by substituting Eqs. (6) and (7) into Eq. (5), and expanding up to the third order of $\phi$, we get
$\frac{\partial^{2}\phi}{\partial x^{2}}+n_+-(1-\delta_{2}+\delta_{3})n_-=(\delta_{2}-\delta_3) +H_1\phi \\ \quad {-0.5cm}+\,H_2 \phi^2+ H_3\phi^3+ \cdots\,,$
where
$H_1=\delta_2(1-\beta)+\delta_3\delta_4\,,\quad H_2=[\delta_2-\delta_3\delta_4^2]/2\,, H_3=[\delta_2(1-\beta+6\beta)+\delta_3\delta_4^3]/6\,.$
We note that the term on the right hand side of Eq. (8) is the contribution of electron and positron species.
3 Derivation of the NLSE
$\xi={\epsilon}(x-v_g t)\,,$
$\tau={\epsilon}^2 t\,,$
$n_+=1+\sum_{m=1}^{\infty}\epsilon^{m}\sum_{l=-\infty}^{\infty} n_{+l}^{(m)}(\xi,\tau) \mbox{exp}[{\rm i} l(kx-\omega t)],$
$u_+=\sum_{m=1}^{\infty}\epsilon^{m}\sum_{l=-\infty}^{\infty} u_{+l}^{(m)}(\xi,\tau) \mbox{exp}[{\rm i} l(kx-\omega t)],$
$n_-=1+\sum_{m=1}^{\infty}\epsilon^{m}\sum_{l=-\infty}^{\infty} n_{-l}^{(m)}(\xi,\tau) \mbox{exp}[{\rm i} l(kx-\omega t)],$
$u_-=\sum_{m=1}^{\infty}\epsilon^{m}\sum_{l=-\infty}^{\infty} u_{-l}^{(m)}(\xi,\tau) \mbox{exp}[{\rm i} l(kx-\omega t)],$
$\phi=\sum_{m=1}^{\infty}\epsilon^{m}\sum_{l=-\infty}^{\infty} \phi_{l}^{(m)}(\xi,\tau) \mbox{exp}[{\rm i} l(kx-\omega t)],$
where $k$ ($\omega$) is real variables representing the carrier wave number (frequency). The derivative operators in above equations are treated as follows:
$\frac{\partial}{\partial t}\rightarrow\frac{\partial}{\partial t}-\epsilon v_g \frac{\partial}{\partial\xi} +\epsilon^2\frac{\partial}{\partial\tau}\,,$
$\frac{\partial}{\partial x}\rightarrow\frac{\partial}{\partial x}+\epsilon\frac{\partial}{\partial\xi}\,.$
Now, by substituting Eqs. (9)-(17) into Eqs. (1)-(4) and Eq. (8), and collecting the terms containing $\epsilon$, the first order ($m=1$ with $l=1$) reduced equations can be written as
$n_{+1}^{(1)}=\frac{k^2}{\omega^2}\phi_1^{(1)}\,,$
$u_{+1}^{(1)}=\frac{k}{\omega}\phi_1^{(1)}\,,$
$n_{-1}^{(1)}=-\frac{k^2\delta_1}{\omega^2}\phi_1^{(1)}\,,$
$u_{-1}^{(1)}=-\frac{k\delta_1}{\omega}\phi_1^{(1)}\,.$
This relations provide the dispersion relation of IAWs
$\omega^2=\frac{k^2[1+\delta_1(1-\delta_2+\delta_3)]}{k^2+H_1}\,.$
The second order ($m=2$ with $l=1$) equations are given by
$n_{+1}^{(1)}=\frac{k^2}{\omega^2}\phi_1^{(2)}+\frac{2{\rm i}k(v_gk-\omega)}{\omega^3}\frac{\partial\phi_1^{(1)}}{\partial\xi},$
$u_{+1}^{(2)}=\frac{k}{\omega}\phi_1^{(2)}+\frac{{\rm i}(v_gk-\omega)}{\omega^2}\frac{\partial\phi_1^{(1)}}{\partial\xi},$
$n_{-1}^{(2)}=-\frac{\delta_1k^2}{\omega^2}\phi_1^{(2)}-\frac{2{\rm i}k\delta_1(v_gk-\omega)}{\omega^3}\frac{\partial\phi_1^{(1)}}{\partial\xi},$
$u_{-1}^{(2)}=-\frac{\delta_1 k}{\omega}\phi_1^{(2)}-\frac{{\rm i}\delta_1(v_gk-\omega)}{\omega^2}\frac{\partial\phi_1^{(1)}}{\partial\xi},$
with the compatibility condition
$v_g=\frac{\omega[1+\delta_1(1-\delta_2+\delta_3)-\omega^2]}{k[1+\delta_1(1-\delta_2+\delta_3)]}.$
The coefficients of $\epsilon$ for $m=2$ and $l=2$ provide the second order harmonic amplitudes, which are found to be proportional to $|\phi_1^{(1)}|^2$
$n_{+2}^{(2)}=H_4|\phi_1^{(1)}|^2\,,$
$u_{+2}^{(2)}=H_5 |\phi_1^{(1)}|^2\,,$
$n_{-2}^{(2)}=H_6|\phi_1^{(1)}|^2\,,$
$u_{-2}^{(2)}=H_7 |\phi_1^{(1)}|^2\,,$
$\phi_{2}^{(2)}=H_8 |\phi_1^{(1)}|^2\,,$
where
$H_4=\frac{3k^4+2H_8\omega^2k^2}{2\omega^4}\,, H_5=\frac{k^3+2\omega^2kH_8}{2\omega^3}\,, H_6=\frac{3\delta_1^2k^4-2\delta_1H_8\omega^2k^2}{2\omega^4}\,, H_7=\frac{\delta_1^2k^3-2\omega^2k\delta_1H_8}{2\omega^3}\,, H_8=\frac{2H_2\omega^4+3\delta_1k^4(1-\delta_2+\delta_3)-3k^4}{2\omega^2k^2-2\omega^4(4k^2+H_1)+2\delta_1\omega^2k^2(1-\delta_2+\delta_3)}\,.$
Now, we consider the expression for ($m=3$ with $l=0$) and ($m=2$ with $l=0$), which leads zeroth harmonic modes. Thus, we obtain
$n_{+0}^{(2)}=H_{9}|\phi_1^{(1)}|^2\,,$
$u_{+0}^{(2)}=H_{10}|\phi_1^{(1)}|^2\,,$
$n_{-0}^{(2)}=H_{11}|\phi_1^{(1)}|^2\,,$
$u_{-0}^{(2)}=H_{12}|\phi_1^{(1)}|^2\,,$
$\phi_0^{(2)}=H_{13} |\phi_1^{(1)}|^2\,,$
where
$H_{9}=\frac{2v_gk^3+\omega k^2+\omega^3H_{13}}{v_g^2\omega^3}\,, H_{10}=\frac{k^2+\omega^2 H_{13}}{v_g\omega^2}\,, H_{11}=\frac{2v_g\delta_1^2k^3+\omega\delta_1^2k^2-\delta_1\omega^3H_{13}}{v_g^2\omega^3}\,, H_{12}=\frac{\delta_1^2k^2-\delta_1\omega^2H_{13}}{v_g\omega^2}\,, H_{13}=\frac{2H_2v_g^2\omega^3+2v_g\delta_1^2k^3(1-\delta_2+\delta_3)+F1}{\omega^3[1+\delta_1(1-\delta_2+\delta_3)-H_1v_g^2]}\,,$
where $F1=\omega k^2\delta_1^2(1-\delta_2+\delta_3)-2v_gk^3-\omega k^2$.Finally, the third harmonic modes ($m=3$) and ($l=1$) and with the help of Eqs. (18)-(40),give a set of equations, which can be reduced to the following NLSE:
$i \frac{\partial\Phi}{\partial\tau}+P\frac{\partial^2\Phi}{\partial\xi^2}+Q |\Phi|^2\Phi=0\,,$
where $\Phi=\phi_1^{(1)}$ for simplicity. In Eq. (38), $P$ is the dispersion coefficient, which can be written as
$P=\frac{3v_g(v_g k-\omega)}{2\omega k}\,,$
and also $Q$ is the nonlinear coefficient, which can be written as
$Q=\frac{\omega^3}{2k^2[1+\delta_1(1-\delta_2+\delta_2)]} [-\frac{2k^3(H_5+H_{10})}{\omega^3} +\,2H_2(H_8+H_{13})+3H_3-\frac{k^2(H_4+H_9)}{\omega^2} -\frac{\delta_1 k^2(1-\delta_2+\delta_2)(H_6+H_{11})}{\omega^2} -\frac{2\delta_1k^3(1-\delta_2+\delta_2)(H_7+H_{12})}{\omega^3}].$
It may be noted here that both $P$ and $Q$ are function of variousplasma parameters such as $\alpha$, $\delta_1$, $\delta_2$, $\delta_3$, $\delta_4$, and $k$. So,all the plasma parameters are used to maintainthe nonlinearity and the dispersion properties of the PIPM.
4 Modulational Instability
The stable and unstable regions of the IAWs are organized by the sign of the dispersion ($P$)and nonlinear ($Q$) coefficients of the standard NLSE (38).The stability of IAWs in four components PIPM is governed by the sign of $P$and $Q$.[22-29]When $P$ and $Q$ have same sign (i.e., $P/Q>0$), the evolution of the IAWs amplitude is modulationallyunstable in presence of the external perturbations. On the other hand, when $P$ and $Q$ haveopposite sign (i.e., $P/Q<0$), the IAWs are modulationally stable.The plot of $P/Q$ against $k$ yields stable and unstable regions for the IAWs.The point, at which transition of $P/Q$ curve intersects with $k$-axis, is known as thresholdor critical wave number $k$ ($=k_c$).[22-29]
We have numerically analyzed the behaviour of $P/Q$ with respect to $k$ for different valuesof $\delta_1$ and $\delta_3$ in Figs. 1 and 2, respectively. It canbe seen from Fig. 1 that (a) the modulationally stable region corresponding to thecriteria (i.e., $P/Q<0$) as well as modulationally unstable region corresponding to the criteria (i.e., $P/Q>0$)can be found from this figure; (b) the $k_c$ decreases with an increase in the value of $\delta_1$; (c) the mass of thenegative ions causes to increase the value of $k_c$ while the mass of the positive ions causes to decrease the value of $k_c$for a constant value of negative and positive ions charge state (via $\delta_1$).Figure 2 demonstrates the presence of the Maxwellian inertialess positronsreduces the $k_c$ as well as allows to generate IARWs associated with IAWs within thecondition $P/Q>0$ in a PIPM for a small value of $k$ when the number density as well asthe charge state of the positive ion remain constant (via $\delta_3$).On the other hand, modulationally stable region (i.e., $P/Q<0$) as well as$k_c$ enhances with increasing the value of number density and charge state of the positive ionfor a constant value of positron population.
Fig. 1 Plot of $P/Q$ vs. $k$ for different values of $\delta_1$ when other parameters are $\alpha=0.7$, $\delta_2=0.4$, $\delta_3=0.4$, and $\delta_4=2.0$. |
Fig. 2 Plot of $P/Q$ vs. $k$ for different values of $\delta_3$ when other parameters are $\alpha=0.7$, $\delta_1=0.8$, $\delta_2=0.4$, and $\delta_4=2.0$. |
5 Rogue Waves
The NLSE (38) has a variety of solutions,among them there is a hierarchy of rational solution that arelocalized in both the $\xi$ and $\tau$ variables. Each solution of thehierarchy represents a unique event in space and time, as itincreases its amplitude quickly along each variable, reachingits maximum value and finally decays, just as quickly as itappeared.[30] Thus, these waves were nicknamed “waves thatappear from nowhere and disappear without a trace”.[31]The first-order rational solution of NLSE (38) is given as[25-34]
$\Phi_1 (\xi, \tau)=\sqrt{\frac{2P}{Q}}\Big[\frac{4+16 {\rm i}\tau P}{1+4 \xi^2 + 16\tau^2 P^2}-1] \mbox{exp} (2{\rm i}\tau P).$
$\Phi_2 (\xi, \tau)=\sqrt{\frac{P}{Q}}\Big[1+\frac{G_2(\xi,\tau)+{\rm i}M_2(\xi,\tau)}{D_2(\xi,\tau)}] \mbox{exp} ({\rm i}\tau P),$
where
$G_2(\xi,\tau)=\frac{3}{8}-6(P\xi\tau)^2-10(P\tau)^4-\frac{3\xi^2}{2}-9(P\tau)^2-\frac{\xi^4}{2}, M_2(\xi,\tau)=-P\tau\Big[\xi^4+4(P\xi\tau)^2+4(P\tau)^4-3\xi^2+2(P\tau)^2-\frac{15}{4}\Big], D_2(\xi,\tau)=\frac{\xi^6}{12}+\frac{\xi^4(P\tau)^2}{2}+\xi^2(P\tau)^4+\frac{2(P\tau)^6}{3}+\frac{\xi^4}{8}+\frac{9(P\tau)^4}{2}-\frac{3(P\xi\tau)^2}{2}+\frac{9\xi^2}{16}+\frac{33(P\tau)^2}{8}+\frac{3}{32}.$
The solutions (39) and (40) represent the profile of the first-order and second-order RWswithin the modulationally unstable region, which concentrate a significant amount of energy into a relatively small area. The properties of the first-order rational solution (by using Eq. (39)) of NLSE (38) in a PIPM can beobserved in Figs. 3-5 corresponding different values of $\alpha$, $\delta_2$,and $\delta_4$. Figure 3 shows the structure of the first-order IARWs according to the value ofelectrons non-thermality ($\alpha$). The increase in the value of $\alpha$ does not only cause toincrease the height of the first-order IARWs associated IAWs in the modulationally unstable region(i.e., $P/Q>0$) but also cause to increase the thickness of the first-order IARWs associated IAWsin the modulationally unstable region (i.e., $P/Q>0$).
Fig. 3 Plot of $|\Phi_1|$ vs. $\xi$ for different values of $\alpha$ when other parameters are $k=0.7$, $\tau=0$, $\delta_1=0.8$, $\delta_2=0.4$, $\delta_3=0.4$, and $\delta_4=2.0$. |
The effects of the ratio of the number density of electron to positive ion as well as the chargestate of the positive ion (via $\delta_2$) can be seen from Fig. 4,and it is obvious from this figure that (a) the increase in the value of $\delta_2$ causes todecrease the height of the first-order IARWs in the modulationally unstable region (i.e., $P/Q>0$)as well as causes to decrease the thickness of the first-order IARWs in the modulationally unstable region (i.e., $P/Q>0$);(b) physically, the presence of the excess number of electrons reduces the nonlinearity ofthe plasma medium as well as the height and thickness of the first-order IARWs while the presence of theexcess number of positive ions increases the nonlinearity of the PIPM as well as the height andthickness of the first-order IARWs when the charge state of the positive ion remains constant (via $\delta_2$).
Fig. 4 Plot of $|\Phi_1|$ vs. $\xi$ for different values of $\delta_2$ when other parameters are $k=0.7$, $\tau=0$, $\alpha=0.7$, $\delta_1=0.8$, $\delta_3=0.4$, and $\delta_4=2.0$. |
The nature of the first-order IARWs with the variation of $\delta_4$ can be observed from Fig. 5 which clearly indicates that (a) the height and thickness of the first-order IARWs associated withIAWs in the modulationally unstable region (i.e., $P/Q>0$) decreases with increasing the value of $\delta_4$; (b) the temperature of the positron in the modulationally unstable region of IAWsenhances the nonlinearity of the plasma medium as well as the height and thickness of the first-order IARWs while the temperature of the electron in the modulationally unstable region of IAWsreduces the nonlinearity of the plasma medium as well as the height and thickness of the first-order IARWs.
Fig. 5 Plot of $|\Phi_1|$ vs. $\xi$ for different values of $\delta_4$ when other parameters are $k=0.7$, $\tau=0$, $\alpha=0.7$, $\delta_1=0.8$, $\delta_2=0.4$, and $\delta_3=0.4$. |
The space and time evolution of the first-order and second-order rational solutions of the NLSE (38) can be observed from Figs. 6(a) and 6(b), respectively. Figure 7 indicates the first-order and second-order solution at $\tau=0$, and it is clear from this figure that (a) the second-order rational solution has double structures compared with the first-order rational solution;(b) the height of the second-order rational solution is always greater than the height of the first-order rational solution; (c) the potential profile of the second-order rational solution becomes more spiky (i.e., the taller height and narrower width) than the first-order rational solution; (d) the second (first) order rational solution has four (two) zeros symmetrically located on the $\xi$-axis; (e) the second (first) order rational solution has three (one) local maxima.
Fig. 6 (Color online) Profile of the first-order solution and (b) second-order rational solutions. |
Fig. 7 (Color online) Plot of the first-order (dashed green curve) and the second-order (solid blue curve) rational solutions of NLSE (38) at $\tau=0$. |
The existence of highly energetic rogue waves has already been confirmed experimentally[35-38]and theoretically.[39] The second-order rational solution was experimentally observed by Chabchoub et al.[36] in a “Water Wave Tank”,and the experimental result regarding the amplification is a nice agreement with the theoretical result. Bailung et al.[37] demonstrated an experiment in a multi-component plasma medium to observe RWs, and found a slowly amplitude modulated perturbation undergoes self modulation and gives rise to a high amplitude localized pulse.Rogue waves are also observed in fiber optics.[38]
6 Conclusion
The amplitude modulation of IAWs has been theoretically investigated in an unmagnetized four components PIPMconsisting of inertial positively and negatively charged ions as well as inertialess non-thermal electrons and iso-thermal positrons.A standard NLSE, which governs the MI of IAWs and the formation of electrostatic first-order and second-order RWs in PIPM, isderived by using the RPM. Finally, it is worth to mention that our results may be useful in understanding thenonlinear phenomena (viz., MI of IAWs and IARWs, etc.) in space PIPM (viz., cometary comae[5-7] andEarth's ionosphere[8-12])and laboratory PIPM (viz., plasma processing reactor[13] and neutral beam sources,[14] etc.).