Propagation of Dust-Ion Acoustic Shocks in Unmagnetized Four Species Dusty Plasmas Having Double Index Generalized Distributed Electrons

Abstract

This work investigates the nonlinear propagation of dust ion-acoustic (DIA) shock waves (SWs) in a collisionless, four-component, unmagnetized, dusty multi-ion plasma. The four components are considered as the generalized $(\alpha ,q)$-distributed electrons, positively charged Maxwell-Boltzmann distributed light ions, negatively charged inertial heavy ions, and negatively charged stationary dusts. To do so, several nonlinear evolution equations are formulated by implementing the reductive perturbation technique. The appropriate solutions to the obtained equations are provided. This work also reinvestigates the electrostatic DIA SWs in the dusty multi-ion nonextensive plasma because of the misleading information for the nonlinear coefficients and analytical solutions of higher-order Burgers equations (BEs) in Ref. [Eur Phys J Plus (2015) 130: 46, https://doi.org/10.1140/epjp/i2015-15046-0]. It is found from the electrostatic potential obtained by BE that the rarefactive electrostatic shocks are only supported in the presence of superthermal $(0.33< q < 1,\;\alpha = 0)$, isothermal $(q = 1,\; \alpha = 0)$, and nonthernal electrons $(0< \alpha < 0.35,\;q = 1)$, but both compressive and rarefactive shocks are supported in the presence of subthermal $(q > 1,\; \alpha = 0)$ electrons. The effects of parameters on the DIA SWs are reported. It is also investigated the nature of electrostatic shocks and double layers around the critical values and at the critical values of any specific plasma parameters described by modified BE and mixed modified BE. The outcomes from this work would be helpful for better understanding the dynamics of DIA SWs in space environments and plasma laboratories.

1 Introduction

Most of the astrophysical and space plasma systems (ASPs), e.g., Earth’s ionosphere, planetary environments, interstellar media, protostellar disks, molecular clouds, asteroid regions, comet tails, and nebulae [1,2,3,4], justify the existence of dust particles. As a result, one may not only study the ion-, electron-, and positron-acoustic wave phenomena but also the dust-ion acoustic (DIA) or DA wave phenomena with the existence of various charged particles in ASPs. In addition, the development of dusty plasmas (DPs) is still mostly focused on the analysis of the propagation of DIA waves [1,2,3,4,5,6,7,8,9,10]. The existence of low-frequency DIA waves was first proposed theoretically by Shukla in 1992 [10]. Shukla and Silin [10] came to the important conclusion in his fundamental research that perhaps the inclusion of dense, enormous, and immovable charged dust species in electron-ion plasmas, which extensively affects the dynamics of the waves. Furthermore, Mamun et al. [11] investigate the formation of low-frequency DIA waves, which take place over a duration of time that is also considerably smaller than the period of DPs. However, Shukla and Mamun [12] have reported that the influence of dust grains, which are considered to be immobile, impacts equilibrium quasi-neutrality. Further, researchers in plasma physics [13,14,15,16,17,18,19] concentrated a lot of their attention on the study of acoustic waves in dusty multi-ion plasmas. Positive ions (PIs) and negative ions (NIs) are examples of multi-ions, and their presence has recently been well-confirmed by ASPs [12, 15,16,17] and plasma laboratories [18,19,20,21]. For instance, in many plasma environments, such as the Earth’s ionosphere and cometary comae, plasma is produced by the mixing of PIs and NIs in addition to electrons [22]. The sources for the successful generation of PI and NI plasmas are neutral beam sources [20], plasma processing reactors [23], and experiments in laboratories [24]. Additionally, it was shown that negatively charged ions performed better in plasma etching than positively charged ions. As a result, NI plasmas have become more and more significant in the field of plasma physics.

Moreover, the distribution of the velocity of associated plasma components affects the basic features of the plasma system. It is essential to describe the motion of lighter-charged particles by the velocity distribution (VD). The Maxwell-Boltzmann velocity distribution (MBVD) is the most familiar in a collisionless plasma. However, the VD of plasma particles in the laboratory and ASPs differs from the MBVD. The particles follow either nonextensive [25,26,27] and kappa [28] or non-thermal [29, 30] distributions that deviate from MBVD. Based on conventional statistical mechanics, the interaction among the charged particles is mainly short-range. In this case, one can use MBVD to describe the motion of lighter-charged particles. However, numerous plasma particles especially interact via long range (e.g., long-range Coulomb interactions) in many ASPs, where the extensible attribute is usually destroyed. To overcome such dynamic challenges, one can consider the double index generalized distribution, that is, $(\alpha ,q)$-VD [31], which is widely applicable not only with the presence of strength of non-extensivity (q) but also non-thermality index ($\alpha$). For instance, one would be considered q-nonextensive VD by setting $\alpha \rightarrow 0$ proposed by Tsallis [25] for analyzing the circumstances where the MBVD is unsuitable. A real-world example for the existence of the aforementioned complex plasma system is the ring of Saturn [2]. In Ref. [2], the proper ranges of the number densities $N_{J0}$ ($J = i$ for ions, $J = e$ for electrons, and $J = d$ for dust, etc.) and temperature are given, i. e., (i) $N_{i0}\sim 10^{1}cm^{- 3}$, $N_{d0}\sim 10^{- 7} - 10^{- 8}cm^{- 3}$ and $T\sim 10^{5} - 10^{6}K$ exists in E-ring, (ii) $N_{i0}\sim 10^{1}cm^{- 3}$, $N_{d0} < 30\ cm^{- 3}$ and $T\sim 10^{5} - 10^{6}K$ exists in F-ring and (iii) $N_{i0}\sim 0.1 - 10^{2}cm^{- 3}$, $N_{d0}\sim 1\ cm^{- 3}$ and $T\sim 2 \times 10^{4}K$ exists in the Spokes. The plasmas associated with the non-Maxwellian distributions have also recently attracted much attention due to their wide relevance to ASPs such as quark-gluon and hadronic matter plasmas [31], dark matter halos [32], Earth’s bow shock [33], and the magnetospheres of Jupiter and Saturn [34].

Due to the existence of dust and multi-ions in ASPs, some theoretical and experimental research [5, 11, 19, 20, 35,36,37,38,39,40,41] has been focused on the DIA waves in a multi-species collisionless plasma composed of electrons, PIs, and NIs with dust grains. Yasmin et al. [5] analyzed the impact of nonextensive electrons on DIA shock waves (SWs) by taking into account a nonextensive plasma made up of ions, nonextensive electrons, and negatively charged stationary dust (SD). The arbitrary amplitude of DIA SWs in such plasma with PIs and NIs has been explored by Mamun et al. [11]. Ema et al. [41] have reported the propagation of DIA SWs in a nonextensive complex plasma consisting of nonextensive electrons, heavy NIs, Maxwell-Boltzmann distributed light ions having positive charge, and SD having negative charge. They have reported the features of nonlinear propagation DIA SWs by deriving Burgers equation (BE) and a higher-order BE, that is, Gardner equation (GE). To derive GE, they have incorrectly determined the nonlinear coefficient of GE. In addition, the shock and double-layer solutions of GE are incorrectly defined. The detail limitations of Ref. [41] are (i) ignored the effect of nonthermalitiy, (ii) the coefficient $\gamma = \frac{V_{p}^{3}}{2}\left[\frac{15\mu _{h}}{2V_{p}^{6}} - \frac{\mu _{e}}{16}(q + 1)(q - 3)(3q - 5) - \frac{1}{2}\sigma ^{3} \right]$ of Eq. (16) incorrectly represented, and (iii) the solution $\psi = \left[\psi _{m}^{2}\left\{ 1 - \tan h\left( \frac{\xi }{\Delta } \right) \right\} \right]^{\frac{1}{4}}$ of Eq. (16) incorrectly defined. As a result, the investigation on nonlinear propagation DIA SWs in Ref. [41] leads to significant truncation errors, which are unrealistic for further verification in a laboratory. Accordingly, the incorrect shock and double-layer solutions of GE provided in Ref. [41] give misleading information about the exact solutions of GE to the plasma physics community. It is therefore essential to do further work on the article [41] by determining the appropriate nonlinear coefficients of higher-order BEs and the appropriate solutions of higher-order BEs. However, the impact of $(\alpha ,q)$-velocity distributed electrons on dusty multi-ion plasma has not been considered in any previously proposed theoretical research, to the best of our knowledge. Thus, in order to study the minuscule amplitude of DIA SWs by deriving various higher-order BEs along with their appropriate solutions in a complex plasma, the plasma species have been considered as $(\alpha , q)$-velocity distributed electrons, inertial heavy NIs, positively charged Maxwell-Boltzmann distributed light ions, and negatively charged SD. The effects of plasma parameters on the nonlinear propagation of DIA SWs are also investigated with physical interpretations.

2 Theoretical Model Under Plasma Assumptions

As it is known, an ion acoustic mode in the presence of dust charged particles of micrometer size, that is, DIA mode, is applicable for describing the physical scenarios in many ASPs. DIA wave is mainly low frequency electrostatic waves, where the phase speed is larger than the thermal speed of heavy particles but much smaller than the thermal speed of lighter particles, and the temperature of the dust is very much less than the temperature of the other plasma species [10, 12]. In DIA mode, the driving (restoring) force comes from the inertia of heavy NIs (the pressures of the inertialess electrons and ions) and the dust particulates are considered to form a fixed neutralizing background in the DPs [10]. One can study the propagation characteristics of DIA waves by deriving the mass and momentum conservation equations for the inertia of heavy NIs with static charged dust gains. However, the collision time between charged particles is basically very long in comparison with the characteristic timescales, that is, the inverse of the plasma frequency, and therefore the plasma can be treated as collisionless. So, the collective motion of particles is ubiquitous in plasma, resulting in various waves and other types of collective phenomena. Note that DIA waves arise on a time scale which is much shorter than the DP period.

Thus, a collisionless, four-component, unmagnetized, DP system, which is a mixture of the $(\alpha ,q)$-distributed electrons, inertial negatively charged heavy ions, Maxwell-Boltzmann distributed light ions (positively charged), and stationary dust gains (negatively charged), known as immobile dust is considered. As a result, the equilibrium charge neutrality condition is obtained as $N_{i0} - Z_{hi}N_{hi0} - N_{e0} - Z_{d}N_{d0} = 0$, where $N_{s0}$ is the number of unperturbed densities of the species s ($s = i$, hi, e, and d for positive charged light ions, heavy NIs, electrons, and immobile dust, respectively) and $Z_{d}$ is the number of electrons residing on the dust grain surface. Note that Yu and Luo [42] have mentioned that many plasma species occupy in distinct regions of phase space. They have also shown that one can assume the different temperatures of plasma species in the model to construct nonlinear quasi-stationary structures. So, the temperature of interialess ions and electrons may absorb in different regions of the phase space. In addition, the inertial heavy NIs may interact with inertialess ions in comparatively short ranges rather than electrons. However, electrons may act together with other components via long-range interactions (Coulomb) where the features of extensivity are frequently violated [41]. Since the energies of the electrons may be isothermal, nonthermal, or have a smaller (subthermal) or superior (superthermal) amount of isothermality. As a result, the $(\alpha ,q)$-VD and MBVD are assumed for the inertialess electrons and ions, respectively. In Ref. [31], the $(\alpha ,q)$-VD function is defined by the composition of Tsallis and Cairns VD functions as

$$\begin{aligned} y\left( v_{x} \right) = k_N\left( 1 + \alpha \frac{v_{x}^{4}}{v_{t}^{4}} \right) \times \left\{ 1 - (q - 1)\frac{v_{x}^{2}}{{2v}_{t}^{2}} \right\} , \end{aligned}$$

(1)

where

$$\begin{aligned} k_N = \left\{ \begin{array}{l} \frac{N_{e0}\sqrt{\frac{m_{e}}{2\pi k_{B}T_{e}}}\Gamma \left( \frac{1}{1 - q} \right) \times (1 - q)^{\frac{5}{2}}}{\Gamma \left( \frac{1}{1 - q} - \frac{5}{2} \right) \ \times \left[3\alpha + \left( \frac{1}{1 - q} - \frac{3}{2} \right) \left( \frac{1}{1 - q} - \frac{5}{2} \right) (1 - q)^{2} \right]}, - 1< q < 1 \\ \frac{N_{e0}\sqrt{\frac{m_{e}}{2\pi k_{B}T_{e}}}\Gamma \left( \frac{1}{1 - q} + \frac{3}{2} \right) (1 - q)^{\frac{5}{2}}\left( \frac{1}{1 - q} + \frac{3}{2} \right) \left( \frac{1}{1 - q} + \frac{5}{2} \right) }{\Gamma \left( \frac{1}{1 - q} + 1 \right) \ \times \left[3\alpha + (q - 1)^{2}\left( \frac{1}{1 - q} + \frac{3}{2} \right) \left( \frac{1}{1 - q} + \frac{5}{2} \right) \right]},\ q > 1, \end{array} \right. \end{aligned}$$

where $v_{t} = \left( \frac{k_{B}T_{e}}{m_{e}} \right) ^{1/2}$ is the electron thermal velocity, $v_{x}$ is the velocity vector, q is the nonextensivity strength, $\alpha$ represents the population of faster electrons, $k_{B}$ is defined as the Boltzmann constant, $T_{e}$ is the electron temperature, and $k_N$ is the normalized constant (see details in Ref. [31]). When $q > 1$, a thermal cut-off on the maximum value allowed for electrons velocity is defined [31] as $v_{\max } = \sqrt{2k_{B}T_{e}/m_{e}(q - 1)}$ beyond which no probable states exist. Note that Eq. (1) is applicable if nonthermality and nonextensivity may act concurrently on the nature (rarefactive or compressive) of the acoustic wave mode [31]. Hence, the electron density ($N_{e}$) function can be obtained by integrating Eq. (1) over velocity space in which includes an additional potential term of interacting electrons, that is, $v_{x} = \sqrt{2k_{B}e\Phi /m_{e}}$ [43]. That is,

$$N_{e} = \left\{ \begin{array}{l} \int _{- \infty }^{\infty }{y\left( v_{x} \right) dv_{x}, - 1< q < 1} \\ \int _{- v_{\max }}^{v_{\max }}{y\left( v_{x} \right) dv_{x},\ q > 1}\end{array}. \right.$$

This implies that

$$\begin{aligned} N_{e} = \left\{ \begin{array}{c} N_{e0}\left[1 + (q - 1)\left( \frac{e\Phi }{k_{B}T_{e}} \right) \right]^{\frac{(q + 1)}{2(q - 1)}} \\ \times \left[1 - \frac{16q\alpha }{3 - 14q + 15q^{2} + 12\alpha }\left( \frac{e\Phi }{k_{B}T_{e}} \right) + \frac{16q(2q - 1)\alpha }{3 - 14q + 15q^{2} + 12\alpha }\left( \frac{e\Phi }{k_{B}T_{e}} \right) ^{2} \right] \end{array},\right. \end{aligned}$$

(2)

where $\Phi$, $T_{e}$, and e are the electrostatic potential function, electron temperature, and magnitude of the electron charge, respectively. Note that Eq. (2) is applicable if nonthermality and nonextensivity may act concurrently on the nature (rarefactive or compressive) of the acoustic wave mode [31]. Later, the proper ranges of $\alpha$ and q are defined in Refs. [44, 45] based on the physical cut-off obligatory by $q \ge 5/7$, and $\alpha _{Max} = (2q - 1)/4$ as (i) $q = 1$, $0< \alpha < 0.35$ (nonthermality case), (ii) $q = 1$, $\alpha = 0$ (isothermality case), (iii) $0.33< q < 1$, $\alpha = 0$ (superthermality case), and (iv) $q > 1$, $\alpha = 0$ (subthermality case), respectively. In contrast, one can determine only well-known Cairns et al. [29] nonthermal electron density functions by taking $q \rightarrow 1$ from the above expression as

$$\begin{aligned} N_{e} = N_{e0}\left[1 - \left( \frac{4\alpha }{1 + 3\alpha } \right) \left( \frac{e\Phi }{k_{B}T_{e}} \right) + \left( \frac{4\alpha }{1 + 3\alpha } \right) \left( \frac{e\Phi }{k_{B}T_{e}} \right) ^{2} \right]\exp \left( \frac{e\Phi }{k_{B}T_{e}} \right) . \end{aligned}$$

(3)

In this case, one obtains $N_{e} = N_{e0}\exp \left( \frac{e\Phi }{k_{B}T_{e}} \right)$ if $\alpha = 0$. Again, one can determine only well-known nonextensive [25] electron density function by taking $\alpha \rightarrow 0$ from Eq. (2) as

$$\begin{aligned} N_{e} &= N_{e0}\left[1 + (q - 1)\left( \frac{e\Phi }{k_{B}T_{e}} \right) \right]^{\frac{(q + 1)}{2(q - 1)}}\\&= N_{e0}\left[1 + \frac{(q + 1)}{2}\left( \frac{e\Phi }{k_{B}T_{e}} \right) + \frac{(q + 1)(3 - q)}{8}\left( \frac{e\Phi }{k_{B}T_{e}} \right) ^{2} + \cdots \right]. \end{aligned}$$

(4)

In this case, one obtains $N_{e} = N_{e0}\exp \left( \frac{e\Phi }{k_{B}T_{e}} \right)$ if $q = 1$. It is provided that the $(\alpha ,q)$-VD function is very useful to describe the energy of electrons in all cases of thermality. However, the light ions density ($N_{i}$) function can be written [41] based on MBVD function as

$$\begin{aligned} N_{i} = N_{i0}\exp \left( - \frac{e\Phi }{k_{B}T_{i}} \right) . \end{aligned}$$

(5)

To study the basic features of nonlinear DIA SWs in the aforementioned plasmas, the dimensionless continuity and momentum equations are obtained by implementing the mass and momentum conservation laws, respectively, in the following forms [41]:

$$\begin{aligned} \frac{\partial N_{hi}}{\partial t} + \frac{\partial \left( N_{hi}U_{hi} \right) }{\partial x} = 0,\end{aligned}$$

(6)

$$\begin{aligned} \frac{\partial U_{hi}}{\partial t} + U_{hi}\frac{\partial U_{hi}}{\partial x} = \frac{\partial \Phi }{\partial x} + \mu \frac{\partial ^{2}U_{hi}}{\partial x^{2}}, \end{aligned}$$

(7)

Eqs. (6) and (7) are normalized by introducing $N_{hi} \rightarrow N_{hi}/N_{hi0}$, $U_{hi} \rightarrow U_{hi}/C_{hi}$ ($C_{hi} = \left( \frac{k_{B}T_{e}}{m_{hi}} \right) ^{\frac{1}{2}}$), $\Phi \rightarrow \Phi /\left( \frac{k_{B}T_{e}}{e} \right)$, $x \rightarrow x/\lambda _{Dhi}$ ($\lambda _{Dhi} = \left( \frac{k_{B}T_{e}}{4\pi e^{2}n_{hi0}} \right) ^{\frac{1}{2}}$), $t \rightarrow t/\omega _{phi}^{- 1}$ ($\omega _{phi}^{- 1} = \left( \frac{m_{hi}}{4\pi e^{2}N_{hi0}} \right) ^{\frac{1}{2}}$) and $\mu \rightarrow \mu /m_{hi}N_{hi0}\omega _{phi}\lambda _{Dhi}^{2}$, where $U_{hi}$ is the heavy NIs fluid speed, $C_{hi}$ is the heavy NIs acoustic speed, t(x) is the time (space) variable, $\mu$ is the viscosity coefficient, and q($\alpha$) is the strength of nonextensivity (measuring the population of nonthermal electrons), respectively. Since the plasma particles are interconnected to the electric field ($E = - \nabla \Phi$), Eqs. (6) and (7) are supplemented with the following equation that obtained by Gauss Law:

$$\begin{aligned} \nabla \bullet E =- 4\pi \rho , \end{aligned}$$

(8)

where $\rho = \left( {eN}_{e} + Z_{h}N_{hi} - N_{i} + Z_{d}N_{d} \right)$ is the overall charge density on the surface, $N_{hi}$ is the heavy NIs density, and $N_{d}$ is the SDs density. Based on the charge neutrality condition, the following dimensionless equation is obtained from the above Maxwell’s equation:

$$\begin{aligned} \frac{\partial ^{2}\Phi }{\partial x^{2}} = &\; N_{r2}\left\{ \left[1 + (q - 1)\Phi \right]^{\frac{q + 1}{2(q - 1)}} \times \left[1 + B_{1}\Phi + B_{2}\Phi ^{2} \right]\right\} \\&+ N_{r1}N_{hi} - e^{- \delta \Phi } + N_{r3}, \end{aligned}$$

(9)

where

$$\begin{aligned} B_{1} = - \frac{16q\alpha }{3 - 14q + 15q^{2} + 12\alpha }\ and\ B_{2} = \frac{16q(2q - 1)\alpha }{3 - 14q + 15q^{2} + 12\alpha }. \end{aligned}$$

(10)

Due to the normalization of the above equation, $N_{r2} = \frac{Z_{hi}N_{hi0}}{N_{i0}}$ (heavy-to-light ion number density ratio), $N_{r1} = \frac{N_{e0}}{N_{i0}}$ (electron-to-light ion number density ratio), and $\delta = \frac{T_{e}}{T_{i}}$ (electron-to-light ion temperature ratio) are obtained. It is noted that Eq. (9) can be converted to the similar form as in Eq. (3) of Ref. [41] with the presence of nonextensivity, that is, $\alpha = 0$. In order to avoid any encumbering effect like the damping effect, the phase speed of the DIA waves is assumed to lie between the ion and electron thermal speed [10]. The proposed plasma system is in good agreement with the plasma system that was proposed in Ref. [41] for the case of $\alpha = 0$.

3 Mathematical Analysis

To investigate the nonlinear wave propagation of DIA SWs, one can derive the evolution equations involving various nonlinearities from Eqs. (6), (7), and (9) by using the appropriate stretching coordinates and only the expansion of perturb quantities ($N_{hi}, U_{hi}$ and $\Phi$) but not any expansion of arbitrary quantities, like $\mu$. Based on reductive perturbation theory, one expands $N_{hi}, U_{hi}$, and $\Phi$ in terms of a perturbation parameter $\varepsilon$. The next thing to determine is the scaling of space (x) and time (t) variables. Suppose x is scaled in a certain way, then it is the dispersion/dissipation relation which provides the information of how the time part of the physical system reacts. The dispersion/dissipation relation for linear waves can easily found from the linearized version of Eqs. (6), (7), and (9) by the solutions in the form $N_{hi},U_{hi},\ \Phi \sim \exp \left( i\Omega (x,t)\right)$ [41, 46] with‘ $\Omega (x,t) = Kx - \omega (K)t, i=\sqrt{-1}$. Note that $\omega (K)$ satisfies the dispersion/dissipation relation by the wave number K. If we are looking for long waves however, then this corresponds to waves with small wave number K and hence long wave length. Even though such waves are not harmonic waves, one can nevertheless use the limiting form of the dispersion/dissipation relation in the long wave limit. Consequently, one can define $K = \varepsilon ^{j}k,$ where k is the new wave number of O(1) and j is some unknown number which is to be determined later. As a result, $\Omega (x,t)$ becomes $\Omega (x,t) = k\varepsilon ^{j}x - \omega \left( \varepsilon ^{j}k \right) t$. For purely dissipative media, the Taylor expansion of $\omega \left( \varepsilon ^{j}k \right)$yields $\omega \left( \varepsilon ^{j}k \right) = \omega ^{'}(0)\varepsilon ^{j}k + \omega ^{''}(0)\varepsilon ^{2j}k^{2} + \ldots$. In such case, $\Omega (x,t)$ is obtained as $\Omega (x,t) = k\varepsilon ^{j}\left( x - \omega ^{'}(0)t \right) - k^{2}\omega ^{''}(0)\varepsilon ^{2j}t$, where the other terms are neglected because $\varepsilon$ is small quantity. Finally, one may consider the stretching for x and t as $X = \varepsilon ^{j}\left( x - V_{p}t \right)$ and $T = \varepsilon ^{2j}t$ since $\omega '(0)$ and $\omega ''(0)$ are constants (see details in Ref. [46]). Thus, one can derive various evolution equations by taking the stretching $X = \varepsilon ^{j}\left( x - V_{p}t \right)$ and $T = \varepsilon ^{2j}t$. When the basic equations are expanded in powers of $\varepsilon$ and space and time are also rescaled, then a suitable choice for the value of j often becomes apparent. For instance, if the nonlinear evolution equation (NLEE), like Burgers equations obtained by taking $j = 1$, is unable to examine the basic features of acoustic wave phenomena, then one can derive another NLEE by taking $j = 2$ to overcome such difficulty.

3.1 Formation of BE and Its Solution

To derive a NLEE from Eqs. (6), (7), and (9), one inserts $X = \varepsilon ^{j}\left( x - V_{p}t \right)$, $T = \varepsilon ^{2j}t$ with $j = 1$ ($V_{p}$ is linear phase speed) [8, 47] and the following expanded perturb quantities [41]

$$\begin{aligned} \left[ \begin{array}{l} N_{hi} \\ U_{hi} \\ \Phi \end{array}\right] = \left[ \begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right] + \sum _{r=1}^{\infty }\varepsilon ^{r}\left[ \begin{array}{l} N_{hi}^{(r)} \\ U_{hi}^{(r)} \\ \Phi ^{(r)} \end{array}\right] , \end{aligned}$$

(11)

into Eqs. (6), (7), and (9). As a result, one converts Eqs. (6), (7), and (9) by including the different orders of $\varepsilon$, that is, $O(\varepsilon ^r)$, $r = 2,3,4,5,.........$. Now, the solutions for $O(\varepsilon ^2)$ equations (ignored for simplicity) yield [41]

$$\begin{aligned} \left. \ \begin{array}{l} N_{hi}^{(1)} = - \frac{1}{V_{p}^{2}}\Phi ^{(1)} \\ U_{hi}^{(1)} = - \frac{1}{V_{p}}\Phi ^{(1)} \\ \end{array} \right\} , \end{aligned}$$

(12)

and $V_{p}$ is obtained as

$$\begin{aligned} V_{p} = \sqrt{\frac{N_{r1}}{N_{r2}\Omega _{1} + \delta }}\, \end{aligned}$$

(13)

where

$$\begin{aligned} \Omega _{1} = \frac{q + 1}{2} + B_{1}. \end{aligned}$$

From Eq. (13), it is clearly predicted that $V_{p}$ is strongly dependent on $N_{r1}$, $\delta$, $N_{r2}$, $\alpha$, and q but not on $\mu$ and validated only if $N_{r1}/(N_{r2}\Omega _{1} + \delta ) \ge 0$. Note that $V_{p}$ is obtained in the similar form of $V_{p}$ that obtained in Ref. [41] for only $\alpha = 0$.

For $O(\varepsilon ^3)$:

$$\begin{aligned} \frac{\partial N_{hi}^{(1)}}{\partial T} - V_{p}\frac{\partial N_{hi}^{(2)}}{\partial X} + \frac{\partial U_{hi}^{(2)}}{\partial X} + \frac{\partial }{\partial X}\left( N_{hi}^{(1)}U_{hi}^{(1)} \right) = 0,\end{aligned}$$

(14)

$$\begin{aligned} \frac{\partial U_{hi}^{(1)}}{\partial T} - V_{p}\frac{\partial U_{hi}^{(2)}}{\partial X} + U_{hi}^{(1)}\frac{\partial U_{hi}^{(1)}}{\partial X} - \frac{\partial \Phi ^{(2)}}{\partial X} - \mu \frac{\partial ^{2}U_{hi}^{(1)}}{\partial X^{2}} = 0, \end{aligned}$$

(15)

and

$$\begin{aligned} N_{r2}\Omega _{1}\Phi ^{(2)} + N_{r2}\Omega _{2}\left[\Phi ^{(1)} \right]^{2} + \delta \Phi ^{(2)} - \delta ^{2}\left[\Phi ^{(1)} \right]^{2} + N_{r1}N_{hi}^{(2)} = 0, \end{aligned}$$

(16)

where

$$\begin{aligned} \Omega _{2} = \frac{q + 1}{2}B_{1} + \frac{(q + 1)(3 - q)}{8} + B_{2}.\end{aligned}$$

Finally, reducing the second order perturb quantities from Eqs. (14)–(16), the BE is obtained as

$$\begin{aligned} \frac{\partial \Phi }{\partial T} + B\Phi \frac{\partial \Phi }{\partial X} = C\frac{\partial ^{2}\Phi }{\partial X^{2}}\, \end{aligned}$$

(17)

where

$$\begin{aligned} B = \left[\frac{V_{p}^{3}\delta ^{2}}{2N_{r1}} - \frac{V_{p}^{3}N_{r2}\Omega _{2}}{N_{r1}} - \frac{3}{{2V}_{p}} \right],\ C = \frac{\mu }{2}\, \end{aligned}$$

(18)

and $\Phi ^{(1)} \approx \Phi$(for simplicity). It is noted that the nonlinear coefficient B is in good agreement with Ref. [41] with the presence of nonextensivity only, that is, $\alpha = 0$.

The useful solution of Eq. (17) is obtained as

$$\begin{aligned} \Phi = \Phi _{A}\left[1 - \tan h\left( \frac{\xi }{\Phi _{W}} \right) \right], \end{aligned}$$

(19)

where $\xi = X - V_{r}T$, $V_{r}$ is the reference speed, $\Phi _{A} =V_{r}/B$ is the amplitude, and $\Phi _{W} =2C/V_{r}$ is width of the shocks. Based on the above solution, Eq. (17) is only applicable to study SWs in the considered plasmas for $B > 0$or $B < 0$. But it failed to address the SWs when $B \rightarrow 0$, which provides one needs to derive another NLEE by taking more higher-order correction into account.

3.2 Formation of Modified BE and Its Solution

In order to study electrostatic DIA SWs approximately to the critical values (CVs), the stretching coordinates are considered as

$$\begin{aligned} X = \varepsilon ^{j}\left( x - V_{p}t \right) ,\quad T = \varepsilon ^{2j}t, \end{aligned}$$

(20)

with $J = 2$. By applying Eqs. (20) and (11) addicted to Eqs. (6), (7), and (9) in a systematic way, one can covert Eqs. (6), (7), and (9) by including the different orders of$\varepsilon$, that is, $O(\varepsilon ^r),\,r = 3,4,5,.........{ The}O(\varepsilon ^3)$ equations yield the similar solutions as in Eq. (12) and $V_{p}$ as in Eq. (13).

For $O(\varepsilon ^4)$:

$$\begin{aligned} - V_{p}\frac{\partial N_{hi}^{(2)}}{\partial X} + \frac{\partial U_{hi}^{(2)}}{\partial X} + \frac{\partial }{\partial X}\left( N_{hi}^{(1)}U_{hi}^{(1)} \right) = 0,\end{aligned}$$

(21)

$$\begin{aligned} - V_{p}\frac{\partial U_{hi}^{(2)}}{\partial X} + U_{hi}^{(1)}\frac{\partial U_{hi}^{(1)}}{\partial X} - \frac{\partial \Phi ^{(2)}}{\partial X} = 0, \end{aligned}$$

(22)

and

$$\begin{aligned} N_{r2}\Omega _{1}\Phi ^{(2)} + N_{r2}\Omega _{2}\left[\Phi ^{(1)} \right]^{2} + \delta \Phi ^{(2)} - \delta ^{2}\left[\Phi ^{(1)} \right]^{2} + N_{r1}N_{hi}^{(2)} = 0. \end{aligned}$$

(23)

For $O(\varepsilon ^5)$:

$$\begin{aligned} \frac{\partial N_{hi}^{(1)}}{\partial T} - V_{p}\frac{\partial N_{hi}^{(3)}}{\partial X} + \frac{\partial U_{hi}^{(3)}}{\partial X} + \frac{\partial }{\partial X}\left( N_{hi}^{(1)}U_{hi}^{(2)} \right) + \frac{\partial }{\partial X}\left( N_{hi}^{(2)}U_{hi}^{(1)} \right) = 0,\end{aligned}$$

(24)

$$\begin{aligned} \frac{\partial U_{hi}^{(1)}}{\partial T} - V_{p}\frac{\partial U_{hi}^{(3)}}{\partial X} + U_{hi}^{(1)}\frac{\partial U_{hi}^{(2)}}{\partial X} + U_{hi}^{(2)}\frac{\partial U_{hi}^{(1)}}{\partial X} - \frac{\partial \Phi ^{(3)}}{\partial X} - \mu \frac{\partial ^{2}U_{hi}^{(1)}}{\partial X^{2}} = 0, \end{aligned}$$

(25)

and

$$\begin{aligned} N_{r2}\Omega _{1}\Phi ^{(3)} + 2N_{r2}\Omega _{2}\Phi ^{(1)}\Phi ^{(2)} + N_{r2}\Omega _{3}\left[\Phi ^{(1)} \right]^{3} + \delta ^{3}\left[\Phi ^{(1)} \right]^{3} + \delta \Phi ^{(3)} - 2\delta ^{2}\Phi ^{(1)}\Phi ^{(2)} + N_{r1}N_{hi}^{(3)} = 0, \end{aligned}$$

where

$$\begin{aligned} \Omega _{3} = \frac{q + 1}{2}B_{2} + \frac{(q + 1)(3 - q)(5 - 3q)}{48} + \frac{(q + 1)(3 - q)}{8}B_{1}.\end{aligned}$$

(26)

Now, one can easily derive the following solutions from Eqs. (24)–(26):

$$\begin{aligned} N_{hi}^{(2)} = \frac{1}{V_{p}^{2}}\left[\frac{3}{2V_{p}^{2}}{{\{\Phi }^{(1)}\}}^{2} - \Phi ^{(2)} \right],\end{aligned}$$

(27)

$$\begin{aligned} U_{hi}^{(2)} = \frac{1}{V_{p}}\left[\frac{1}{2V_{p}^{2}}{{\{\Phi }^{(1)}\}}^{2} - \Phi ^{(2)} \right], \end{aligned}$$

(28)

and

$$\begin{aligned} \left[N_{r2}\Omega _{2} - \frac{1}{2}\delta ^{2} + \frac{{3N}_{r1}}{2V_{p}^{4}} \right]\left\{ \Phi ^{(1)} \right\} ^{2} + \left[N_{r2}\Omega _{1} + \delta - \frac{N_{r1}}{V_{p}^{2}} \right]\Phi ^{(2)} = 0 \end{aligned}$$

yields

$$\begin{aligned} - C_{f}\left\{ \Phi ^{(1)} \right\} ^{2} = 0, \end{aligned}$$

(29)

where

$$\begin{aligned} C_{f} = N_{r2}\Omega _{2} - \frac{1}{2}\delta ^{2} + \frac{{3N}_{r1}}{2V_{p}^{4}}. \end{aligned}$$

It is clearly observed that the critical values (CVs) can be easily evaluated by putting $C_{f} = 0$. Figure 1 shows the variation of CVs for $q = q_{c}$ with regard to $N_{r1}$ and $\delta$ by considering the other parameters constant. For instance, $C_{f}$ with the parametric values $\alpha = 0$, $N_{r1} = 0.5$, $N_{r2} = 0.05$, and $\delta = 0.01$ yields the CV for $q$ as $q_{c} = 5.066562171$. Also, Fig. 1 shows CVs with the variation of $\delta$ and $N_{r1}$. This figure clearly indicates that CVs are only supported in the case of subthermality, which is in good agreement with the Ref. [41].

Fig. 1

3D shape by $B=0$ taking which represents the variation of $q_c$ with regards to $\delta$ and $N_{r1}$. The other parameters are chosen as $\alpha=0$ and $N_{r2}=0.05$ 

Finally, eliminating the third order perturb quantities from Eqs. (24)–(26), the following modified BE is derived:

$$\begin{aligned} \frac{\partial \Phi ^{(1)}}{\partial T} + B^{'}\left\{ \Phi ^{(1)} \right\} ^{2}\frac{\partial \Phi ^{(1)}}{\partial X} = C\frac{\partial ^{2}\Phi ^{(1)}}{\partial X^{2}}. \end{aligned}$$

(30)

The nonlinear coefficient of Eq. (30) is determined as

$$\begin{aligned} B' = \frac{V_{p}^{3}}{2N_{r1}}\left( \frac{15N_{r1}}{2V_{p}^{6}} - {\frac{1}{2}}\delta ^{3} - 3\Omega _{3}N_{r2} \right) , \end{aligned}$$

(31)

With the presence of nonextensivity only, that is $\alpha = 0$, Eq. (31) reduces to

$$\begin{aligned} B^{'} = \frac{V_{p}^{3}}{2N_{r1}}\left[\frac{15N_{r1}}{2V_{p}^{6}} - {\frac{1}{2}}\delta ^{3} - \frac{N_{r2}}{16}(q + 1)(3 - q)(5 - 3q) \right]. \end{aligned}$$

(32)

The coefficient $B'$ is determined as

$$\begin{aligned} \gamma = \frac{V_{p}^{3}}{2}\left[\frac{15\mu _{h}}{2V_{p}^{6}} - \frac{\mu _{e}}{16}(q + 1)(q - 3)(3q - 5) - \frac{1}{2}\sigma ^{3} \right], \end{aligned}$$

(33)

in Ref. [41], which is incorrect. In our present work, we introduce $\mu _{h} \rightarrow N_{r1}$, $\mu _{e} \rightarrow N_{r2}$, $\sigma \rightarrow \delta$. It is provided that one needs to do further study with the presence of nonextensivity on SW excitations described by Eq. (30) to minimize the truncation errors of Ref. [41]. The useful stationary SW solution of Eq. (30) is obtained as

$$\begin{aligned} \Phi ^{(1)} = \sqrt{\Phi _{A}^{'}\left\{ 1 - \tan h\left( \frac{\xi }{\Phi _{W}^{'}} \right) \right\} }\, \end{aligned}$$

(34)

where $\Phi _{A}^{'} = \left( \frac{3V_{r}}{2B'} \right)$ ($\Phi _{W}^{'} = \left( \frac{C}{V_{r}} \right)$) is the amplitude (thickness) of DIA SWs approximately to CVs. But it failed to address the DIA SWs when $B' \rightarrow 0$, which provides one needs to derive another NLEE by taking more higher-order correction into account.

3.3 Formation of Mixed Modified BE and Its Solution

Noted that Eq. (31) is not valuable for reporting the propagation of DIA SWs at the critical composition of the CVs. To prevail over such obscurity, one can take $C_{f} = C_{f}^{0}$ for $q$ (say) around its $q_{c}$ as

$$\begin{aligned} C_{f}^{0} = h\left( \frac{\partial C_{f}}{\partial q} \right) _{q = q_{c}}\left| q - q_{c} \right| = SG {\varepsilon }\, \end{aligned}$$

(35)

where $\left| q - q_{c} \right| \equiv \varepsilon$ (because $\left| q - q_{c} \right|$ is small quantity). $S = 1( - 1)$ for $q > q_{c}(q < q_{c})$ and $G = \left( \frac{\partial C_{f}}{\partial q} \right) _{q = q_{c}}$. As a result, one can re-evaluate Eq. (26) by adding $\rho ^{(2)} = \varepsilon ^{3}\frac{1}{2}SG\Phi ^{2}$ with the $O(\varepsilon ^5)$ equation and yields

$$\begin{aligned} \begin{array}{l}N_{r2}\Omega _{1}\frac{\partial \Phi ^{(3)}}{\partial X} + SG\Phi ^{(1)}\frac{\partial \Phi ^{(1)}}{\partial X} + 2N_{r2}\Omega _{2}\frac{\partial }{\partial X}\left[\Phi ^{(1)}\Phi ^{(2)} \right]+\\ 3N_{r2}\Omega _{3}\left[\Phi ^{(1)} \right]^{2}\frac{\partial \Phi ^{(1)}}{\partial X} + \delta \frac{\partial \Phi ^{(3)}}{\partial X} - 2\delta ^{2}\frac{\partial }{\partial X}\left[\Phi ^{(1)}\Phi ^{(2)} \right]+ N_{r1}\frac{\partial N_{hi}^{(3)}}{\partial X} = 0.\end{array} \end{aligned}$$

(36)

Eliminating the third order perturbs quantities from Eqs. (24), (25), and (36), the following mixed modified BE is derived:

$$\begin{aligned} \frac{\partial \Phi ^{(1)}}{\partial T} + SD\Phi ^{(1)}\frac{\partial \Phi ^{(1)}}{\partial X} + B{'\left\{ \Phi ^{(1)} \right\} }^{2}\frac{\partial \Phi ^{(1)}}{\partial X} = C\frac{\partial ^{2}\Phi ^{(1)}}{\partial X^{2}}, \end{aligned}$$

(37)

where

$$\begin{aligned} D = \frac{V_{p}^{3}}{2N_{r1}}\left( \frac{\partial C_{f}}{\partial q} \right) _{q = q_{c}}\, \end{aligned},$$

(38)

One can easily convert the mixed modified BE not only to modified BE but also to BE. Noted that Eq. (35) is applicable to studying DIA SWs not only around CVs but also at the critical composition of CVs. The correct solution of Eq. (37) is defined [47] as

$$\begin{aligned} \Phi ^{(1)} = \sqrt{\Phi _{mA}\left\{ 1 - \tan h\left( \frac{\xi }{\Phi _{mW}} \right) \right\} } - \frac{SD}{2B}, \end{aligned}$$

(39)

where

$$\begin{aligned} \Phi _{mA} = \frac{3}{2B'}\left( V_{r} + \frac{S^{2}D^{2}}{4B'} \right) ,\ \ \Phi _{mW} = \frac{C}{V_{r} + \frac{S^{2}D^{2}}{4B'}}\ . \end{aligned}$$

(40)

The details formulation of Eq. (39) are given in Appendix. Noted that the solution of Eq. (37) is incorrectly defined in Ref. [41] as follows:

$$\begin{aligned} \psi = \left[\psi _{m}^{2}\left\{ 1 - \tan h\left( \frac{\xi }{\Delta } \right) \right\} \right]^{\frac{1}{4}}, \end{aligned}$$

(41)

where $\psi _{m} = \frac{3U_{0}}{4\Upsilon }$ and $\Delta = \sqrt{\frac{4B}{U_{0}}}$. In our present work, we introduce $\psi _{m} \rightarrow \Phi _{mA}$, $\Delta {\rightarrow \Phi }_{mW}$, ${\Upsilon \rightarrow B}^{'}$, $B \rightarrow C$, $U_{0} \rightarrow V_{r}$. The verification of the solution to Eq. (37) can be found in Appendix.

4 Result and Concluding Remarks

An unmagnetized dusty plasma system composed of negatively charged SD, $(\alpha ,q)$-distributed electrons, positively charged Maxwellian light ions, and negatively charged inertial heavy ions has been considered. If $\alpha = 0$, then the considered plasma system becomes non-extensive dusty multi-ion plasma, which is in good agreement with the earlier investigation in Ref. [41]. The nonlinear propagation of DIA SWs phenomena in such plasmas has been investigated by changing the parametric values of the parameters. In addition, the article mentions the conditions under which DIA SWs phenomena are supported. To do so, the reductive perturbation technique has been employed to formulate BE, modified BE, and mixed modified BE. Noted that there is a lot of misleading information regarding the nonlinear coefficients of these evolution equations and the solutions of higher-order BE that provided in Ref. [41]. For instance, the solutions for higher-order BE that provided in Ref. [41] are completely wrong. That is why, the nonlinear propagation of DIA SWs around CVs and at the critical composition of the CVs has been reinvestigated in the presence of subthermal electrons. The outcomes for the nonlinear propagation of electrostatic DIA SWs found in this work are summarized below.

Figures 2 and 3 present the roles of nonthermality, isothermality, superthermality, and subthermality by varying $\alpha$ and $q$ on DIA SWs described by BE. It is clearly observed that the considered plasmas are supported only by rarefactive DIA SWs with the presence of nonthermal, isothermal, and superthermal electrons, whereas both compressive and rarefactive DIA SWs are supported by the presence of subthermal electrons. It is provided that BE is invalid to study the propagation of DIA SWs when $B \rightarrow 0$ only in the case of subthermality. As a result, it is only possible to investigate the fundamental characteristics (polarity, amplitude, width, etc.) of DIA SWs by depending on the higher-order BEs and the strength of electron nonextensivity ($q > 1)$. The influences of $\delta$ and $N_{r1}$ ($N_{r2}$) with the presence of subthermality on the nonlinear propagation of DIA SWs are displayed by the useful solution of BE in Figs. 4 and 5, respectively. These figures are clearly shown that the amplitude of electrostatic shock increases (decreases) with the increase in subthermal electron temperature and heavy ion number density (subthermal electron number density). This means that the driving force (restoring force) dominates rather than the restoring force (driving force) with the increase of heavy ion number density (with the increase of subthermal electron temperature but minimizing its density). Figure 6 shows the variation of the width of electrostatic shocks with regards to $\mu$ and $V_{r}$ by the useful solution of BE. It is observed from Fig. 6 that the widths of monotonic shocks increase with the increase of the viscosity coefficient of heavy-ions, whereas the amplitude (figure ignored for simplicity) and width (Fig. 6b) increase with the increase of the reference speed, as expected.

Fig. 2

Effect of a population of nonthermal electrons ($q=1$) and b superthermal electrons ($\alpha =0$) on DIA SW profiles. The other fixed parametric values are $N_{r1}=0.5$, $N_{r2}=0.05$, $\mu =0.1$, $\delta =0.1$, and $V_r=0.01$

Fig. 3

Effect of subthermal electrons ($\alpha =0$) on a compressive DIASW and a rarefactive DIA SW profiles. The remaining parameters are selected as $N_{r1}=0.5$, $N_{r2}=0.05$, $\mu =0.1$, $\delta =0.1$, and $V_r=0.01$

Fig. 4

Variation of positive (negative) electrostatic shocks with regards to $\xi$ and $\delta$ for a $q=7$ and b $q=3.5$. The other parameters are chosen as $\alpha = 0$, $N_{r1}=0.5$, $N_{r2}=0.05$, $\mu =0.1$, and $V_r=0.01$

Fig. 5

Electrostatic DIA shocks for different values of a $N_{r1}$ with $\alpha =0$, $\delta =0.01$, $\mu =0.1$, $q=6$, $N_{r2}=0.05$, and $V_r=0.01$, b $N_{r2}$ with $\alpha =0$, $\delta =0.01$, $\mu =0.1$, $q=9$, $N_{r1}=0.6$, and $V_r=0.01$

Fig. 6

DIA shocks width with regards to $V_r$ and the variation of $\mu$. The remaining parameters are considered as $\alpha =0$, $N_{r1}=0.01$, $N_{r2}=0.05$, and $q=7$

Figures 7a and b and  8a and b show the electrostatic DIA SWs with regards to $\xi$and $q$, and $\xi$ and $\delta$ ($\xi$ and $N_{r1}$, and $\xi$ and $N_{r2}$), respectively, around the CVs. It is found from these figures that the positive formation of DIA SWs is produced in the considered plasmas in the cases of (i) $q$ less than from its CVs ($q_{c}$), (ii) $\delta$ greater than from its CVs ($\delta _{c}$), (iii) $N_{r1}$ less than from its CVs ($N_{r1C}$), and (iv) $N_{r2}$ greater than from its CVs ($N_{r2C}$). Otherwise, it is not possible to predict what happens with the electrostatic DIA SWs in the considered plasma system because the amplitude of DIA SWs around CVs becomes complex. That is why, the mixed modified BE is needed to overcome such complexity. It is also found from Figs. 7 and 8 that the maximum amplitude of shocks occurs very close to the CVs.

Fig. 7

Variation of mB shock profiles with regards to a $\xi$ and q around CV ($q_c=5.066562171$) and b $\xi$ and $\delta$ around CV ($\delta _c$)

Fig. 8

Variation of mB shock profiles with regards to a $\xi$ and $N_{r1}$ around CV ($N_{r1C}$) and b $\xi$ and $N_{r2}$ around CV ($N_{r2C}$)

Figure 9 displays the electrostatic shocks and double-layer very close to the critical composition. It is found that the considered plasma system is supported by electrostatic shocks very close to the critical composition and at the critical composition, but the double layer is only produced in the case of $\delta = 0.195 > \delta _{c}$. It is also found that the electrostatic shocks and double layer described by the mixed modified BE are supported very close to the critical composition and at the critical composition by depending on all the related parameters, except the viscosity coefficient ($\mu )$ and reference speed ($V_{r}$). That is why the influence of $\mu$ and $V_{r}$ on the electrostatic shocks and double layer is displayed in Fig. 10. It is interesting to find that the thickness of monotonic shocks and double layers is increased, but the amplitude remains unchanged with the increase of $\mu$. In addition, both the amplitude and thickness of monotonically induced shocks and double layersare remarkably affected by the variation of $V_{r}$. Because both the amplitude and thickness of electrostatic shocks (double layer) are decreased (increased) with the increase of $V_{r}$ with the presence of subthermal electrons.

Fig. 9

Electrostatic DIA shock profiles with X and T for a $\delta =0.009<\delta _c=0.01$ (blue surface) and $\delta =0.2>\delta _c=0.01$ with $\alpha = 0$, $q=5.066562171$, $N_{r1}=0.5$, $N_{r2}=0.05$, $\mu =0.1$, and $V_r=0.01$ and b $q=5<q_c=5.066562171$ (green surface) and $q=5.5>q_c=5.066562171$ with $\alpha = 0$, $\delta =0.01$, $N_{r1}=0.5$, $N_{r2}=0.05$, $\mu =0.1$, and $V_r=0.01$

Fig. 10

Effect of a viscosity coefficient ($V_r=0.01$) and b reference speed ($\mu =0.1$) on the electrostatic DIA shock profiles by considering $\delta =0.009<\delta _c=0.01$ (blue surface) and double layer by considering $\delta =0.2>\delta _c=0.01$. The remaining parametric values are $\alpha = 0$, $q=5.066562171$, $N_{r1}=0.5$, and $N_{r2}=0.05$

It is concluded that the findings of this study would help comprehend nonthermality and nonextensivity effects in laboratory plasmas as well as interstellar and space plasmas (particularly in proto-neutron stars, dark-matter halos, stellar polytropes, hadronic matter, quark-gluon plasma, and other objects). It is remarkable to note that the reinvestigating outcomes with the presence of subthermal electrons represent the actual scenarios of the dynamics of electrostatic shocks and double layers not only around the CVs but also at the critical composition CVs. It is also provided in this article that the appropriate solutions of higher-order BEs are essential to predicting not only the nature of electrostatic shocks and double layers but also the propagation of electrostatic shocks and double layers in further laboratory verification. Thus, it may be suggested to conduct a laboratory experiment that will be able to distinguish the unique new features of the DIA SWs propagating in the dusty multi-ion plasmas in the presence of all electron energy cases.

Appendix

To determine the correct stationary SW solution of Eq. (37), one can convert Eq. (37) by considering $\Phi ^{(1)}=\Psi -SD/2B'$ as

$$\begin{aligned} \frac{\partial \Psi }{\partial T} -\left( \frac{SD}{2\sqrt{B'}}\right) ^2\frac{\partial \Psi }{\partial X} + B'{\left\{ \Psi \right\} }^{2}\frac{\partial \Psi }{\partial X} = C\frac{\partial ^{2}\Psi }{\partial X^{2}} \end{aligned}.$$

(42)

Using $\Psi =F(\xi ), \xi =X-V_r T$ along with the conditions

$$\begin{aligned} F(\xi )\rightarrow 0, \frac{dF(\xi )}{d\xi }\rightarrow 0, \frac{d^2F(\xi )}{d\xi ^2}\rightarrow 0,..... as\;\xi \rightarrow \pm \infty \end{aligned}$$

into (42), one obtains

$$\begin{aligned} -V_r\frac{dF(\xi )}{d \xi } -\left( \frac{SD}{2\sqrt{B'}}\right) ^2\frac{dF(\xi )}{d\xi } + B'{\left\{ F(\xi ) \right\} }^{2}\frac{dF(\xi )}{d\xi } = C\frac{d^2F(\xi )}{d\xi ^2} \end{aligned}$$

(43)

Integrating with regards to $\xi$, one obtains

$$\begin{aligned} -V_rF(\xi ) -\left( \frac{SD}{2\sqrt{B'}}\right) ^2F(\xi ) + \frac{B'}{3}{\left\{ F(\xi ) \right\} }^{3} = C\frac{dF(\xi )}{d\xi } \end{aligned}.$$

(44)

This implies that

$$\begin{aligned} \frac{C}{2\left( V_r+\frac{S^2D^2}{4B'}\right) } \left[ \frac{2F(\xi )}{F^2(\xi )-\frac{3}{B'} \left( V_r+\frac{S^2D^2}{4B'}\right) }-\frac{2}{F(\xi )}\right] dF(\xi ) =d\xi . \end{aligned}$$

(45)

Again, integrating and then simplifying Eq. (45), one determines

$$\begin{aligned} F(\xi )=\sqrt{\frac{3}{B'}\left( V_r+\frac{S^2D^2}{4B'}\right) \left\{ 1-\tanh \left( \frac{V_r+\frac{S^2D^2}{4B'}}{C}\xi \right) \right\} }. \end{aligned}$$

(46)

Finally, one can easily determine the solution as in Eq. (39) by combining $\Phi ^{(1)}=\Psi -SD/2B'$, $\Psi =F(\xi )$ and Eq. (46). The verification code for the analytical solution of mixed modified BE as Eq. (39) with the aid of Maple 18 is given below:

$$> \xi : = X - V[r]*\ T$$

$$> \Phi [mA]: = \left( \frac{3}{2B^{'}} \right) *\left( V[r]+ \frac{S^{2}*D^{2}}{4*B'} \right)$$

$$> \Phi [mW]: = \left( \frac{C}{V[r]+ \frac{S^{2}*D^{2}}{4B'}} \right)$$

$$> \Phi ^{(1)}: = \sqrt{\Phi [mA]*\left( 1 - \tan h\left( \frac{\xi }{\Phi [mW]} \right) \right) } - \frac{S*D}{2*B'}$$

$$> diff(\Phi ^{(1)},T)$$

$$> diff(\Phi ^{(1)},X)$$

$$> diff(\Phi ^{(1)},x\$ 2)$$

$$>diff\left( \Phi ^{(1)},T \right) + S*D*\Phi ^{(1)}*\ diff\left( \Phi ^{(1)},X \right) + B^{'}*\left( \Phi ^{(1)} \right) ^{2}*\ diff\left( \Phi ^{(1)},X \right) = C*diff(\Phi ^{(1)},X\$ 2)$$

$$> simplify(\%)$$

$>0$ It is noticeable that Eqs. (17), (30), and (37) are derived by using Maple software.