Nonlinear pulsational mode dynamics in strongly correlated dust molecular clouds

Abstract

We theoretically study the nonlinear pulsational mode dynamics in strongly correlated (viscoelastic) self-gravitating complex charge-fluctuating dust molecular clouds (DMCs) on the astrophysical spatiotemporal scales. A nonlinear normal mode (local) analysis results in a unique pair of extended Kortweg de-Vries-Burgers (KdV-B) equations on the conjugational gravito-electrostatic potential fluctuations in a mixed form. The KdV-B system is numerically analysed in a wide-range parametric window relevant to realistic astronomical DMC circumstances. It is found that the fluctuation dynamics evolves as solitary-chain patterns. The electrostatic fluctuation amplitude increases with the referral frame velocity; whereas, the gravitational fluctuations are insensitive to this velocity. The fluctuation dynamics are found to be independent of variation in the equilibrium dust density, equilibrium dust charge, dust mass, and so forth. The validation and reliability checkup of our results is highlighted in fair corroboration with the literature. Our results could be useful in understanding the mechanism behind bounded structure formation via the non-local self gravitational collapse dynamics.

Keywords:

• Astroclouds
• gravito-electrostatics
• viscoelasticity
• dusty plasma
• soliton

PACS Nos:

• 52.27.Lw
• 52.27.Gr
• 52.35.Fp
• 52.35.Sb
• 52.65.Vv
• 04.40.-b

1. Introduction

The fact that intense locations of the interstellar dense molecular clouds, which are basically self-gravitating partially ionized plasmas, are active sites triggering the formation of stars, planets and other bounded structures in galaxies, is now widely acknowledged [1–4]. These relatively complex formation processes are prompted by a wide variety of collective waves, instabilities and oscillations [5–10]. Various studies [11–14] have brought to light that partially charged clouds display a comparatively new type of collective mode, christened as the pulsational mode.

The dusty plasma plays a pivotal role in astrophysical bodies and space environments like planetary ring systems, cometary tails, interstellar and circumstellar clouds and asteroid zones [15] apart from laboratory plasmas [16]. The ubiquitous nature of the dusty plasmas and its vital role in the space environment results in new interesting salient features effected by dust-charged grains [15,17–21]. Astronomical and satellite observations [17,22,23] have brought to the fore the presence of the massive dust grains in the interstellar dust molecular clouds, in the vicinity of comets and other astrophysical objects. The presence of massive grains in astrophysical plasmas results in the considerable importance of the gravitational effects. The self-gravitational and electrostatic repulsive effects balance each other [12,24,25] in plasmas with medium-sized charged grains.

Numerous studies on nonlinear waves in multispecies self-gravitating dusty plasmas have been carried out over the last few decades in the light of the significant role played by them in the formation processes of varied galactic structures [15,26,27]. As a consequence of the nonlinear and dispersive properties of the considered self-gravitating medium [27], Korteweg–de Vries (K-dV) type of equations and their derivative forms usually govern the evolution of such waves in a nonlinear regime. It is a well-known fact that there exists a wide spectrum of different nonlinear waves and collective eigen modes in star-forming dusty molecular clouds. The excitation of perturbations and waves in such interstellar fluid media saturates into various eigen mode structures like shock, solitons and vertices [9,12–14,28,29]. Karmakar et al. [30] developed a theoretical model to investigate the formation of nonlinear conjugational eigen structure in complex magnetized collisional astrofluid and showed that the fluid perturbation dynamics is administered by a unique bimodal pair forced Korteweg-de Vries (f-KdV) equations. Gohain and Karmakar [31] have come up with an analytical model to investigate the weakly nonlinear gravito-electrostatic waves in a field-free polytropic dust molecular cloud. Later, a classical formalism for the weakly nonlinear instability analysis of a gravitating rotating viscoelastic gaseous cloud in the presence of gyratory dark matter [32].

Of late, Dutta and Karmakar [33] have come up with an evolutionary model analysis with a view to studying the linear pulsational mode dynamics in viscoelastic complex self-gravitating clouds. Various stabilizing and destabilizing agencies have been identified and characterized systematically. However, it has completely ignored any kind of quassi-linear or nonlinear gravito-electrostatic mode-mode coupling and evolution thereof in such visco-elastic media. Subsequently, the nonlinear fluctuation dynamics in the case of self-gravitating nonthermal viscoelastic complex fluid was also investigated in a semi-analytical manner [34]. Later El-Taibany et al. [35] have carried out dust-acoustic solitary and periodic wave investigations in a magnetized self-gravitating dusty plasma. In the recent past, Yang et al.[36] have studied the behaviours of Jeans modes for gravitational systems composed of dark and baryonic matters.

Motivated by this property of the star-forming nonlinear molecular clouds, we herein propose a theoretical model formalism to investigate the nonlinear dynamics of collective pulsational mode excitation and propagation contributing to the triggered initiation of structure formalism in such complex viscoelastic fluids with full incorporation of dust charge fluctuations. Application of multiple scaling techniques in the molecular cloud results in a conjugational pair extended KdV-B equation of complex non-integrable mathematical construct. The KdV-B system is numerically integrated for the exact solutions evolving as diversified eigen – patterns of broadband K-dV family. The key factors, stabilizing and destabilizing the perturbed cloud dynamics, are investigated in detail followed by a realistic comparison with the real astronomical circumstances as already reported in the literature.

2. Basic governing equations

The model under consideration encompasses a four-component isothermal strongly coupled dusty plasma consisting of thermal electrons, ions and inertial dust grains with partial ionization. The comparatively lighter electrons and ions, with the non-consideration of inertial effects, are treated to be Boltzmannian in case of the low-frequency fluctuations on the gravitational scales of space and time; on the contrary, the heavier dust species under the impact of high inertia are taken as fluids. The dust grains are deemed to be viscoelastic fluids because of their low thermal velocities and as a consequence, the mean-free path of the gravitationally interacting grains is practically negligible. It is evident from observation that the massive grains in the cloud which interact gravitationally under the impact of long-range Newtonian coupling processes, add to the net mass of the cloud. The random interaction of the electron and ion thermal currents on the surface of the dust grains results in their charge fluctuation [12], which is taken into consideration. Moreover, the effects of collisional momentum transfer among all the constituent species are also taken care of.

The classical dynamics of the dust cloud constituents, i.e. thermal electrons and ions are controlled by the equations of continuity and momentum respectively [9–12] in flat space-time $(x,t)$ under non-relativistic conditions as (1) $\begin{array}{rl}& {\mathrm{\partial }}_t{n}_j+{\mathrm{\partial }}_x\left(n_j{v}_j\right)=-v_{j\mathrm{d}c}\left(n_j-n_{j0}\right),\end{array}$(1) (2) $\begin{array}{rl}& -T_j{\mathrm{\partial }}_x{n}_j+e{n}_j{\mathrm{\partial }}_x\varphi -m_j{n}_j{v}_{j\mathrm{d}c}{v}_j=0.\end{array}$(2) where $n_{j0}$, $n_j$, $m_j$, $v_j$ and $T_j$ represent respectively the equilibrium number density, non-equilibrium number density, mass, velocity and temperature of the jth particles (j = e, i where “e” and “i” stands for electrons and ions respectively). $v_{j\mathrm{d}c}$ stand for the collision frequencies of the j-th particles with the charged dust grains and ϕ denotes the electrostatic plasma potential.

The dynamics of the viscoelastic fluid comprising of uncharged dust grains is taken care of by the equation of continuity and viscoelastic momentum equation which respectively are (3) $\begin{array}{rl}& {\mathrm{\partial }}_t{n}_{\mathrm{d}n}+{\mathrm{\partial }}_x\left(n_{\mathrm{d}n}{v}_{\mathrm{d}n}\right)=0,\end{array}$(3) (4) $\begin{array}{rl}& \left[1+{\tau }_{m\mathrm{d}n}({\mathrm{\partial }}_t+v_{\mathrm{d}n}{\mathrm{\partial }}_x)\right][m_{\mathrm{d}n}{n}_{\mathrm{d}n}({\mathrm{\partial }}_t+v_{\mathrm{d}n}{\mathrm{\partial }}_x)v_{\mathrm{d}n}\\ & +T_{\mathrm{d}n}{\mathrm{\partial }}_x{n}_{\mathrm{d}n}+m_{\mathrm{d}n}{n}_{\mathrm{d}n}{\mathrm{\partial }}_x\psi +m_{\mathrm{d}n}{n}_{\mathrm{d}n}{v}_{\mathrm{nc}}\left(v_{\mathrm{d}n}-v_{\mathrm{d}c}\right)]\\ & =\left({\xi }_{\mathrm{d}n}+\frac{4{\tau }_{\mathrm{d}n}}{3}\right){\mathrm{\partial }}_x^2{v}_{\mathrm{d}n}.\end{array}$(4) On the other hand, the dynamics of the viscoelastic fluid consisting of the charged dust grains is described by the equation of continuity and viscoelastic momentum equations as follows (5) $\begin{array}{rl}& {\mathrm{\partial }}_t{n}_{\mathrm{d}c}+{\mathrm{\partial }}_x\left(n_{\mathrm{d}c}{v}_{\mathrm{d}c}\right)=0,\end{array}$(5) (6) $\begin{array}{rl}& \left[1+{\tau }_{m\mathrm{d}c}({\mathrm{\partial }}_t+v_{\mathrm{d}c}{\mathrm{\partial }}_x)\right]\\ & \times [m_{\mathrm{d}c}{n}_{\mathrm{d}c}({\mathrm{\partial }}_t+v_{\mathrm{d}c}{\mathrm{\partial }}_x)v_{\mathrm{d}c}+q_d{n}_{\mathrm{d}c}{\mathrm{\partial }}_x\varphi \\ & +T_{\mathrm{d}c}{\mathrm{\partial }}_x{n}_{\mathrm{d}n}+m_{\mathrm{d}c}{n}_{\mathrm{d}c}{\mathrm{\partial }}_x\psi \\ & +m_{\mathrm{d}c}{n}_{\mathrm{d}c}{v}_{\mathrm{cn}}\left(v_{\mathrm{d}c}-v_{\mathrm{d}n}\right)]=\left({\xi }_{\mathrm{d}c}+\frac{4{\tau }_{\mathrm{d}c}}{3}\right){\mathrm{\partial }}_x^2{v}_{\mathrm{d}c}.\end{array}$(6) Here $n_{\mathrm{d}n(\mathrm{d}c)}$, $m_{\mathrm{d}n(\mathrm{d}c)}$, $v_{\mathrm{d}n(\mathrm{d}c)}$ and $T_{\mathrm{d}n(\mathrm{d}c)}$ stand for population density, mass, flow velocity and temperature of neutral (charged) dust grains respectively [33]. $q_d$ is the charge of the dust grain. Moreover, ${\tau }_{m\mathrm{d}n(\mathrm{d}c)}$, ${\upsilon }_{\mathrm{d}n(\mathrm{d}c)}$, ${\xi }_{\mathrm{d}n(\mathrm{d}c)}$ represent respectively viscoelastic relaxation time, shear viscosity co-efficient and bulk viscosity co-efficient of neutral (charged) dust particles. ${\upsilon }_{\mathrm{nc}}$ and ${\upsilon }_{\mathrm{cn}}$ designate the neutral-charged and charged-neutral collision frequencies respectively. ϕ and ψ respectively denote the electrostatic potential and self-gravitating potential generated by fluid density fields.

Poisson equations for these potential distributions are respectively given by (7) $\begin{array}{rl}{\mathrm{\partial }}_x^2\varphi & =4\mathrm{\pi e}\left[\left(n_e-n_i\right)-q_d\frac{n_{\mathrm{d}c}}{e}\right],\end{array}$(7) (8) $\begin{array}{rl}{\mathrm{\partial }}_x^2\psi & =4\mathrm{\pi G}\left(m_{\mathrm{d}n}{n}_{\mathrm{d}n}+m_{\mathrm{d}c}{n}_{\mathrm{d}c}\right),\end{array}$(8) where $G=6.67\times {10}^{-11}N{m}^{2}k{g}^{-2}$ is the universal gravitational constant.

Based on the assumption that $m_{\mathrm{d}n}\approx m_{\mathrm{d}c}=m_d$, Equation (8(8) $\begin{array}{rl}{\mathrm{\partial }}_x^2\psi & =4\mathrm{\pi G}\left(m_{\mathrm{d}n}{n}_{\mathrm{d}n}+m_{\mathrm{d}c}{n}_{\mathrm{d}c}\right),\end{array}$(8) ) takes the form (9) ${\mathrm{\partial }}_x^2\psi =4\mathrm{\pi G}{m}_d\left(n_{\mathrm{d}n}+n_{\mathrm{d}c}\right).$(9) The charge fluctuation equation which dictates the dynamics of the fluctuating dust charge is (10) $({\mathrm{\partial }}_t+v_d{\mathrm{\partial }}_x)q_d=e\left[v_{\mathrm{ed}}\frac{\left(n_e-n_{e0}\right)}{n_{d0}}-v_{\mathrm{id}}\frac{\left(n_i-n_{i0}\right)}{n_{d0}}\right].$(10) It may be noted that Equation (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_t{n}_j+{\mathrm{\partial }}_x\left(n_j{v}_j\right)=-v_{j\mathrm{d}c}\left(n_j-n_{j0}\right),\end{array}$(1) ) represents the equation of continuity for the flux conservation of the thermal species (electrons and ions). It is not a conservation law for mass but for population density. As this model considers charge fluctuating dust molecular clouds (DMCs), therefore the net flux conservation is in correlation with the fluctuating charge dynamical Equation (10(10) $({\mathrm{\partial }}_t+v_d{\mathrm{\partial }}_x)q_d=e\left[v_{\mathrm{ed}}\frac{\left(n_e-n_{e0}\right)}{n_{d0}}-v_{\mathrm{id}}\frac{\left(n_i-n_{i0}\right)}{n_{d0}}\right].$(10) ). As a consequence, the net results revealed by Equation (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_t{n}_j+{\mathrm{\partial }}_x\left(n_j{v}_j\right)=-v_{j\mathrm{d}c}\left(n_j-n_{j0}\right),\end{array}$(1) ) in association with Equation (10(10) $({\mathrm{\partial }}_t+v_d{\mathrm{\partial }}_x)q_d=e\left[v_{\mathrm{ed}}\frac{\left(n_e-n_{e0}\right)}{n_{d0}}-v_{\mathrm{id}}\frac{\left(n_i-n_{i0}\right)}{n_{d0}}\right].$(10) ) can represent the conservation laws of thermal particle population density and dust charge conservation laws, respectively. If the dust charge is conserved to be static, then the RHS of Equation (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_t{n}_j+{\mathrm{\partial }}_x\left(n_j{v}_j\right)=-v_{j\mathrm{d}c}\left(n_j-n_{j0}\right),\end{array}$(1) ) would be exactly zero, thereby representing incompressible fluids associated with the thermal species.

It is clearly evident from Equations (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_t{n}_j+{\mathrm{\partial }}_x\left(n_j{v}_j\right)=-v_{j\mathrm{d}c}\left(n_j-n_{j0}\right),\end{array}$(1) )–(10(10) $({\mathrm{\partial }}_t+v_d{\mathrm{\partial }}_x)q_d=e\left[v_{\mathrm{ed}}\frac{\left(n_e-n_{e0}\right)}{n_{d0}}-v_{\mathrm{id}}\frac{\left(n_i-n_{i0}\right)}{n_{d0}}\right].$(10) ) that the plasma thermal species (hot, mutually thermalized) constitute here inertialess fluids (Equations (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_t{n}_j+{\mathrm{\partial }}_x\left(n_j{v}_j\right)=-v_{j\mathrm{d}c}\left(n_j-n_{j0}\right),\end{array}$(1) )–(2(2) $\begin{array}{rl}& -T_j{\mathrm{\partial }}_x{n}_j+e{n}_j{\mathrm{\partial }}_x\varphi -m_j{n}_j{v}_{j\mathrm{d}c}{v}_j=0.\end{array}$(2) )) and cold dusty species form the inertial fluids (Equations (3(3) $\begin{array}{rl}& {\mathrm{\partial }}_t{n}_{\mathrm{d}n}+{\mathrm{\partial }}_x\left(n_{\mathrm{d}n}{v}_{\mathrm{d}n}\right)=0,\end{array}$(3) )–(6(6) $\begin{array}{rl}& \left[1+{\tau }_{m\mathrm{d}c}({\mathrm{\partial }}_t+v_{\mathrm{d}c}{\mathrm{\partial }}_x)\right]\\ & \times [m_{\mathrm{d}c}{n}_{\mathrm{d}c}({\mathrm{\partial }}_t+v_{\mathrm{d}c}{\mathrm{\partial }}_x)v_{\mathrm{d}c}+q_d{n}_{\mathrm{d}c}{\mathrm{\partial }}_x\varphi \\ & +T_{\mathrm{d}c}{\mathrm{\partial }}_x{n}_{\mathrm{d}n}+m_{\mathrm{d}c}{n}_{\mathrm{d}c}{\mathrm{\partial }}_x\psi \\ & +m_{\mathrm{d}c}{n}_{\mathrm{d}c}{v}_{\mathrm{cn}}\left(v_{\mathrm{d}c}-v_{\mathrm{d}n}\right)]=\left({\xi }_{\mathrm{d}c}+\frac{4{\tau }_{\mathrm{d}c}}{3}\right){\mathrm{\partial }}_x^2{v}_{\mathrm{d}c}.\end{array}$(6) )). In limiting case of $m_d\to 0$, $v_{\mathrm{nc}}\to 0$ and $v_{\mathrm{cn}}\to 0$, all the species can collectively be treated in a common footing of Boltzmannian equilibrium. Now, we adopt a standard astrophysical normalization procedure [33] as follows

$\xi =x/{\lambda }_J$, $\tau =t/{\omega }_J^{-1}$,$F_{\alpha }={\nu }_{\alpha }/{\omega }_J$ ($\alpha =\mathrm{ed},\mathrm{id},\mathrm{nc},\mathrm{cn}$), $N_{\gamma }=n_{\gamma }/n_{\gamma 0}$, $M_{\gamma }=v_{\gamma }/c_{\mathrm{ss}}$ ($\gamma =e,i,\mathrm{d}n,\mathrm{d}c)$, $\mathrm{\Phi }=\mathrm{\varphi e}/T$, $\mathrm{\Psi }=\psi /c_{\mathrm{ss}}^2$, $Q_d=q_d/q_{d0}$

where ${\lambda }_J$ is the Jeans length, ${\omega }_J^{-1}$ is the Jeans time, ${\omega }_J$is the Jeans frequency, $n_{\gamma 0}$ is the equilibrium density, $c_{\mathrm{ss}}$ is the acoustic phase speed, $T/e$ is the plasma thermal potential and $q_{d0}$ is the equilibrium dust charge. The typical values of the various normalizing parameters adopted here are the same as that already reported in the literature [33].

The normalized form of Equations (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_t{n}_j+{\mathrm{\partial }}_x\left(n_j{v}_j\right)=-v_{j\mathrm{d}c}\left(n_j-n_{j0}\right),\end{array}$(1) )–(10(10) $({\mathrm{\partial }}_t+v_d{\mathrm{\partial }}_x)q_d=e\left[v_{\mathrm{ed}}\frac{\left(n_e-n_{e0}\right)}{n_{d0}}-v_{\mathrm{id}}\frac{\left(n_i-n_{i0}\right)}{n_{d0}}\right].$(10) ) are (11) $\begin{array}{rl}& {\mathrm{\partial }}_{\tau }{N}_j+{\mathrm{\partial }}_{\xi }(N_j{M}_j)=-F_{j\mathrm{d}c}(N_j-1),\end{array}$(11) (12) $\begin{array}{rl}& {\mathrm{\partial }}_{\xi }{N}_j+N_j{\mathrm{\partial }}_{\xi }\mathrm{\Phi }+F_{j\mathrm{d}c}\left(\frac{m_j}{m_d}\right)N_j{M}_j=0,\end{array}$(12) (13) $\begin{array}{rl}& {\mathrm{\partial }}_{\tau }{N}_{\mathrm{d}n}+{\mathrm{\partial }}_{\xi }\left(N_{\mathrm{d}n}{M}_{\mathrm{d}n}\right)=0,\end{array}$(13) (14) $\begin{array}{rl}& \left[1+{\tau }_{m\mathrm{d}n}^{\ast }({\mathrm{\partial }}_{\tau }+M_{\mathrm{d}n}{\mathrm{\partial }}_{\xi })\right]\\ & \times [N_{\mathrm{d}n}({\mathrm{\partial }}_{\tau }+M_{\mathrm{d}n}{\mathrm{\partial }}_{\xi })M_{\mathrm{d}n}+\left(\frac{T_{\mathrm{d}n}}{T}\right){\mathrm{\partial }}_{\xi }{N}_{\mathrm{d}n}\\ & +N_{\mathrm{d}n}{\mathrm{\partial }}_{\xi }\mathrm{\Psi }+F_{\mathrm{nc}}\left\{N_{\mathrm{d}n}\left(M_{\mathrm{d}n}-M_{\mathrm{d}c}\right)\right\}]\\ & ={\chi }_{\mathrm{rxn}}^2{\tau }_{m\mathrm{d}n}^{\ast }{\mathrm{\partial }}_{\xi }^2{M}_{\mathrm{d}n},\end{array}$(14) (15) $\begin{array}{rl}& {\mathrm{\partial }}_{\tau }{N}_{\mathrm{d}c}+{\mathrm{\partial }}_{\xi }\left(N_{\mathrm{d}c}{M}_{\mathrm{d}c}\right)=0,\end{array}$(15) (16) $\begin{array}{rl}& \left[1+{\tau }_{m\mathrm{d}c}^{\ast }({\mathrm{\partial }}_{\tau }+M_{\mathrm{d}c}{\mathrm{\partial }}_{\xi })\right]\\ & \times [N_{\mathrm{d}c}({\mathrm{\partial }}_{\tau }+M_{\mathrm{d}c}{\mathrm{\partial }}_{\xi })M_{\mathrm{d}c}+\left(\frac{T_{\mathrm{d}c}}{T}\right){\mathrm{\partial }}_{\xi }{N}_{\mathrm{d}c}\\ & \phantom{[N_{\mathrm{d}c}({\mathrm{\partial }}_{\tau }+M_{\mathrm{d}c}{\mathrm{\partial }}_{\xi })M_{\mathrm{d}c}+\left(\frac{T_{\mathrm{d}c}}{T}\right){\mathrm{\partial }}_{\xi }{N}_{\mathrm{d}c}}+Q_d{N}_{\mathrm{d}n}{\mathrm{\partial }}_{\xi }\mathrm{\Phi }+N_{\mathrm{d}c}{\mathrm{\partial }}_{\xi }\mathrm{\Psi }+F_{\mathrm{cn}}\left\{N_{\mathrm{d}c}\left(M_{\mathrm{d}c}-M_{\mathrm{d}n}\right)\right\}]\\ & ={\chi }_{\mathrm{rxc}}^2{\tau }_{m\mathrm{d}c}^{\ast }{\mathrm{\partial }}_{\xi }^2{M}_{\mathrm{d}c},\end{array}$(16) (17) $\begin{array}{rl}& {\mathrm{\partial }}_{\xi }^2\mathrm{\Phi }=\frac{4\pi e^2}{m_d{\omega }_j^2}\left[\left(n_{e0}{N}_c-n_{i0}{N}_i\right)-n_{\mathrm{d}c0}{Q}_d{N}_{\mathrm{d}c}\frac{q_{d0}}{e}\right],\end{array}$(17) (18) $\begin{array}{rl}& {\mathrm{\partial }}_{\xi }^2\psi =[\left(\frac{{\omega }_{\mathrm{jc}}^2}{{\omega }_j^2}\right)N_{\mathrm{d}c}+\left(\frac{{\omega }_{\mathrm{jn}}^2}{{\omega }_j^2}\right)N_{\mathrm{d}n}],\end{array}$(18) (19) $\begin{array}{rl}& {\mathrm{\partial }}_{\tau }{Q}_d+M_{\mathrm{d}c}{\mathrm{\partial }}_{\xi }{Q}_d\\ & =\frac{e}{q_{d0}}[F_{\mathrm{ed}}\left(\frac{n_{e0}}{n_{d0}}\right)\left(N_e-1\right)-F_{\mathrm{id}}\left(\frac{n_{i0}}{n_{d0}}\right)\left(N_i-1\right)].\end{array}$(19) Here, β = “dn” for neutral dust and “dc” for charged dust.

We propose to study the weakly nonlinear bimodal fluctuation dynamics with a procedurally derived pair extended KdV-B equations. We apply a multi-scale analysis in a new stretched co-ordinate system defined by a pair of coordinates as [29] (20) ${\xi }^{\ast }={ϵ}^{\frac{1}{2}}\left(\xi -\mathrm{\mu \tau }\right),{\tau }^{\ast }={ϵ}^{\frac{3}{2}}\tau ;$(20) where ϵ is a smallness (order) parameter, designating the strength of nonlinearity $(O({10}^{-2}))$ arising due to the fluid convection dynamics. μ is the normalized referral frame velocity in the new coordinate space. The dependent relevant physical variable (Z) is now nonlinearly perturbed and expanded in the various ϵ- powers around the hydrostatic homogeneous cloud equilibrium ($Z_0$) as (21) $\begin{array}{rl}Z& =Z_0+\sum _{k=1}^{\mathrm{\infty }}{ϵ}^k{Z}_k,\end{array}$(21) (22) $\begin{array}{rl}Z& ={\left[\begin{array}{ccccccc}{N}_e& N_i& M_e& M_i& N_{\mathrm{d}n}& N_{\mathrm{d}c}& M_{\mathrm{d}n}\\ M_{\mathrm{d}c}& \varphi & \psi & Q_d\end{array}\right]}^T,\end{array}$(22) (23) $\begin{array}{rl}Z& ={\left[\begin{array}{ccccccccccc}1& 1& 0& 0& 1& 1& 0& 0& 0& 0& 1\end{array}\right]}^T,\end{array}$(23) (24) $\begin{array}{rl}{Z}_k& ={\left[\begin{array}{ccccccc}{N}_{\mathrm{ek}}& N_{\mathrm{ik}}& M_{\mathrm{ek}}& M_{\mathrm{ik}}& N_{\mathrm{d}\mathrm{nk}}& N_{\mathrm{d}\mathrm{ck}}& M_{\mathrm{d}\mathrm{nk}}\\ M_{\mathrm{d}\mathrm{ck}}& {\varphi }_k& {\psi }_k& Q_{\mathrm{d}k}\end{array}\right]}^T.\end{array}$(24) We apply the perturbation series expansion Equations (21(21) $\begin{array}{rl}Z& =Z_0+\sum _{k=1}^{\mathrm{\infty }}{ϵ}^k{Z}_k,\end{array}$(21) )–(24(24) $\begin{array}{rl}{Z}_k& ={\left[\begin{array}{ccccccc}{N}_{\mathrm{ek}}& N_{\mathrm{ik}}& M_{\mathrm{ek}}& M_{\mathrm{ik}}& N_{\mathrm{d}\mathrm{nk}}& N_{\mathrm{d}\mathrm{ck}}& M_{\mathrm{d}\mathrm{nk}}\\ M_{\mathrm{d}\mathrm{ck}}& {\varphi }_k& {\psi }_k& Q_{\mathrm{d}k}\end{array}\right]}^T.\end{array}$(24) ) in Equations (11(11) $\begin{array}{rl}& {\mathrm{\partial }}_{\tau }{N}_j+{\mathrm{\partial }}_{\xi }(N_j{M}_j)=-F_{j\mathrm{d}c}(N_j-1),\end{array}$(11) )–(19(19) $\begin{array}{rl}& {\mathrm{\partial }}_{\tau }{Q}_d+M_{\mathrm{d}c}{\mathrm{\partial }}_{\xi }{Q}_d\\ & =\frac{e}{q_{d0}}[F_{\mathrm{ed}}\left(\frac{n_{e0}}{n_{d0}}\right)\left(N_e-1\right)-F_{\mathrm{id}}\left(\frac{n_{i0}}{n_{d0}}\right)\left(N_i-1\right)].\end{array}$(19) ) for order by order analysis. Then we equate the like terms in various powers of ϵ and after simplification we get:

(25) $\begin{array}{rl}{N}_{\mathrm{d}n1}& =\frac{{\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}}{\begin{array}{c}\left({\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}+{\mu }^2-\frac{T_{\mathrm{d}n}}{T}\right)\left({\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}+{\mu }^2-\frac{T_{\mathrm{d}c}}{T}\right)\\ -({\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}})({\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}})\end{array}}{\varphi }_1\\ & +\frac{\left({\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}+{\mu }^2-\frac{T_{\mathrm{d}c}}{T}\right)}{\begin{array}{c}\left({\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}+{\mu }^2-\frac{T_{\mathrm{d}n}}{T}\right)\left({\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}+{\mu }^2-\frac{T_{\mathrm{d}c}}{T}\right)\\ -({\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}})({\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}})\end{array}}{\psi }_1\\ & =A{\varphi }_1+B{\psi }_1,\end{array}$(25) (26) $\begin{array}{rl}{N}_{\mathrm{d}c1}& =\frac{{\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}+{\mu }^2-\frac{T_{\mathrm{d}n}}{T}}{\begin{array}{c}\left({\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}+{\mu }^2-\frac{T_{\mathrm{d}n}}{T}\right)\left({\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}+{\mu }^2-\frac{T_{\mathrm{d}c}}{T}\right)\\ -({\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}})({\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}})\end{array}}{\varphi }_1\\ & +\frac{\left({\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}+{\mu }^2-\frac{T_{\mathrm{d}n}}{T}\right)}{\begin{array}{c}\left({\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}+{\mu }^2-\frac{T_{\mathrm{d}n}}{T}\right)\left({\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}+{\mu }^2-\frac{T_{\mathrm{d}c}}{T}\right)\\ -({\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}})({\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}})\end{array}}{\psi }_1\\ & =P{\varphi }_1+Q{\psi }_1.\end{array}$(26)

with (27) $\begin{array}{rl}A& =\frac{{\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}}{\begin{array}{c}\left({\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}+{\mu }^2-\frac{T_{\mathrm{d}n}}{T}\right)\left({\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}+{\mu }^2-\frac{T_{\mathrm{d}c}}{T}\right)\\ -({\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}})({\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}})\end{array}},\end{array}$(27) (28) $\begin{array}{rl}B& =\frac{\left({\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}+{\mu }^2-\frac{T_{\mathrm{d}c}}{T}\right)}{\begin{array}{c}\left({\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}+{\mu }^2-\frac{T_{\mathrm{d}n}}{T}\right)\left({\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}+{\mu }^2-\frac{T_{\mathrm{d}c}}{T}\right)\\ -({\mu }^2{\tau }_{\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}})({\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}})\end{array}},\end{array}$(28) (29) $\begin{array}{rl}P& =\frac{{\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}+{\mu }^2-\frac{T_{\mathrm{d}n}}{T}}{\begin{array}{c}\left({\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}+{\mu }^2-\frac{T_{\mathrm{d}n}}{T}\right)\left({\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}+{\mu }^2-\frac{T_{\mathrm{d}c}}{T}\right)\\ -({\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}})({\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}})\end{array}},\end{array}$(29) (30) $\begin{array}{rl}Q& =\frac{\left({\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}+{\mu }^2-\frac{T_{\mathrm{d}n}}{T}\right)}{\begin{array}{c}\left({\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}+{\mu }^2-\frac{T_{\mathrm{d}n}}{T}\right)\left({\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}+{\mu }^2-\frac{T_{\mathrm{d}c}}{T}\right)\\ -({\mu }^2{\tau }_{\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}})({\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}})\end{array}}.\end{array}$(30) Equation (17(17) $\begin{array}{rl}& {\mathrm{\partial }}_{\xi }^2\mathrm{\Phi }=\frac{4\pi e^2}{m_d{\omega }_j^2}\left[\left(n_{e0}{N}_c-n_{i0}{N}_i\right)-n_{\mathrm{d}c0}{Q}_d{N}_{\mathrm{d}c}\frac{q_{d0}}{e}\right],\end{array}$(17) ) leads to: (31) $\begin{array}{rl}{\mathrm{\partial }}_{{\xi }^{\ast }}^3{\varphi }_1& =\frac{4\pi e^2}{M_d{\omega }_j^2}[n_{e0}{\mathrm{\partial }}_{{\xi }^{\ast }}{N}_{e2}-n_{i0}{\mathrm{\partial }}_{{\xi }^{\ast }}{N}_{i2}-\frac{n_{d0}{q}_{d0}}{e}\\ & \{{\mathrm{\partial }}_{{\xi }^{\ast }}(N_{\mathrm{d}c2}+Q_{d2})+{\mathrm{\partial }}_{{\xi }^{\ast }}(N_{\mathrm{d}c1}{Q}_{d1})\}].\end{array}$(31) Equation (18(18) $\begin{array}{rl}& {\mathrm{\partial }}_{\xi }^2\psi =[\left(\frac{{\omega }_{\mathrm{jc}}^2}{{\omega }_j^2}\right)N_{\mathrm{d}c}+\left(\frac{{\omega }_{\mathrm{jn}}^2}{{\omega }_j^2}\right)N_{\mathrm{d}n}],\end{array}$(18) ) leads to: (32) ${\mathrm{\partial }}_{{\xi }^{\ast }}^3{\psi }_1=\frac{{\omega }_{\mathrm{jc}}^2}{{\omega }^2}{\mathrm{\partial }}_{{\xi }^{\ast }}{N}_{\mathrm{d}c2}+\frac{{\omega }_{\mathrm{jn}}^2}{{\omega }^2}{\mathrm{\partial }}_{{\xi }^{\ast }}{N}_{\mathrm{d}n2}.$(32) Finally we derive the pair extended KdV-B equations as: (33) ${A_1}^{\ast }{\mathrm{\partial }}_{{\tau }^{\ast }}{\varphi }_1+{B_1}^{\ast }{\varphi }_1{\mathrm{\partial }}_{{\xi }^{\ast }}{\varphi }_1+{\mathrm{\partial }}_{{\xi }^{\ast }}^3{\varphi }_1=S_e({\xi }^{\ast },{\tau }^{\ast }),$(33) where, (34) $\begin{array}{rl}{S}_e({\xi }^{\ast },{\tau }^{\ast })& =C_1{\mathrm{\partial }}_{{\tau }^{\ast }}{\psi }_1+{D_1}^{\ast }{\psi }_1{\mathrm{\partial }}_{{\xi }^{\ast }}{\psi }_1+{E_1}^{\ast }{\mathrm{\partial }}_{{\xi }^{\ast }}({\varphi }_1{\psi }_1)\\ & +{F_1}^{\ast }{\varphi }_1{\mathrm{\partial }}_{{\xi }^{\ast }}{\psi }_1,\end{array}$(34) and (35) ${A_2}^{\ast }{\mathrm{\partial }}_{{\tau }^{\ast }}{\psi }_1+{B_2}^{\ast }{\psi }_1{\mathrm{\partial }}_{{\xi }^{\ast }}{\psi }_1+{\mathrm{\partial }}_{{\xi }^{\ast }}^3{\psi }_1=S_g({\xi }^{\ast },{\tau }^{\ast }),$(35) and (36) $\begin{array}{rl}{S}_g({\xi }^{\ast },{\tau }^{\ast })& ={C_2}^{\ast }{\mathrm{\partial }}_{{\tau }^{\ast }}{\varphi }_1+{D_2}^{\ast }{\varphi }_1{\mathrm{\partial }}_{{\xi }^{\ast }}{\varphi }_1+{E_2}^{\ast }{\mathrm{\partial }}_{{\xi }^{\ast }}({\varphi }_1{\psi }_1)\\ & +{F_2}^{\ast }{\psi }_1{\mathrm{\partial }}_{{\xi }^{\ast }}{\varphi }_1.\end{array}$(36) where

(37) $\begin{array}{rl}{A_1}^{\ast }& =-[(P+1)(\mu -\frac{T_{\mathrm{d}c}}{\mathrm{\mu T}})-2\mathrm{\mu P}+\mu {\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}(P-A)][\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\\ & +\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}B(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^2\}+{\omega }_j^2]\\ & -[{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}(2\mathrm{\mu P}+\mu {\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}P-\mu {\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}A)+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}(2\mathrm{\mu A}+\mu {\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}A+\mu {\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}P)]\\ & -[{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2-\frac{m_d{\omega }_j^2}{4\mathrm{\pi e}{N}_{\mathrm{d}c0}{Q}_{d0}}({\mu }^2-\frac{T_{\mathrm{d}c}}{T})]\\ & \times [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})\end{array}$(37) (37) $\begin{array}{rl}& +{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}B(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^2\}\\ & +{\omega }_j^2+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\left\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}A(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})\right\}],\\ {B_1}^{\ast }& =-[(P^2+P+1)({\mu }^2-\frac{T_{\mathrm{d}c}}{T})-2{\mu }^2{P}^2-{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}{P}^2-{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}\mathrm{PA}+1]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}\\ & +\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}B(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^2\}\\ & +{\omega }_j^2]-[{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}(2{\mu }^2{P}^2+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}{P}^2+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}\mathrm{PA}-1)\\ & +\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}(2{\mu }^2{A}^2+{\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}{A}^2-{\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}\mathrm{PA})]\\ & -[{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2-\frac{m_d{\omega }_j^2}{4\mathrm{\pi e}{N}_{\mathrm{d}c0}{Q}_{d0}}({\mu }^2-\frac{T_{\mathrm{d}c}}{T})]\\ & \times [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})\\ & +{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}B(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^2\}\\ & +{\omega }_j^2+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\left\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}A(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})\right\}],\end{array}$(37) (39) $\begin{array}{rl}{C_1}^{\ast }& =-[2{\mu }^{2}Q+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}(Q-B)-({\mu }^2-\frac{T_{\mathrm{d}c}}{T})Q][\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}\\ & +\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}B(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^2\}+{\omega }_j^2]\\ & +[\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}(2\mathrm{\mu Q}+\mu {\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}Q-\mu {\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}B)\\ & +\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}(2\mathrm{\mu B}-\mu {\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}B-\mu {\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}Q)][{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2]\\ & -[{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2-\frac{m_d{\omega }_j^2}{4\mathrm{\pi e}{N}_{\mathrm{d}c0}{Q}_{d0}}({\mu }^2-\frac{T_{\mathrm{d}c}}{T})]\\ & \times [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})\\ & +{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}B(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^2\}\\ & +{\omega }_j^2+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}A(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})\}],\end{array}$(39) (40) $\begin{array}{rl}{D_1}^{\ast }& =-[2{\mu }^2{Q}^2+Q+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}{Q}^2+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}\mathrm{BQ}-Q^2][\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})\\ & +{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}B(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^2\}+{\omega }_j^2]\\ & +[\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}(2{\mu }^2{Q}^2+Q+\mu {\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}{Q}^2\\ & +\mu {\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}\mathrm{BQ})+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}(2{\mu }^2{B}^2+B+\mu {\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}{B}^2-\mu {\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}\mathrm{BQ})]\\ & \times [{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2]\\ & -[{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2-\frac{m_d{\omega }_j^2}{4\mathrm{\pi e}{N}_{\mathrm{d}c0}{Q}_{d0}}({\mu }^2-\frac{T_{\mathrm{d}c}}{T})]\\ & \times [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})\\ & +{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}B(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^2\}\\ & +{\omega }_j^2+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\left\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}A(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})\right\}],\end{array}$(40) (41) $\begin{array}{rl}{E_1}^{\ast }& =-[2{\mu }^2\mathrm{PQ}+P+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}\mathrm{PQ}-({\mu }^2-\frac{T_{\mathrm{d}c}}{T})\mathrm{PQ}+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}\mathrm{AQ}]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}B(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})\\ & +{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^2\}+{\omega }_j^2]+[{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}(2{\mu }^2\mathrm{PQ}+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}\mathrm{PQ}+P)+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}(2{\mu }^2\mathrm{AB}+{\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}\mathrm{AB})]\end{array}$(41) (41) $\begin{array}{r}\\ & -[{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2-\frac{m_d{\omega }_j^2}{4\mathrm{\pi e}{N}_{\mathrm{d}c0}{Q}_{d0}}({\mu }^2-\frac{T_{\mathrm{d}c}}{T})]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})\\ & +{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}B(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^2\}\\ & +{\omega }_j^2+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\left\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}A(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})\right\}],\\ {F_1}^{\ast }& =-[{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}\mathrm{PB}-{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}\mathrm{AQ}-({\mu }^2-\frac{T_{\mathrm{d}c}}{T})Q][\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2\\ & +\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}B(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^2\}+{\omega }_j^2]\\ & -[\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}({\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}\mathrm{PB}-{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}\mathrm{AQ})+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}(A-{\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}\mathrm{AQ}+\mathrm{BP})]\\ & [{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2]\\ & -[{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2-\frac{m_d{\omega }_j^2}{4\mathrm{\pi e}{N}_{\mathrm{d}c0}{Q}_{d0}}({\mu }^2-\frac{T_{\mathrm{d}c}}{T})]\\ & \times [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})\\ & +{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}B(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^2\}\\ & +{\omega }_j^2+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\left\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}A(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})\right\}],\end{array}$(41) Also (43) $\begin{array}{rl}A{{}_2}^{\ast }& =[{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2-\frac{m_d{\omega }_j^2}{4\mathrm{\pi e}{N}_{\mathrm{d}c0}{Q}_{d0}}({\mu }^2-\frac{T_{\mathrm{d}c}}{T})]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}(2\mathrm{\mu Q}+\mu {\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}Q-\mu {\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}B)+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}(2\mathrm{\mu B}-\mu {\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}B-\mu {\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}Q)]\\ & -[2{\mu }^{2}Q+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}(Q-B)-({\mu }^2-\frac{T_{\mathrm{d}c}}{T})Q][\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}\\ & +\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\left\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}A(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})\right\}]\\ & -[{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2-\frac{m_d{\omega }_j^2}{4\mathrm{\pi e}{N}_{\mathrm{d}c0}{Q}_{d0}}({\mu }^2-\frac{T_{\mathrm{d}c}}{T})]\\ & \times [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})\\ & +{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}B(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^2\}\\ & +{\omega }_j^2+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\left\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}A(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})\right\}],\end{array}$(43) (44) $\begin{array}{rl}{B_2}^{\ast }& =[{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2-\frac{m_d{\omega }_j^2}{4\mathrm{\pi e}{N}_{\mathrm{d}c0}{Q}_{d0}}({\mu }^2-\frac{T_{\mathrm{d}c}}{T})]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}(2{\mu }^2{Q}^2+Q+\mu {\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}{Q}^2+\mu {\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}\mathrm{BQ})+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}(2{\mu }^2{B}^2+B\\ & +\mu {\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}{B}^2-\mu {\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}\mathrm{BQ})]-[2{\mu }^2{Q}^2+Q+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}{Q}^2+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}\mathrm{BQ}-Q^2]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\left\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}A(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})\right\}]\\ & -[{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2-\frac{m_d{\omega }_j^2}{4\mathrm{\pi e}{N}_{\mathrm{d}c0}{Q}_{d0}}({\mu }^2-\frac{T_{\mathrm{d}c}}{T})]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})\\ & +{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}B(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^2\}\\ & +{\omega }_j^2+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\left\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}A(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})\right\}],\end{array}$(44) (45) $\begin{array}{rl}{C_2}^{\ast }& =-[{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2-\frac{m_d{\omega }_j^2}{4\mathrm{\pi e}{N}_{\mathrm{d}c0}{Q}_{d0}}({\mu }^2-\frac{T_{\mathrm{d}c}}{T})]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}(2\mathrm{\mu P}+\mu {\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}P-\mu {\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}A)+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}(2\mathrm{\mu A}+\mu {\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}A+\mu {\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}P)]\\ & [(P+1)(\mu -\frac{T_{\mathrm{d}c}}{\mathrm{\mu T}})-2\mathrm{\mu P}+\mu {\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}(P-A)][\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}\\ & +\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\left\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}A(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})\right\}]\\ & -[{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2-\frac{m_d{\omega }_j^2}{4\mathrm{\pi e}{N}_{\mathrm{d}c0}{Q}_{d0}}({\mu }^2-\frac{T_{\mathrm{d}c}}{T})]\\ & \times [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})\\ & +{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}B(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^2\}\\ & +{\omega }_j^2+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\left\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}A(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})\right\}],\end{array}$(45) (46) $\begin{array}{rl}{D_2}^{\ast }& =-[{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2-\frac{m_d{\omega }_j^2}{4\mathrm{\pi e}{N}_{\mathrm{d}c0}{Q}_{d0}}({\mu }^2-\frac{T_{\mathrm{d}c}}{T})]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}(2{\mu }^2{P}^2+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}{P}^2+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}\mathrm{PA}-1)+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}(2{\mu }^2{A}^2+{\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}{A}^2\\ & -{\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}\mathrm{PA})]-[(P^2+P+1)({\mu }^2-\frac{T_{\mathrm{d}c}}{T})-2{\mu }^2{P}^2-{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}{P}^2-{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}\mathrm{PA}+1]\\ & \times [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\left\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}A(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})\right\}]\\ & -[{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2-\frac{m_d{\omega }_j^2}{4\mathrm{\pi e}{N}_{\mathrm{d}c0}{Q}_{d0}}({\mu }^2-\frac{T_{\mathrm{d}c}}{T})]\\ & \times [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})\\ & +{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}B(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^2\}\\ & +{\omega }_j^2+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\left\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}A(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})\right\}],\end{array}$(46) (47) $\begin{array}{rl}{E_2}^{\ast }& =-[{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2-\frac{m_d{\omega }_j^2}{4\mathrm{\pi e}{N}_{\mathrm{d}c0}{Q}_{d0}}({\mu }^2-\frac{T_{\mathrm{d}c}}{T})]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}(2{\mu }^2\mathrm{PQ}+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}\mathrm{PQ}+P)+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}(2{\mu }^2\mathrm{AB}+{\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}\mathrm{AB})]\\ & +[2{\mu }^2\mathrm{PQ}+P+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}\mathrm{PQ}-({\mu }^2-\frac{T_{\mathrm{d}c}}{T})\mathrm{PQ}+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}\mathrm{AQ}]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\left\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}A(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})\right\}]\\ & -[{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2-\frac{m_d{\omega }_j^2}{4\mathrm{\pi e}{N}_{\mathrm{d}c0}{Q}_{d0}}({\mu }^2-\frac{T_{\mathrm{d}c}}{T})]\\ & \times [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})\\ & +{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}B(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^2\}\\ & +{\omega }_j^2+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\left\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}A(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})\right\}],\end{array}$(47) (48) $\begin{array}{rl}{F_2}^{\ast }& =-[{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2-\frac{m_d{\omega }_j^2}{4\mathrm{\pi e}{N}_{\mathrm{d}c0}{Q}_{d0}}({\mu }^2-\frac{T_{\mathrm{d}c}}{T})]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}(2{\mu }^2\mathrm{PQ}+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}\mathrm{PQ}+P)+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}(2{\mu }^2\mathrm{AB}+{\mu }^2{\tau }_{m\mathrm{d}n}^{\ast }{F}_{\mathrm{nc}}\mathrm{AB})]\\ & +[2{\mu }^2\mathrm{PQ}+P+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}\mathrm{PQ}-({\mu }^2-\frac{T_{\mathrm{d}c}}{T})\mathrm{PQ}+{\mu }^2{\tau }_{m\mathrm{d}c}^{\ast }{F}_{\mathrm{cn}}\mathrm{AQ}]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\left\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}A(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})\right\}]\\ & -[{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2-\frac{m_d{\omega }_j^2}{4\mathrm{\pi e}{N}_{\mathrm{d}c0}{Q}_{d0}}({\mu }^2-\frac{T_{\mathrm{d}c}}{T})]\\ & \times [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})\\ & +{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}B(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^2\}\\ & +{\omega }_j^2+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}Q(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2]\\ & [\frac{{\omega }_{\mathrm{jc}}^2}{({\mu }^2-\frac{T_{\mathrm{d}c}}{T})}\{{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^{2}P(X_{\mathrm{rxc}}^2-{\mu }^2+\frac{T_{\mathrm{d}c}}{T})+{\mu }^2{{\tau }_{m\mathrm{d}c}^{\ast }}^2\}+\frac{{\omega }_{\mathrm{jn}}^2}{({\mu }^2-\frac{T_{\mathrm{d}n}}{T})}\left\{{\mu }^2{{\tau }_{m\mathrm{d}n}^{\ast }}^{2}A(X_{\mathrm{rxn}}^2-{\mu }^2+\frac{T_{\mathrm{d}n}}{T})\right\}],\end{array}$(48)

The steady-state pulsational fluctuations can now be well realized by transforming Equations (33(33) ${A_1}^{\ast }{\mathrm{\partial }}_{{\tau }^{\ast }}{\varphi }_1+{B_1}^{\ast }{\varphi }_1{\mathrm{\partial }}_{{\xi }^{\ast }}{\varphi }_1+{\mathrm{\partial }}_{{\xi }^{\ast }}^3{\varphi }_1=S_e({\xi }^{\ast },{\tau }^{\ast }),$(33) ) and (35(35) ${A_2}^{\ast }{\mathrm{\partial }}_{{\tau }^{\ast }}{\psi }_1+{B_2}^{\ast }{\psi }_1{\mathrm{\partial }}_{{\xi }^{\ast }}{\psi }_1+{\mathrm{\partial }}_{{\xi }^{\ast }}^3{\psi }_1=S_g({\xi }^{\ast },{\tau }^{\ast }),$(35) ) into a time-stationary form of ordinary differential equations (ODEs) in a comoving Galilean frame $\rho =({\xi }^{\ast }-{\tau }^{\ast })$ as (49) $\begin{array}{rl}& -{A_1}^{\ast }{\mathrm{\partial }}_{\rho }{\varphi }_1+{B_1}^{\ast }{\varphi }_1{\mathrm{\partial }}_{\rho }{\varphi }_1+{\mathrm{\partial }}_{\rho }^3{\varphi }_1\\ & =-{C_1}^{\ast }{\mathrm{\partial }}_{\rho }{\psi }_1+{D_1}^{\ast }{\psi }_1{\mathrm{\partial }}_{\rho }{\psi }_1+{E_1}^{\ast }{\mathrm{\partial }}_{\rho }({\varphi }_1{\psi }_1)\\ & +{F_1}^{\ast }{\varphi }_1{\mathrm{\partial }}_{\rho }{\psi }_1,\end{array}$(49) (50) $\begin{array}{rl}& -{A_2}^{\ast }{\mathrm{\partial }}_{\rho }{\psi }_1+{B_2}^{\ast }{\psi }_1{\mathrm{\partial }}_{\rho }{\psi }_1+{\mathrm{\partial }}_{\rho }^3{\psi }_1\\ & =-{C_2}^{\ast }{\mathrm{\partial }}_{\rho }{\varphi }_1+{D_2}^{\ast }{\mathrm{\partial }}_{\rho }{\varphi }_1+{E_2}^{\ast }{\mathrm{\partial }}_{\rho }({\varphi }_1{\psi }_1)\\ & +{F_2}^{\ast }{\psi }_1{\mathrm{\partial }}_{\rho }{\varphi }_1.\end{array}$(50) It is seen that Equations (49(49) $\begin{array}{rl}& -{A_1}^{\ast }{\mathrm{\partial }}_{\rho }{\varphi }_1+{B_1}^{\ast }{\varphi }_1{\mathrm{\partial }}_{\rho }{\varphi }_1+{\mathrm{\partial }}_{\rho }^3{\varphi }_1\\ & =-{C_1}^{\ast }{\mathrm{\partial }}_{\rho }{\psi }_1+{D_1}^{\ast }{\psi }_1{\mathrm{\partial }}_{\rho }{\psi }_1+{E_1}^{\ast }{\mathrm{\partial }}_{\rho }({\varphi }_1{\psi }_1)\\ & +{F_1}^{\ast }{\varphi }_1{\mathrm{\partial }}_{\rho }{\psi }_1,\end{array}$(49) ) and (50(50) $\begin{array}{rl}& -{A_2}^{\ast }{\mathrm{\partial }}_{\rho }{\psi }_1+{B_2}^{\ast }{\psi }_1{\mathrm{\partial }}_{\rho }{\psi }_1+{\mathrm{\partial }}_{\rho }^3{\psi }_1\\ & =-{C_2}^{\ast }{\mathrm{\partial }}_{\rho }{\varphi }_1+{D_2}^{\ast }{\mathrm{\partial }}_{\rho }{\varphi }_1+{E_2}^{\ast }{\mathrm{\partial }}_{\rho }({\varphi }_1{\psi }_1)\\ & +{F_2}^{\ast }{\psi }_1{\mathrm{\partial }}_{\rho }{\varphi }_1.\end{array}$(50) ) are analytically non-integrable due to the involvement of non-linear complications. A numerical platform is therefore constructed for analysing the patterns of the fluctuations of interest.

3. Results and discussions

The presented work explores the nonlinear gravito-electrostatic (pulsational) properties of a complex astrophysical viscoelastic cloud with the help of derived extended pair KdV-B equations. The various multi-parametric inputs for our numerical analysis are taken from the different sources available in the literature [5, 30, 33, 34]. The different input initial values used are ${\varphi }_1=-1\times {10}^{-3}$, ${\varphi }_{1\rho }=-1\times {10}^{-4}$, ${\varphi }_{1\mathrm{\rho \rho }}=-1\times {10}^{-5}$, ${\psi }_1=-1\times {10}^{-3}$, ${\psi }_{1\rho }=-1\times {10}^{-4}$, ${\psi }_{1\mathrm{\rho \rho }}=-1\times {10}^{-5}$. The results so obtained are focally indicative of diversified interstellar modified wave spectra which are graphically displayed in Figures 1–8.

Figure 1. (Colour online) Profile of the normalized (a) electrostatic potential (${\mathrm{\Phi }}_1$) varying with the normalized distance (ρ), (b) potential gradient ($\frac{\mathrm{d}{\mathrm{\Phi }}_1}{\mathrm{d}\rho }$) with the normalized distance (ρ), (c) potential gradient ($\frac{\mathrm{d}{\mathrm{\Phi }}_1}{\mathrm{d}\rho }$) over the potential (${\mathrm{\Phi }}_1$), and (d) potential curvature ($\frac{d^2{\mathrm{\Phi }}_1}{\mathrm{d}{\rho }^2}$) with the normalized distance (ρ). The various lines refer to different μ values. Various lines refer to (i) $\mu =0.9$ (blue curve), (ii) $\mu =1.2$ (red curve), (iii) $\mu =1.5$ (black curve). The fine input details are described in the text.

In Figure 1, we show the profile of the normalized (a) electrostatic potential (${\mathrm{\Phi }}_1$) varying with the normalized distance (ρ ), (b) potential gradient ($\frac{\mathrm{d}{\mathrm{\Phi }}_1}{\mathrm{d}\rho }$) with the normalized distance (ρ ), (c) potential gradient ($\frac{\mathrm{d}{\mathrm{\Phi }}_1}{\mathrm{d}\rho }$) over the potential (${\mathrm{\Phi }}_1$), and (d) potential curvature ($\frac{d^2{\mathrm{\Phi }}_1}{\mathrm{d}{\rho }^2}$) with the normalized distance (ρ ). Various lines refer to (i) $\mu =0.9$ (blue curve), (ii) $\mu =1.2$ (red curve), (iii) $\mu =1.5$ (black curve). The other input parameters kept fixed are $n_{e0}=10{\mathrm{m}}^{-3}$, $n_{i0}=5{\mathrm{m}}^{-3}$, $n_{\mathrm{d}c0}=1\times {10}^1{\mathrm{m}}^{-3}$, $n_{\mathrm{d}n0}=1\times {10}^1{\mathrm{m}}^{-3}$, $m_d=2.0\times {10}^{-9}\mathrm{kg}$, $q_{d0}=100e$, $T_{\mathrm{d}n}=0.01\mathrm{eV}$, $T_{\mathrm{d}c}=0.01\mathrm{eV}$, T = 1.0. It is found that the electrostatic fluctuations evolve as hybrid eigen-wave structures composed of compressive and rarefactive soliton pairs with amplitude increasing with referral frame velocity (Figure 1(a)). The corresponding amplitude gradients evolve as similar hybrid patterns with opposite phase but with wave amplitude varying negligibly with the referral frame velocity (Figure 1(b)). The nonlinear wave evolution, despite the Burger terms, evolve as conservative systems as evidenced confirmatorily with the help of phase space geometrical trajectories (Figure 1(c)). The wave amplitude curvature evolves as admixtures of compressive and rarefactive solitonic structures with wave amplitude decreasing with the referral frame velocity (Figure 1(d)), and so forth. The basic physical mechanism behind such fluctuation behaviour may be attributable to the complex non-local long-range gravito-electrostatic interplay on the astrophysical fluid scales of space and time as already reported in the literature [5,30,33,34].

In Figure 2, we plot the profile of the normalized (a) gravitational potential (${\mathrm{\Psi }}_1$) varying with the normalized distance (ρ), (b) gravitational gradient ($\frac{\mathrm{d}{\mathrm{\Psi }}_1}{\mathrm{d}\rho }$) with the normalized distance (ρ ), (c) gravitational gradient ($\frac{\mathrm{d}{\mathrm{\Psi }}_1}{\mathrm{d}\rho }$) over the potential (${\mathrm{\Psi }}_1$), and (d) gravitational potential curvature ($\frac{d^2{\mathrm{\Psi }}_1}{\mathrm{d}{\rho }^2}$) with the normalized distance (ρ). The reference of various lines as well as the different input parameters are the same as in Figure 1. From Figure 2, we find that gravitational fluctuations evolve as a typical hybrid structures composed of compressive and rarefactive soliton admixtures. The resultant wave amplitude is found to decrease with the referral frame velocity μ and vice versa (Figure 2(a)). The corresponding field fluctuations evolve accordingly but with an opposite phase (Figure 2(b)). This fluctuation evolution is still a conservative system indicated with the help of closed phase space (geometrical) trajectories (Figure 2(c)). The gravitational potential curvature also evolves as hybrid structures composed of compressive and rarefactive solitonic disturbances. The overall potential curvature amplitude is found to enhance with the referral frame velocity (Figure 2(d)).

Figure 2. (Colour online) Profile of the normalized (a) gravitational potential (${\mathrm{\Psi }}_1$) varying with the normalized distance (ρ), (b) gravitational gradient ($\frac{\mathrm{d}{\mathrm{\Psi }}_1}{\mathrm{d}\rho }$) with the normalized distance (ρ), (c) gravitational gradient ($\frac{\mathrm{d}{\mathrm{\Psi }}_1}{\mathrm{d}\rho }$) over the potential (${\mathrm{\Psi }}_1$), and (d) gravitational potential curvature ($\frac{d^2{\mathrm{\Psi }}_1}{\mathrm{d}{\rho }^2}$) with the normalized distance (ρ). The various lines refer to different μ values. Various lines refer to (i) $\mu =0.9$ (blue curve), (ii) $\mu =1.2$ (red curve), (iii) $\mu =1.5$ (black curve). The fine input details are described in the text.

In Figure 3, we depict the same as in Figure 1, but now for different $n_{\mathrm{d}c0}$ values. Various lines refer to (i) $n_{\mathrm{d}c0}=1.0\times {10}^2{\mathrm{m}}^{-3}$ (blue curve), (ii) $n_{\mathrm{d}c0}=2.0\times {10}^2{\mathrm{m}}^{-3}$ (red curve), (iii) $n_{\mathrm{d}c0}=3.0\times {10}^2{\mathrm{m}}^{-3}$ (black curve).The other input parameters kept fixed are $n_{e0}=10{\mathrm{m}}^{-3}$, $n_{i0}=5{\mathrm{m}}^{-3}$, $n_{\mathrm{d}c0}=1.2\times {10}^2{\mathrm{m}}^{-3}$, $n_{\mathrm{d}n0}=1\times {10}^2{\mathrm{m}}^{-3}$, $m_d=2.0\times {10}^{-9}\mathrm{kg}$, $q_{d0}=100e$, $T_{\mathrm{d}n}=0.01\mathrm{eV}$, $T_{\mathrm{d}c}=0.01\mathrm{eV}$, T = 1.0. Here we find the same fluctuation scenarios as in Figure 1 but a notable feature in this case is that the fluctuation dynamics are oblivious of variation in the equilibrium dust density. As a result, the expected distinct lines are no more distinguishable and so forth. The microphysical insight [37] behind this feature lies in the gravity-induced electrostatic polarization effects on the astrophysical scales.

Figure 3. (Colour online) Profile of the normalized (a) electrostatic potential (${\mathrm{\Phi }}_1$) varying with the normalized distance (ρ), (b) potential gradient ($\frac{\mathrm{d}{\mathrm{\Phi }}_1}{\mathrm{d}\rho }$) with the normalized distance (ρ), (c) potential gradient ($\frac{\mathrm{d}{\mathrm{\Phi }}_1}{\mathrm{d}\rho }$) over the potential (${\mathrm{\Phi }}_1$), and (d) potential curvature ($\frac{d^2{\mathrm{\Phi }}_1}{\mathrm{d}{\rho }^2}$) with the normalized distance (ρ). The various lines refer to different $n_{\mathrm{d}c0}$ values. Various lines refer to (i) $n_{\mathrm{d}c0}=1.0\times {10}^2{\mathrm{m}}^{-3}$ (blue curve), (ii) $n_{\mathrm{d}c0}=2.0\times {10}^2{\mathrm{m}}^{-3}$ (red curve), (iii) $n_{\mathrm{d}c0}=3.0\times {10}^2{\mathrm{m}}^{-3}$ (black curve). The fine input details are described in the text.

Figure 4 is similar to Figure 2, but with different $n_{\mathrm{d}c0}$ values. The reference of the various lines and the input parameters used are analogous to those of Figure 3. It is evident from this figure that the gravitational fluctuation amplitude decreases with $n_{\mathrm{d}c0}$. Furthermore the potential gradient also withnesses a downward slide with increase of equillibrium dust density.

Figure 4. (Colour online) Profile of the normalized (a) gravitational potential (${\mathrm{\Psi }}_1$) varying with the normalized distance (ρ), (b) gravitational gradient ($\frac{\mathrm{d}{\mathrm{\Psi }}_1}{\mathrm{d}\rho }$) with the normalized distance (ρ), (c) gravitational gradient ($\frac{\mathrm{d}{\mathrm{\Psi }}_1}{\mathrm{d}\rho }$) over the potential (${\mathrm{\Psi }}_1$), and (d) gravitational potential curvature ($\frac{d^2{\mathrm{\Psi }}_1}{\mathrm{d}{\rho }^2}$) with the normalized distance (ρ). The various lines refer to different $n_{\mathrm{d}c0}$ values. Various lines refer to (i) $n_{\mathrm{d}c0}=1.0\times {10}^2{\mathrm{m}}^{-3}$ (blue curve), (ii) $n_{\mathrm{d}c0}=2.0\times {10}^2{\mathrm{m}}^{-3}$ (red curve), (iii) $n_{\mathrm{d}c0}=3.0\times {10}^2{\mathrm{m}}^{-3}$ (black curve). The fine input details are described in the text.

Figure 5 depicts the same as Figure 1, but for different $q_{d0}$ values. Various lines refer to (i) $q_{d0}=100e$ (blue curve), (ii) $q_{d0}=200e$ (red curve), (iii) $q_{d0}=300e$ (black curve). The other input parameters kept fixed are $n_{e0}=10{\mathrm{m}}^{-3}$, $n_{i0}=5{\mathrm{m}}^{-3}$, $n_{\mathrm{d}c0}=1.0\times {10}^1{\mathrm{m}}^{-3}$, $n_{\mathrm{d}n0}=1\times {10}^1{\mathrm{m}}^{-3}$, $m_d=2.0\times {10}^{-9}\mathrm{kg}$, $\mu =1.5$, $T_{\mathrm{d}n}=0.01\mathrm{eV}$, $T_{\mathrm{d}c}=0.01\mathrm{eV}$, T = 1.0. It is clear from this figure that the equilibrium grain charge has no role to play in modulating the wave amplitude of the electrostatic fluctuations under the action of gravito-electrostatic coupling. It might happen only because of the consideration of weak nonlinearity (negligible wave steepening with $q_{d0}$) in our weakly nonlinear perturbative analysis (with very smallness order parameter, $ϵ(O({10}^{-2}))$).

Figure 5. (Colour online) Profile of the normalized (a) electrostatic potential (${\mathrm{\Phi }}_1$) varying with the normalized distance (ρ), (b) potential gradient ($\frac{\mathrm{d}{\mathrm{\Phi }}_1}{\mathrm{d}\rho }$) with the normalized distance (ρ), (c) potential gradient ($\frac{\mathrm{d}{\mathrm{\Phi }}_1}{\mathrm{d}\rho }$) over the potential (${\mathrm{\Phi }}_1$), and (d) potential curvature ($\frac{d^2{\mathrm{\Phi }}_1}{\mathrm{d}{\rho }^2}$) with the normalized distance (ρ). The various lines refer to different $q_{c0}$ values. Various lines refer to (i) $q_{d0}=100e$ (blue curve), (ii) $q_{d0}=200e$ (red curve), (iii) $q_{d0}=300e$ (black curve). The fine input details are described in the text.

Figure 6 illustrates the same as Figure 2 but with different values of $q_{d0}$. The reference of the different lines and the input parameters made use of here are similar to those of Figure 5. This figure reveals that the gravitational fluctuation amplitude decreases with $q_{d0}$.

Figure 6. (Colour online) Profile of the normalized (a) gravitational potential (${\mathrm{\Psi }}_1$) varying with the normalized distance (ρ), (b) gravitational gradient ($\frac{\mathrm{d}{\mathrm{\Psi }}_1}{\mathrm{d}\rho }$) with the normalized distance (ρ), (c) gravitational gradient ($\frac{\mathrm{d}{\mathrm{\Psi }}_1}{\mathrm{d}\rho }$) over the potential (${\mathrm{\Psi }}_1$), and (d) gravitational potential curvature ($\frac{d^2{\mathrm{\Psi }}_1}{\mathrm{d}{\rho }^2}$) with the normalized distance (ρ). The various lines refer to different $q_{c0}$ values. Various lines refer to (i) $q_{d0}=100e$ (blue curve), (ii) $q_{d0}=200e$ (red curve), (iii) $q_{d0}=300e$ (black curve). The fine input details are described in the text.

Figure 7 is analogous to Figure 1 with different $m_d$ values. Various lines refer to (i) $m_d=3\times {10}^{-11}\mathrm{kg}$ (blue curve), (ii) $m_d=4\times {10}^{-11}\mathrm{kg}$ (red curve), (iii) $m_d=5\times {10}^{-11}\mathrm{kg}$ (black curve). The other input parameters kept fixed are $n_{e0}=10{\mathrm{m}}^{-3}$, $n_{i0}=5{\mathrm{m}}^{-3}$, $n_{\mathrm{d}c0}=1.0\times {10}^1{\mathrm{m}}^{-3}$, $n_{\mathrm{d}n0}=1\times {10}^1{\mathrm{m}}^{-3}$, $m_d=2.0\times {10}^{-9}\mathrm{kg}$, $\mu =1.1$, $T_{\mathrm{d}n}=0.01\mathrm{eV}$, $T_{\mathrm{d}c}=0.01\mathrm{eV}$, T = 1.0. The results obtained in this case are also akin to those of Figure 1. The fluctuation dynamics is found to be independent of variation in the dust mass. The physical mechanism responsible for this is attributable to gravity-induced electrostatic polarization effects as already discussed in the context of Figure 3.

Figure 7. (Colour online) Profile of the normalized (a) electrostatic potential (${\mathrm{\Phi }}_1$) varying with the normalized distance (ρ), (b) potential gradient ($\frac{\mathrm{d}{\mathrm{\Phi }}_1}{\mathrm{d}\rho }$) with the normalized distance (ρ), (c) potential gradient ($\frac{\mathrm{d}{\mathrm{\Phi }}_1}{\mathrm{d}\rho }$) over the potential (${\mathrm{\Phi }}_1$), and (d) potential curvature ($\frac{d^2{\mathrm{\Phi }}_1}{\mathrm{d}{\rho }^2}$) with the normalized distance (ρ). The various lines refer to different $m_d$ values. Various lines refer to (i) $m_d=3\times {10}^{-11}\mathrm{kg}$ (blue curve), (ii) $m_d=4\times {10}^{-11}\mathrm{kg}$ (red curve), (iii) $m_d=5\times {10}^{-11}\mathrm{kg}$ (black curve). The fine input details are described in the text.

Figure 8. (Colour online) Profile of the normalized (a) gravitational potential (${\mathrm{\Psi }}_1$) varying with the normalized distance (ρ), (b) gravitational gradient ($\frac{\mathrm{d}{\mathrm{\Psi }}_1}{\mathrm{d}\rho }$) with the normalized distance (ρ), (c) gravitational gradient ($\frac{\mathrm{d}{\mathrm{\Psi }}_1}{\mathrm{d}\rho }$) over the potential (${\mathrm{\Psi }}_1$), and (d) gravitational potential curvature ($\frac{d^2{\mathrm{\Psi }}_1}{\mathrm{d}{\rho }^2}$) with the normalized distance (ρ). The various lines refer to different$m_d$ values. Various lines refer to (i) $m_d=3\times {10}^{-11}\mathrm{kg}$ (blue curve), (ii) $m_d=4\times {10}^{-11}\mathrm{kg}$ (red curve), (iii) $m_d=5\times {10}^{-11}\mathrm{kg}$ (black curve). The fine input details are described in the text.

Figure 8 is similar to Figure 2 but with various values of $m_d$. The values of $m_d$ in the different coloured lines and the input parameters are as those in Figure 7. It is seen from the figure that gravitational fluctuation amplitude increases with $m_d$.

The phase portraits presented here (Figures 1(c)–8(c)) represent the geometrical trajectories of the weakly nonlinear local gravito-electrostatic fluctuations under consideration. This phase space is constructed with the defined phase variables -- potential and potential gradient. The closed nature of the phase trajectories physically ensures the fact that the fluctuation dynamics under consideration are quite conservative in nature in the considered plasma model configuration. It is quite in accordance with the phase plane analyses already extensively reported in the literature [37].

It is worth repetition that the fluctuation dynamics investigated here have a unique feature that the net fluctuation effects are independent of equilibrium dust density, equilibrium dust charge and dust grain mass. The basic physical mechanism behind this is the gravity-induced electrostatic phenomena triggering non-local non-uniform polarization influences on the astrophysical scales of space and time as already discussed above. It is in qualitative accord with the literature [38]. It is in addition to the rest of the figures presented above, although founded in the same physical setting.

4. Conclusions

We report a theoretical model development to speculate the nonlinear pulsational mode dynamics excitable in star-forming dust molecular clouds exhibiting strong collective cooperative effects. The basic governing equations are accordingly constructed in standard astronomical fluid scales of space and time. Application of multiple scaling techniques therein yields an extended pair KdV-B equation having gravito-electrostatic coupling. A numerical illustrative platform shows that the nonlinear gravito-electrostatic fluctuations evolve as a typical hybrid structures composed of compressive and rarefactive solitonic admixtures. The associated field fluctuations, geometrical trajectories, and the fluctuation curvatures are portrayed illustratively. We see that the electrostatic fluctuation amplitude increases with referral frame velocity (Figure 1) but remains totally insensitive to equilibrium dust charge density (Figure 3), equilibrium dust charge (Figure 5) and dust mass (Figure 7), whereas the gravitational amplitude decreases with referral frame velocity (Figure 2), equilibrium dust charge density (Figure 4), equilibrium dust charge (Figure 6) and increases with dust mass (Figure 8). The authenticity of our adopted calculation schemes is ensured in light of the similar gravito-electrostatic fluctuation results already reported in the literature [5,30,33,34] The analysis could be helpful in understanding astrophysical structure formation dynamics initiated via material transport processes trigerred by waves and instabilities of the non-local pulsational origin. In addition to the above, it can extensively be useful to understand flow-induced instability dynamics and saturation mechanism in different astrophysical situations in the presence of strong collective correlation effects [39].

Disclosure statement

No potential conflict of interest was reported by the author(s).