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# Zeitschrift für Naturforschung A

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# Dust-Ion-Acoustic Solitary and Shock Structures in Multi-Ion Plasmas with Super-Thermal Electrons

Md. Masum Haider
• Corresponding author
• Department of Physics, Mawlana Bhashani Science and Technology University, Santosh, Tangail-1902, Bangladesh, E-mail:
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• Other articles by this author:
/ Aynoon Nahar
Published Online: 2017-06-16 | DOI: https://doi.org/10.1515/zna-2017-0108

## Abstract

The propagation of dust-ion-acoustic (DIA) solitary and shock waves in multi-ion (MI) unmagnetised and magnetised plasmas have been studied theoretically. The plasma system contains positively and negatively charged inertial ions, opposite polarity dusts, and high energetic super-thermal electrons. The fluid equations in the system are reduced to a Korteweg-de Vries (K-dV) and Korteweg-de Vries Burger (K-dVB) equations in the limit of small amplitude perturbation. The effect of super-thermal electrons, the opposite polarity of ions, and dusts in the solitary and shock waves are presented graphically and numerically. Present investigations will help to astrophysical and laboratory plasmas.

## 1 Introduction

Generally, ions are considered as a positively charged particles, not only in the field of plasma physics but also in other fields. But the presence of negative ions is found in both laboratory [1], [2], [3] and space [4], [5] palsmas. Cooney et al. [6] have shown experimentally the existence of positive and negative ion in plasma. The formation of ion-acoustic shocks may also support the above work, when the ratio of the negative ion to positive ion number density exceeds about 0.9 [7]. The collective interactions in positive and negative ion in plasmas have potential applications in natural and technological environments like the D-region of the Earth’s ionosphere, the Earth’s mesosphere, the solar photosphere and the microelectronics plasma processing reactors [8]. These negative ions can modify the basic properties of ion-acoustic (IA), dust-ion-acoustic (DIA), and electron-acoustic (EA) (with the presence of ions) solitary waves (SWs). Positive and negative ions species oscillate 180° out of phase in fast mode and in phase in slow mode of multi-ion (MI) plasmas. It is well-established by some observations that in the case of low temperature plasmas there are some micron or sub-micron sized dust particles might be present [9], [10], [11], [12], [13], [14], [15], [16], [17] in laboratory (plasma crystal) and space plasmas (Earth’s ionosphere, planetary atmospheres, planerary rings, interstellar and circumstellar clouds, asteroid zones, protostellar disks, cometary tails, nebula, etc.). Some dusty plasma parameters are shown in Table 1 [18], [19]. Due to size effect on secondary emission [20] these dust particles can have positively charged (smaller particles) and negatively charged (larger particles).

Table 1:

Dusty plasma parameters [18], [19].

The collective interactions of positive and negative ions in plasmas have potential applications in natural and technological environments including the D-region of the Earth’s ionosphere, the Earth’s mesosphere, the solar photosphere, and the microelectronics plasma processing reactors [8], [21]. Recently Rahman [22] has studied the effect of super-thermal electrons in solitary and shock waves in four component unmagnetised plasmas considering positive ions as mobile and negative ions as Maxwellian with static positive dust. Earlier, Shah et al. [23] have theoretically investigated heavy ion-acoustic solitary and shock waves in unmagnetised plasmas with super-thermal electrons and light Maxwellian light ions. Ema et al. [24] have studied the DIA shock structures in MI unmagnetised plasmas containing negatively charged heavy ions, positively charged light ions, non-extensive electrons, and negatively charged stationary dusts. But these inertial positive and negative ions might have similar mass and both of these might be considered as mobile [21], [25], [26], [27], [28], [29], [30], [31]. Many authors have studied the basic properties of DIA solitary and shock waves in MI unmagnetised [21] and magnetised [25], [26], [27], [28], [29], [30], [31] plasma system containing inertial positive and negative mobile ions and opposite polarity dusts for different electron distribution like Maxwellian [21], [27], vortex [28], [29], and non-thermal [30], [31]. Most of the space plasmas are magnetised and in recent year the existence of energetic super-thermal particles has been found in laboratory and space plasmas [22], [23], [32], [33]. Vasyliunas [34] introduced kappa distribution [32], [34], [35], [36] to fit phenomenologically the power law-like dependence of electron distribution functions observed in space.

$ne=[1−ψκ−32]−κ+12$(1)

where κ is the spectral index which modifies the effective thermal speed in the distribution function. At low values of κ, distributions exhibit strong super-thermality; by this, we mean that there is an excess in the super-thermal component of the distribution compared to that of a Maxwellian. At very large values of κ, the distribution function approaches a Maxwellian distribution. It is commonly fitted to observational data [32], [35], [37], [38].

In the present work, the propagation of DIA SWs has been studied in magnetised dusty plasma system consisting of negatively and positively charged inertial mobile ions with high energetic supertharmal electrons and opposite polarity dusts. When both the nonlinear and the dispersive effects are positive, the Kortewegde Vries (K-dV) equation [39] may result in the compressive (hump-like) soliton. A medium with dispersive and significant dissipative properties supports the existence of shock waves instead of solitons. In order to study such waves (liquid flow containing gas bubbles, fluid flow in elastic tubes, etc.) the governing equation can be reduced to the so-called Korteweg-de Vries Burgers (K-dVB) equation [40]. It is a linear combination of K-dV equation [39] and the Burgers equation [41]. The K-dVB equation has been obtained when including electron inertia effects in the description of weak nonlinear plasma waves [42]. The K-dVB equation has also been used in a study of wave propagation through liquid field elastic tube [43] and for a description of shallow water waves on viscous fluid. In peculiar circumstances the Burger’s term (due to the dissipative mechanisms such as wave-particle interaction, the effects of turbulence, dustcharge fluctuations in a dusty plasma, MI streaming, Landau damping, anomalous viscosity, etc.) in the K-dVB equation becomes negligibly small. The K-dVB equation transforms to the K-dV equation, which admits the soliton solutions. In this manuscript the well-established reductive perturbation method [44] has been employed to derive the K-dV [39] and K-dVB [40] equations for studying solitary and shock waves, respectively.

## 2 Plasma Model

To study the nonlinear dynamics of DIA solitary and shock waves, we have considered a collisionless MI plasmas, containing

1. Negatively and positively charged mobile ions,

2. Oppositely charged stationary dusts, and

3. Super-thermal electrons.

First of all we have studied nonlinear dynamics of DIA SWs in unmagnetised plasma situation, after that studied the solitary and shock waves in magnetised dusty plasmas. It has been considered, while studying solitary and shock profiles in magnetised dusty plasmas, that there is an external static magnetic field B0 acting along the z-direction $\left({B}_{0}=\stackrel{^}{k}{B}_{0}\right),$ where $\stackrel{^}{k}$ is the unit vector along the z-direction which is very strong, so that the electrons and dusts are moving along the magnetic field direction very quickly, i.e. the response of electrons and dusts look like that in the unmagnetised plasma.

## 3 Solitary Wave in Unmagnetised Plasmas

The nonlinear dynamics of the DIA SWs propagating in such a multi-component dusty plasma is governed by

$∂nni∂t+∇⋅(nniuni)=0,$(2)

$∂npi∂t+∇⋅(npiupi)=0,$(3)

$∂uni∂t+(uni⋅∇)uni=∇ψ,$(4)

$∂upi∂t+(upi⋅∇)upi=−β∇ψ,$(5)

$∇2ψ=μ0ne+nni−μpinpi−jμd,$(6)

where nni(npi) is the negative (positive) ion number density normalised by its equilibrium value nni0(npi0), uni(upi) is the negative (positive) ion fluid speed normalised by Cni=(kBTe/mni)1/2, with kB as the Boltzman constant, Te is the temperature of electrons, and mni being the rest mass of negative ions. ψ is the DIA wave potential normalised by kBTe/e, with e being the magnitude of the charge of an electron. ωcni is the ion (negative) cyclotron frequency (eB0/mnic) normalised by plasma frequency ωp=(4πnn0e2/mni)1/2 with c being the speed of light. β is the mass ratio of negative ions to positive ion (mni/mpi). The time variable (t) is normalised by ωp−1 and the space variables are normalised by Debye radius λD=(kB/Te/4πnnie2)1/2.

At equilibrium,

${n}_{pi}+j{n}_{d0}={n}_{e0}+{n}_{ni0},$

where, jnd0=npdnnd with npd being the positive dust number density and nnd being the number density of negative dust. j=1 for npd>nnd and j=−1 for npd<nnd, i.e. the value of j is dependent on the net charge of the dust grain. We can also write

$μ0=μpi+jμd−1,$(7)

where, μ0=ne0/nni0, μpi=npi0/nni0, and μd=nd/nni0.

To derive a dynamical equation for the nonlinear propagation of the electrostatic waves in a dusty plasma using the reductive perturbation technique [44] the following stretched coordinates [21] are introduced

$ξ=ϵ12(x−V0t),$(8)

$τ=ϵ32t,$(9)

where ϵ is a smallness parameter (0<ϵ<1) measuring the weakness of the dispersion and V0 is the Mach number (the phase speed normalised by Cni). nni, uni, and ψ can be expanded about their equilibrium values in a power series of ϵ, viz.,

$nni=1+ϵnni(1)+ϵ2nni(2)+⋯,$(10)

$npi=1+ϵnpi(1)+ϵ2npi(2)+⋯,$(11)

$uni=ϵuni(1)+ϵ2uni(2)+⋯,$(12)

$upi=ϵupi(1)+ϵ2upi(2)+⋯,$(13)

$ψ=ϵψ(1)+ϵ3/2ψ(2)+⋯.$(14)

Using the stretched coordinates and power series expansion [as shown in (10)–(14)] in (2)–(6); and equating the coefficients of ϵ3/2 from the continuity and momentum equation and coefficients of ϵ from Poisson’s equation one can obtain the linear dispersion relation for the DIA SWs in unmagnetised plasmas as

$V0=1μ0(k−32k−12)(βμpi+1).$(15)

It can be seen from (15) that the presence of the super-thermal electrons significantly modifies the linear dispersion relation. Compering with the dispersion relation (16) of Haider et al. [27] it can be said that the presence of super-thermal electrons reduce the phase speed of the DIA waves.

Equating the next higher order co-efficient of ϵ from above equations and using the parameters we can finally obtain a K-dV equation describing the nonlinear propagation of the DIA SWs in the dusty plasma

$∂ψ(1)∂τ+Aψ(1)∂ψ(1)∂ξ+B∂3ψ(1)∂ξ3=0,$(16)

where the nonlinear coefficient A and the dissipation coefficient B are given by

$A=3(βμpi−1)2V0(βμpi+1)−V02(κ+12),$(17)

$B=12μ0V0(κ−32)(κ−12).$(18)

The stationary SW solution of (16) can be found by introducing a new coordinate

$ζ=ξ−uτ,$(19)

where, u is the wave speed (in the reference frame) normalised by Cni and ζ is normalised by λD.

By imposing the appropriate boundary conditions, namely ψ(1)→0, (1)/→0, d2ψ(1)/2→0 at ζ→±∞. Thus, the steady state solution [21] of the K-dV equation, shown in (16), can be expressed as

$ψ(1)=ψmsech2(ζ/Δ),$(20)

where, amplitude (ψm) and width (Δ) of the SWs respectively are

$ψm=2V0uβμpi−1βμpi+1−V023(κ+12),$(21)

$Δ=2μ0V0u(κ−32)(κ−12).$(22)

## 4 Solitary Wave in Magnetised Plasmas

The nonlinear dynamics of the DIA SWs propagating in such a MI dusty plasma is governed by equations of continuity and poission’s equation denoted in (2), (3), and (6), respectively and momentum equations for magnetised case are given below

$∂uni∂t+(uni⋅∇)uni=∇ψ−ωcniuni×k^,$(23)

$∂upi∂t+(upi⋅∇)upi=−β∇ψ+βωcniupi×k^,$(24)

where, ωcni is the ion (negative) cyclotron frequency (eB0/mnic) normalised by plasma frequency ωp=(4πnn0e2/mni)1/2 with c being the speed of light.

To derive a dynamical equation for the nonlinear propagation of the electrostatic waves in a dusty plasma using the reductive perturbation technique [44] the following stretched coordinates [30], [45] are introduced

$ξ=ϵ1/2(lxx+lyy+lzz−V0t),$(25)

$τ=ϵ3/2t,$(26)

where ϵ is a smallness parameter (0<ϵ<1) measuring the weakness of the dispersion, lx, ly, and lz are the direction cosine of the wave vector k along the x, y, and z directions, respectively so that ${l}_{x}^{2}+{l}_{y}^{2}+{l}_{z}^{2}=1.$

nni, npi, and ψ can be expanded as in (10), (11), (14), respectively. The components of uni and upi can be expressed as about their equilibrium values in a power series of ϵ, viz.,

$unix=ϵ3/2unix(1)+ϵ2unix(2)+⋯,$(27)

$upix=ϵ3/2upix(1)+ϵ2upix(2)+⋯,$(28)

$uniy=ϵ3/2uniy(1)+ϵ2uniy(2)+⋯,$(29)

$upiy=ϵ3/2upiy(1)+ϵ2upiy(2)+⋯,$(30)

$uniz=ϵuniz(1)+ϵ2uniz(2)+⋯,$(31)

$upiz=ϵupiz(1)+ϵ2upiz(2)+⋯,$(32)

Using the stretched coordinates [given in (25) and (26)] and (10), (11), (14), and (27)–(32) in (2), (3), (6), (23), and (24); and also equating the first order coefficients of ϵ and rearranging them one can obtain the dispersion relation for the DIA SWs containing external magnetic field as

$V0=lz1μ0(κ−32κ−12)(βμpi+1).$(33)

Now using equating the second order coefficient of ϵ, one can obtain a K-dV equation describing the nonlinear propagation of the DIA SWs in the dusty plasma

$∂ψ(1)∂τ+Aψ(1)∂ψ(1)∂ξ+B∂3ψ(1)∂ξ3=0,$(34)

where the nonlinear coefficient A and the dissipation coefficient B are given by

$A=3lz2(βμpi−1)2V0(βμpi+1)−V02(κ+12),$(35)

$B=12μ0V0(κ−32)(κ−12)[1+1−lz2ωcni2(1+μpiβ)].$(36)

The stationary SW solution of K-dV equation shown in (20) with amplitude (ψm) and width (Δ) of the SWs, respectively, are

$ψm=2V0ulz2(βμpi−1βμpi+1)−V023(κ+12),$(37)

$Δ=2μ0V0u(κ−32)(κ−12)[1+1−lz2ωcni2(1+μpiβ)].$(38)

## 5 Shock Wave in Magnetised Plasmas

The nonlinear dynamics of the DIA shock waves propagating in such a multi-component dusty plasma is governed by equations of continuity and poission’s equation denoted in (2), (3), and (6), respectively, and momentum equations for magnetised case are given below.

$∂uni∂t+(uni⋅∇)uni=∇ψ−ωcniuni×k^+η∇2uni,$(39)

$∂upi∂t+(upi⋅∇)upi=−β∇ψ+βωcniupi×k^+η∇2upi,$(40)

where, η is the viscuss term, i.e. coefficient of viscosity normalised by its equilibrium value η0.

One can obtain a K-dVB equation describing the nonlinear propagation of the DIA shock waves in MI dusty plasma [46]

$∂ψ(1)∂τ+Aψ(1)∂ψ(1)∂ξ+B∂3ψ(1)∂ξ3=C∂2ψ(1)∂ξ2,$(41)

where the nonlinear coefficient A is same as SWs in magnetised plasmas denoted in (35) and C are given by

$C=η2.$(42)

The stationary shock wave solution of (41) can be found by introducing a new coordinate ζ=ξ (where u is the wave speed (in the reference frame) normalised by Cni and ζ is normalised by λD) and by imposing the appropriate boundary conditions, namely ψ(1)→0, (1)/→0, d2ψ(1)/2→0 at ζ→±∞. Thus, the steady state solution of the K-dVB equation [as shown in (41)] [46], can be expressed as

$ψ(1)=ψm[1−tanh(ζ/Δ)],$(43)

where, amplitude (ψm) and width (Δ) of the shocks, are respectively

$ψm=2V0ulz2(βμpi−1βμpi+1)−V023(κ+12),Δ=ηu.$(44)

It is seen that the amplitude of the shock waves is similar that of SWs, as shown in (37).

## 6 Results and Discussions

The nonlinear propagation of DIA solitary and shock waves in multi-component dusty plasmas have been analyzed theoretically, containing positively and negatively charged inertial ions, opposite polarity dusts, and high energetic super-thermal electrons. We have derived K-dV and K-dVB equations using a reductive perturbation method and their solutions for unmagnetised and magnetised plasmas system.

It can be find out form the solutions of unmagnetised SWs and magnetised solitary and shock waves that both the solitary and shock waves might be associated with positive or negative potentials depending on the the nonlinear coefficient (A). There must be a critical value of each parametric regimes, considering other are constant, below or after which the solitary or shocks to be rarefactive or compressive. It is found that the amplitude is same for solitary and shock waves. Equation (37) represents the amplitude of the solitary and shock wave with presence of mangetic field. If we consider the value of direction cosine (lz)=1, (37) reduced to (21), i.e. nonlinear coefficient for the case of unmagnetised system. Figure 1 represents the A=0 surface plot for magnetised plasma system showing the variation of κ with μpi and μd for for u=0.1, δ=60°, and β=1. The upper surface is for j=1 and the lower one is for j=−1. Below the surfaces the solitary and shock waves might be associate with positive potentials and above the surfaces the solitary and shock waves might associate with negative potentials. The surface plot indicates the critical values of the parametric regimes. From the numerical analysis and figures it is found that the amplitude goes infinity at the surface. The reflection of the statement found in Figures 5 and 6 (describe latter) in which amplitude goes to infinity at the critical value. On the other hand, same situation is found at δ → 90°. That is, at the critical values of parametric regimes no solitary or shock waves are found according to our present theory. Modified K-dV [47] or Gardnar [48] or modified Gardnar [49] equations or fully electromagnetic structure estrade of electrostatic structure might applicable to explain the situations.

Figure 1:

A=0 surface plot, showing the variation of super-thermal index (κ), with μd and μpi for u=0.1, δ=60°, β=1, j=1 (smooth surface) and −1 (lined surface).

The effects of the parametric regimes on solitary and shock waves are discussed below.

## Super-Thermal Electrons

The super-thermal electron which follows the kappa distribution [32], [34], [35], [36] is responsible for the solitary and shock structures to be associated with positive or negative potentials. This super-thermality of electrons modifies the wave properties like dispersion relation, amplitude, width, etc. Figure 2 represents the variations of wave potentials with ξ for various value of κ with u=0.1, β=1, μd=1.2, and μpi=3. The solid curve represents the value of j=1 and dashed curve is for the value of j=−1. It is found from the figure that, below the critical value of κ the SWs associated with positive potentials and both amplitude and width of the SWs increases with increasing the super-thermal index κ for j=1. Above the critical value the negative potential the SWs associate and both amplitude and width decreases with increasing the value of κ. The SWs goes negative potentials for the above values of κ with j=−1.

Figure 2:

Showing the variation of ψ and ξ for the values of u=0.1, β=1, ud=1.2, upi=3, with j=1 (solid curve) and j=−1 (dashed curve). Here blue curve represents κ=2, green curve is for κ=2.1, red is for κ=2.2, yellow curve shows κ=2.3, black curve is for κ=2.4 and orange curve shows the variation for κ=2.5.

## Magnetic Field

It is already known that the magnetic field modifies the wave properties in plasmas in various way [21], [27], [28], [29], [30], [31], [45], [46]. In the case of unmagnetised plasmas the wave propagation can be considered one directional. So, the stretched coordinate as shown in (8) is sufficient to derive K-dV equation, but with the presence of magnetic field (8) is not valid and one have to introduced the stretched coordinate shown in (25) which contains direction cosines. This can effect not only the amplitude and width of the wave profile but also the dispersion relation. If we consider lz=1 the amplitude, width and dispersion relation of magnetised plasmas reduced to unmagnetised situation. Figure 3 represents the variation of ψm with δ considering u=0.1, k=2, β=1, μd=1.2, μp=5 and j=1(−1). It has been found from the figure that the amplitude increases with increasing the value of angle (δ) for j=1 but decreases for j=−1. Again Figure 4 shows the variation of width of SWs with angle of propagation for various value of ωcni which indicates that the width increase with δ for lower range and achiving a maximum width it started to decreases and goes to zero at δ=90°.

Figure 3:

Variation of amplitude (ψm) of the solitary and shock waves with δ for u=0.1, κ=2, μd=1.2, μpi=5, β=1, j=1 (solid curve) and j=−1 (dashed curve).

Figure 4:

Variation of the width (Δ) of SWs with and δ for u=0.1, μd=1.2, μpi=4, β=1, κ=2, j=1 (solid curve) and j=−1 (dashed curve). Here ωcni=0.4 (blue), ωcni=0.6 (red), ωcni=0.8 (green) and ωcni=1 (black).

When magnetic field is introduced in plasma system the charged particle starts to gyrate with gyrating frequency or cyclotron frequency (ωcni). From above numerical analysis it is found that there is no direct effect of ωcni in amplitude of SW [as in (35)] but it plays a vital role on the width of SWs [as in (36)]. It is observed from the Figure 4 that increasing the value of ωcni the width of the SWs decreases for both the cases j=1 and j=−1, which means that the magnetic field makes the solitary structure more spiky. Similar observations was found previously for different magnetised plasma system [27], [28], [29], [30], [31], [46], [50]. It is also found that above analysis that the magnetic field only introduces direction cosine in shock waves, but amplitude and width of the SWs are independent of ωcni.

## Ions

Ion number density plays a vital role in the present DIA solitary and shock waves to be rarefactive or compressive. There must be a critical value of μpi, as discussed earlier, for both j=1 and j=−1. Figures 5 and 6 show the variation of amplitude of the solitary and shock waves with μpi and μd for both j=1 (Fig. 5) and j=−1 (Fig. 6). From these figures it can be said that below the critical value of μpi the solitary and shock waves associated with negative potentials and amplitude increases in negative direction with increasing the value of μpi. Similar observations are found earlier by Haider [30] and Haider and Mamun [50]. The solitary and shock waves with positive potentials are found after the critical value and decreases with increasing the value of μpi. The criticality is obtain at the higher value of μpi for j=1 than j=−1. It is also seen from the Figures 7 and 8 that the width decreases with μpi for both j=1 and −1.

Figure 5:

Variation of the amplitude (ψm) of solitary and shock waves with μd and μpi for the values of u=0.1, κ=2, β=1, δ=60° and j=1.

Figure 6:

Variation of the amplitude (ψm) of solitary and shock waves with μd and μpi for the values of u=0.1, κ=2, β=1, δ=60° and j=−1.

Figure 7:

Variation of the width of SWs with μd and μpi for κ=2, j=1, l=0.5, u=0.1, ωcni=0.5 and β=1.

Figure 8:

Variation of the width of SWs with μd and μpi for κ=2, j=−1, l=0.5, u=0.1, ωcni=0.5 and β=1.

## Dusts

There are some heavy charged particles have been considered in the system called dusts. The dusts are positively and negatively charged. The net charged density of the dusts is denoted by j. It is observed that the critical value surface plots have changed, as shown in Figures 1, 5, and 6, depending on the value of j; i.e. the critical value of each parameters is different for j=1 and −1. The critical value surface for j=1 is higher than j=−1 surface plot. So there is a region where solitary or shock waves associate with positive or negative potentials depending only the value of j. From the Figures 58 it is found that the amplitude of the solitary and shock waves increase with μd for both j=1 and −1 but the width decreases with it for j=1 and increases with j=−1. It means that the profile of the solitary or shock waves with positively net charged dusts will be taller and narrower than negatively net charged dusts.

## Coefficient of Viscosity

The amplitude of the shock waves independent of coefficient of viscosity, but it is seen form (44) that the width of the shock waves is linear function of η. It means that steeper shock waves might be found out for the lower value of coefficient of viscosity.

Finally, the present work can provide a guideline to explain the solitary structure of the Earth’s ionosphere, planetary atmospheres, planerary rings, interstellar and circumstellar clouds, asteroid zones, protostellar disks, cometary tails, nebula as well as plasma crystal and microelectronics plasma processing reactors which will be able to detect the DIA solitary structures and to identify their basic features predicted in this theoretical investigation.

## Acknowledgement

The authors would like to thankfully acknowledges “Mawlana Bhashani Science and Technology University Research Cell” for granting a research fund for successfully completing the work. The author would like to thank the referees for their valuable suggestions and comments which help to improve the original manuscript.

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Accepted: 2017-05-15

Published Online: 2017-06-16

Published in Print: 2017-07-26

Citation Information: Zeitschrift für Naturforschung A, Volume 72, Issue 7, Pages 627–635, ISSN (Online) 1865-7109, ISSN (Print) 0932-0784,

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©2017 Walter de Gruyter GmbH, Berlin/Boston.