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Publicly Available Published by De Gruyter January 30, 2018

Dust Ion-Acoustic Shock Waves in a Multicomponent Magnetorotating Plasma

  • Barjinder Kaur and N.S. Saini EMAIL logo

Abstract

The nonlinear properties of dust ion-acoustic (DIA) shock waves in a magnetorotating plasma consisting of inertial ions, nonextensive electrons and positrons, and immobile negatively charged dust are examined. The effects of dust charge fluctuations are not included in the present investigation, but the ion kinematic viscosity (collisions) is a source of dissipation, leading to the formation of stable shock structures. The Zakharov–Kuznetsov–Burgers (ZKB) equation is derived using the reductive perturbation technique, and from its solution the effects of different physical parameters, i.e. nonextensivity of electrons and positrons, kinematic viscosity, rotational frequency, and positron and dust concentrations, on the characteristics of shock waves are examined. It is observed that physical parameters play a very crucial role in the formation of DIA shocks. This study could be useful in understanding the electrostatic excitations in dusty plasmas in space (e.g. interstellar medium).

1 Introduction

It is well known that dust particulates are ubiquitous components of space and astrophysical plasma environments such as cometary tails and comae, interstellar clouds, Earth’s mesosphere and ionosphere, Saturn’s rings, and the gossamer ring of Jupiter, as well as in laboratory devices [1]. The presence of dust in an electron-ion (e-i) plasma generates new modes. Dust ion-acoustic (DIA) mode is one such new mode with high (low) frequency in contrast to dust acoustic (ion-acoustic) mode (i.e. csd,csi<ωk<cse, where csd, csi, and cse are thermal velocities of dust, ions, and electrons). Dust ion-acoustic solitary waves (DIASWs) have been explored widely by a number of researchers [2], [3], [4], [5]. Because of the presence of negatively (positively) charged dust grains, the speed of DIASWs in an e-i dusty plasma is larger (smaller) than the usual ion-acoustic speed. Not only dust but any other species in the plasma system can affect the propagation of the wave. Positron is one of such species that significantly affects the characteristics of plasma waves [6], [7], [8]. Positron concentration decreases the number density of ions and the restoring force on the electron fluid, which modifies the linear and nonlinear wave structures. It was suggested theoretically by Surko and Murphy [9] that the annihilation time of electrons and positrons is much larger than the characteristic time scale of the ion-acoustic wave. Positrons can be used to probe particle transport in tokamaks, and because of its sufficient life time, the two-component, i.e. e-i, plasma becomes a three-component electron-positron-ion (e-p-i) plasma. Kotani et al. [10] confirmed the existence of the e-p-i plasma, which was also confirmed experimentally by various authors [11], [12], [13]. The four-component plasma composed of electrons, ions, dust, and positrons is believed to exist in regions like galactic nuclei, pulsar magnetospheres, interstellar clouds, supernova environments, etc., as well as in laboratory experiments of cluster explosions by intense laser beams [14], [15], [16], [17]. In the past, many researchers have studied the propagation characteristics of solitary structures in four-component e-p-i-d plasmas [4], [18], [19]. Ghosh and Bharuthram [18] observed for the first time the nonlinear propagation of ion-acoustic solitons and double layers in an e-p-i-d plasma with Boltzmann-distributed electrons and positrons. Both compressive/rarefactive solitons and weak double layers were examined. The properties of fully nonlinear DIA waves in a multicomponent plasma in the presence of warm ions, dust, superthermal electrons, and positrons were studied by El-Tantawy et al. [19]. Saini et al. [4] studied the propagation properties of large-amplitude DIA solitary waves and double layers in an e-p-i-d plasma. They observed that compressive and rarefactive solitons were formed in such a plasma system.

Different investigators have reported the study of nonlinear wave propagation in highly rotating and magnetized plasmas because of their applicability in astrophysical and fusion devices [20], [21]. Rotation in plasma dynamics was considered for the first time by Chandrasekhar [20]. It was found that Coriolis force plays a very crucial role in cosmic phenomena, which was indeed supported by subsequent studies on astrophysical plasma environments [21]. Rotation plays a very important role in the interaction of the weak magnetic field and Coriolis force in the interior of Sun for long waves. The magnitude of the Coriolis force is very small but plays an effective role in plasma waves. The effect of the Coriolis force on pulsar radiations was studied by Mamun [22]. He showed that with increase in rotation, narrow wave packets are formed, which give rise to soliton radiation known as pulsar radiation. Several authors have studied the theory of nonlinear wave propagation in rotating plasmas by considering the Coriolis force as part of rotation [23], [24], [25]. Mushtaq [26] reported the effect of the angle of rotation θ in a magnetized e-i rotating plasma in the presence of untrapped and trapped electrons. By using a reductive perturbation technique, the authors derived the KdV and Schamel’s modified KdV equations. It was reported that for a small angle of rotation θ, the rotational frequency and magnetic field significantly affect the width of solitons but have no direct effect on the amplitude of the soliton. Saini et al. [27] investigated the propagation properties of solitary waves and double layers in an obliquely propagating DIA wave in an e-p-i-d magnetized plasma. The authors analysed the influence of the angle of rotation θ and other plasma parameters on solitary waves and double layers.

It is well known that deviations from the Maxwell–Boltzmann distribution may affect and modify the existing domain of nonlinear structures. Nonextensive statistical mechanics, which is based on the deviations of Boltzmann–Gibbs–Shanon (BGS) entropic measure, has been given much attention in last few decades. For the first time, Renyi [28] recognized the nonextensive generalization of BGS entropy, which was extended to the nonlinear nonextensive case by Tsallis [29]. Tsallis modelled nonextensivity by assuming a composition law in the sense that the entropy of the composition (G+H) of two independent systems A and B is equal to Sq(G+H)=Sq(G)+Sq(H)+(1q)Sq(G)Sq(H). The parameter q is related to the underlying dynamics of the system, supports the generalized entropy of Tsallis, and also provides a measure of the degree of its correlation. Lima et al. [30] explained the importance of the q-nonextensive formulism for a system having long-range interactions. The q-nonextensive distribution is a more generalized distribution than the Maxwellian distribution, and the result leads to the Maxwellian limit for q→→ 1. To model the effect of nonextensivity, we refer to the one-dimensional q-distribution function as

(1)fα(vx)=Cqα(1(qα1)[mαvx22Tα±eϕTα])1qα1,

where qα stands for the strength of nonextensivity and α=i, e, and the ± sign stands for ions and electrons.

The constant of normalization Cqα for −1<qα<1 is

Cqα=nα0Γ(11qα)Γ(11qα12)[mα(1qα)2πTα]12,

and for qα>1 it is

Cqα=nα0(qα+12)Γ(1qα1+12)Γ(1qα1)[mα(qα1)2πTα]12.

Here, Γ is the standard gamma function. Using the limit qα→ 1, Γ(1qα1+12)/Γ(1qα1)exp(12ln1qα1)=1, it is seen that Cqα approaches the standard Maxwellian limit with value mα2πTα [31]. The integration of (1) yields the number density of electrons/ions as

nα=nα0[1±(qα1)eϕTα](qα+1)2(qα1).

During the last many years, a large number of investigations have been reported in the framework of Tsallis q-distribution [32], [33], [34], [35], [36]. Hussain et al. [37] studied the solitary wave structures in a two-component nonextensive magnetorotating plasma. They concluded that the the amplitude and width of solitary structures are minimum for q>1 and observed dip solitary structures for negative values of q. Gill et al. [38] performed the stability analysis in a nonextensive magnetized e-p-i plasma. The authors examined the effect of nonextensivity of electrons and positrons by deriving the Zakharov–Kuznetsov (ZK) equation. The balance between nonlinearity and dispersion always supports a solitary wave; however, a medium having significant dispersion and dissipation effects supports the formation of shock waves instead of solitons. Shock waves are also formed as a result of turbulence, by supernovae, by distribution of high-energy charged particles, or by the effects of a magnetic field. Other sources of occurrence of shocks are solar flares, black holes, and high-density objects such as pulsars and merging galaxies. They dissipate their energy and accelerate the electrons and ions present in space and astrophysical regions. Shock waves are also known as kinks because they are monotonic changes in the physical parameters from one extreme value to another. During the last four decades, numerous researchers have shown interest in interpreting the solitons and shocks in different plasma systems [39], [40], [41], [42], [43], [44]. Alam et al. [45] investigated the propagation properties of DIA shock waves in an unmagnetized plasma containing cold ions with kinematic ion viscosity, superthermal electrons, and dust particulates with two distinct temperatures. Shahmansouri and Mamun [46] studied obliquely propagating ion-acoustic shock waves in a magnetized plasma with a cold viscous ion fluid and Maxwellian electrons. From the solution of KdV Burger equation, they illustrated the combined effects of the external magnetic field and obliqueness on amplitude and width of ion-acoustic shocks. The characteristics of dust acoustic shock waves in a magnetized plasma comprising of negatively charged dust fluid with κ-distributed electrons and ions were investigated by Chahal et al. [47]. A nonlinear Korteweg–de Vries–Burgers (KdVB) equation was derived using the reductive perturbation technique, and only negative potential shock structures were observed. Three-dimensional ion-acoustic solitary and shock waves in an e-p-i plasma with high superthermal electrons and positrons were reported by El-Bedwehy and Moslem [48]. They derived Zakharov–Kuznetsov–Burgers (ZKB) equation to examine the dependence of different plasma parameters on the shock waves. Only positive potential shocks were observed. Recently, El-Tantawy [49] examined the features of DIA shock waves in a magnetized plasma containing nonextensive electrons, positive ions, and dust particles. The ZKB equation was derived using the reductive perturbation technique to see the influence of different plasma parameters on DIA shock waves. It was found that both positive and negative polarity shocks exist. To the best of our knowledge, no theoretical investigation focusing on the study of DIA shocks has been reported by considering nonextensive electrons and positrons in a dusty magnetorotating plasma with cold viscous ion fluids. Therefore, the aim of our study is to understand the underlying physics of DIA shock structures in a rotating magnetized plasma containing cold ions with kinematic ion viscosity, nonextensive electrons and positrons, and immobile dust particulates.

The organization of the rest of the paper is as follows. In Section 2, fluid equations are discussed. In Section 3, by adopting a reductive perturbation method, the ZKB equation is derived. Parametric analysis with combined effects of different plasma parameters on the characteristics of DIA shocks is discussed in Section 4. The last section is devoted to conclusions.

2 The Fluid Equations

We consider a three-dimensional, magnetorotating, four-component plasma system consisting of fluid ions, nonextensively distributed electrons and positrons, and negative dust grains. This plasma system is assumed to be immersed in an external magnetic field B=B0. The plasma is rotating with angular frequency Ω0 around an axis in the xz plane, making an angle θ with axis of the magnetic field B under the effect of the Coriolis force. The normalized fluid equations in rotating frame of reference can be written as

(2)nit+(niui)=0,
(3)uit+(ui)ui=ϕ+Ω(ui×z^)+ηi2ui,
(4)2ϕ=nenpni+δd.

Charge neutrality condition at equilibrium yields

μ=1+δpδd

2=(2x2+2y2+2z2).μ=ne0ni0 is the electron to ion number density ratio, δp=np0ni0 is the positron concentration, and δd=zd0nd0ni0 is the dust concentration. ni is the number density of ions normalized by its corresponding equilibrium value, ui is the ion fluid velocity normalized by ion-acoustic speed cs=Timi, and ϕ(=eϕTi) is the normalized electrostatic wave potential. Also, Ω=ωci+2Ω0, where Ω0 is the rotational frequency and ωci=eBmic (mi is the mass of ions, B is the magnitude of ambient magnetic field, and c is the speed of light) is the ion gyrofrequency; both are normalized by ion plasma frequency ωpi=4πni0e2mi. Time and space coordinates have been normalized with respect to the inverse plasma frequency ωpi1 and Debye length λD, respectively. ηi=ηi0ωpiλD2 is the normalized ion kinematic viscosity, where ηi0 is the unnormalized ion kinematic viscosity and λD=(Ti4πni0e2)12. The rotation of the plasma is considered to be slow (Ω0<1) so that quadratic and higher terms in Ω0 [such as centrifugal force Ω0×(Ω0×r)] may be neglected. Further, because of the slow rotation of the plasma, only the influence of the Coriolis force on dynamic system is considered.

In the components form, (2) and (3) can be written as

(5)nit+(niu)x+(niv)y+(niw)z=0,
(6)ut+uux+vuy+wuz=ϕx+Ω0v+η2ux2,
(7)vt+uvx+vvy+wvz=ϕyΩ0u+η2vy2,
(8)wt+uwx+vwy+wwz=ϕz+η2wz2.

The normalized Poisson’s equation can be written as

(9)2ϕx2+2ϕy2+2ϕz2=nenpni+δd.

Further, np and ne are the number densities of positrons and electrons, which are normalized by their equilibrium values. The normalized number densities of q-nonextensive distributed electrons (ne) and positrons (np) are

(10)ne=μ[1+(qe1)ϕ]qe+12(qe1)

and

(11)np=δp[1(qp1)σϕ]qp+12(qp1),

where σ=TeTp.Te and Tp are the temperature of electrons and positrons, respectively. It is reported that for the requirement of finite energy in the superextensive range −1<q<1, the allowed range shrinks to 13<q<1 [50]. Thus, in this study we have considered the range 13<q<1 for numerical analysis. Using Taylor expansion of ne and np from (10) and (11) in (9), Poisson’s equation is modified as

(12)2ϕx2+2ϕz2=1+A1ϕ+A2ϕ2ni,

where

(13)A1=μ[qe+12+σδp(qp+12)]

and

(14)A2=μ[(qe+1)(3qe)8σ2δp(qp+1)(3qp)8].

3 Derivation of Zakharov–Kuznetsov–Burgers Equation

We employ the reductive perturbation technique to derive the ZKB equation and choose following stretching coordinates:

(15)X=ϵ1/2x,   Y=ϵ1/2y,   Z=ϵ1/2(zλt),ηi=ϵ1/2η, and τ=ϵ3/2t,

where ϵ is a small parameter characterizing the strength of the nonlinearity and λ is the phase velocity.

The expansion of the perturbed quantities can be written as

(16)ni=1+ϵni1+ϵ2ni2+u=ϵ32u1+ϵ2u2+v=ϵ32v1+ϵ2v2+w=ϵw1+ϵ2w2+ϕ=ϵϕ1+ϵ2ϕ2+ϵ3ϕ3+

Using the stretching coordinates from (15) and the expanded variables from (16) in (5)–(8) and (12), after collecting the terms of the different order of ϵ, we obtain the evolution equations in different orders. At lowest order of ϵ, we obtain first-order expressions as

ni1=ϕ1λ2, u1=1Ωϕ1Y

and

(17)v1=1Ωϕ1X,w1=ϕ1λ.

From the first-order expressions, the phase velocity of the DIA waves can be determined as

(18)λ=[μ(qe+12+σδp(qp+12))]12.

We see from (18) that the phase velocity explicitly depends on various physical parameters, namely qe, qp, δd, δp, σ, η, and Ω0. So it is important to analyse the effect of these parameters on the phase velocity, which is shown in Figure 1.

Figure 1: Phase velocity λ against dust concentration (δd) and nonextensivity of electrons (qe) for ωci=0.3, Ω0=0.1, η=0.1, qp=0.6, σ=0.9, and δp=0.16.
Figure 1:

Phase velocity λ against dust concentration (δd) and nonextensivity of electrons (qe) for ωci=0.3, Ω0=0.1, η=0.1, qp=0.6, σ=0.9, and δp=0.16.

For the next higher order of ϵ, we obtain

(19)ni1τλni2Z+u2X+v2Y+w2Z+(ni1w1)Z=0,
(20)w1τλw2Z+w1w1Z=ϕ2Z+η2w1Z2,
(21)v2=λΩu1Zandu2=λΩv1Z.

By eliminating second-order quantities from (12) and (19)–(21), we obtain the ZKB equation in a magnetized rotating plasma as

(22)ϕ1τ+Aϕ1ϕ1X+B3ϕ1X3+CX(2ϕ1Y2+2ϕ1Z2)D(2X2+2Y2+2Z2)ϕ1=0,

where

(23)A=B[2A2+3A12],
(24)B=λ32,   C=λ32[1+1Ω2] and  D=η2.

A is the nonlinear coefficient, B is the dispersion coefficient, C is the higher order coefficient, and D is the dissipation coefficient. For the case of no positrons (δp=0, σ=0) and non-rotating fields (Ω0=0), the expressions obtained by Tantawy [49] are retrieved. We observe that in (22) the kinematic viscosity appears only in the last term, and for η=0, (22) becomes the well-known ZK equation. In our case, because of the presence of the dissipative term, some of the system energy gets lost, which leads to the formation of shock waves. So the existence of solitons is considerably affected by the presence of the dissipative coefficient. To obtain the solution of (22), we introduce a single variable transformation χ=lX+mY+nZ for co-moving frame. Here, l, m, and n are the direction cosines, and U is the velocity of the moving frame. By considering Φ(χ)=ϕ1(X, Y, Z, τ), (22) leads to the ordinary differential equation

(25)UdΦdχ+AlΦdΦdχ+lPd3Φdχ3Dd2Φdχ2=0,

where

P=Bl2+C(m2+n2).

For the limiting case, i.e. when dispersion is dominant (when D → 0), and using the boundary conditions Φ → 0, dΦdχ,d2Φdχ20 at χ →±∞, we get the steady-state solitary solution of (25) as

(26)Φ=Φ0sech2(χwm),

where the maximum amplitude Φ0 and width wm of the solitary waves are given, respectively, by

(27)Φ0=3UAl     and     wm=2lPU

Several techniques like the Hirota bilinear formalism, the Backlund transformation, and the hyperbolic tangent approach have been reported to solve nonlinear partial differential equations but the hypebolic tangent (tanh) approach [48] is found to be the most convenient technique in systems involving dissipative and dispersive terms. The ZKB equation belongs to the nonintegrable Hamiltonian system and its exact analytical solution is not possible. But by applying the tanh method, one can find the approximate analytical solution of the ZKB equation (22). In the presence of dispersion, shock structures may also show an oscillating and monotonic behaviour, but the hyperbolic tanh approach yields only the monotonic solution of shock structures and does not provide the oscillatory solution [44]. Employing the hyperbolic tanh approach, the solution of (22) is determined as

(28)Φ(χ)=350η2PAl2[1tanh(χW)+12sech2(χW)].

Here, ϕmax=9100(η2PAl2) is the maximum amplitude of the shocks, and W=20Plη is the width of shocks.

4 Parametric Analysis

It is clear from (18), (23), (24), and (28) that the phase velocity, nonlinear coefficient, dispersion coefficient, and shock profile explicitly depend upon various physical parameters. Hence, it is of paramount importance to study the effect of these physical parameters on the different coefficients and nonlinear structures. The polarity of shock structures depends upon the sign of the nonlinear coefficient A, as the other coefficients B, C, and D are always positive. The negative (positive) potential shock structures are formed when A is negative (positive). Figure 1 shows the contour plot the phase velocity λ with dust concentration (via. δd) and the nonextensivity of electrons (via qe). The numbers on the curves in the contour plot show the value of magnitude of the phase velocity λ. It is observed from that with the increase in nonextensivity of electrons (via qe), the phase velocity (λ) of DIA waves decreases, but it increases with the increase of dust concentration (via δd). This is because, for the superextensive range (q<1), the q-distribution in comparison to Maxwellian distribution describes that the system has particles of large energy, i.e. the particles will move faster than their thermal counterparts. To see the polarity of shock waves that is connected with the nature of nonlinear coefficient A, we plot the contour graph of A in the plane of δdqe (see Fig. 2). From this contour plot, we can judge the existence domain of compressive (positive polarity) and rarefactive (negative polarity) shock waves. By keeping other parameters fixed, one can find the critical values of δd and qe for the formation of compressive and rarefactive shock waves. As depicted in Figure 2, in parametric region above the curves, A is negative (A<0), and hence negative potential shock waves are formed for parametric values in this region, whereas below the curves, A is positive (A>0), and positive potential shocks exist. At the curves, A=0, and hence singularity exists. In this case, one can derive the modified ZKB equation using new stretching coordinates to see the existence of shocks for the critical parameters. It is observed that with the increase in positron concentration (δp), the parametric regime for the existence of compressive shock waves increases. For a given set of parameters with δd=0.768, we find numerically the critical value of qe at which polarity of shock waves shift from positive to negative (see Fig. 2). The critical value of qe is 0.4076 and for qe>0.4076 (<0.4076), positive (negative) potential shock waves are formed. From Figure 3, we analyse the effect of nonextensivity of electrons (qe) on shock profile. An increase in qe shows that the amplitude and width of positive potential shock waves are reduced. However, the effect of the nonextensivity of electrons (via qe) on negative potential shock waves is very strong as compared to positive potential shocks. For negative potential shock waves, both maximum amplitude and width of shock waves are enhanced with increase in nonextensivity of electrons (qe). Numerically, we have also determined the critical value of dust concentration that shows the formation of different polarity shock waves. Keeping the parameters fixed at qe=0.408, δp=0.16, we find that for δd≤0.7686, only positive potential shock waves are observed. For δd>0.7686, negative shock waves are observed. Figure 4 shows the effect of dust concentration (via δd) on the characteristics of shock waves. It is seen that with increase in dust density, the maximum amplitude and width of positive potential shock waves increase, whereas for negative potential shock waves, the maximum amplitude and width decrease with the increase in dust concentration. In Figure 5, it is seen that both positive and negative potential shock waves are observed for different values of the rotational frequency Ω0, cyclotron frequency ωci, and viscosity η with a fixed set of parameters obtained from the contour plot of Figure 2. As ωci increases, the amplitude of positive polarity shocks decreases but that of negative polarity shocks increases. Also with increase in Ω0, the amplitude of both polarity shocks increases. For Ω0=0 (no rotational frequency), the amplitude and width of the shock structures for both polarities decrease significantly (see the long dashed curve in Fig. 5). It is also observed that with increase in viscosity η, the amplitude of the both polarity shocks increases whereas the width reduces. Similarly, we have analysed the effect of nonextensivity of positrons (via qp) and positron concentration (via δp) on the characteristics of shock waves (see Fig. 6). It is observed that with increase in both qp and δp, the amplitude of positive potential shocks reduces whereas that of negative potential shocks increases. In Figure 7, we have presented the variation of 3D profiles of positive potential shock structures with dust concentration δd as well as the nonextensivity of electrons qe. It can be inferred from the graphs that with increase in dust concentration, the amplitude of shocks increases whereas with increase in nonextensivity of electrons, the amplitude of positive potential shocks reduces. Similar results are retrieved from Figure 8. It is remarkable that nonextensivity (qe and qp), dust concentration, and other parameters have profound influence on the profile of DIA shock structure.

Figure 2: Contour plot for A=0; dust concentration (δd) vs. nonextensivity of electrons qe for different values of positron concentration δp, with the other parameters same as in Figure 1. δp=0.12 [solid (blue)], δp=0.14 [dotted (black)], and δp=0.16 [dashed (red)].
Figure 2:

Contour plot for A=0; dust concentration (δd) vs. nonextensivity of electrons qe for different values of positron concentration δp, with the other parameters same as in Figure 1. δp=0.12 [solid (blue)], δp=0.14 [dotted (black)], and δp=0.16 [dashed (red)].

Figure 3: Opposite polarity shock wave profile Φ plotted against χ for different values of nonextensivity of electrons qe with δd=0.76, δp=0.16, and fixed values of other parameters as in Figure 1. Positive potential shock wave profile for qe=0.41 [solid (blue)], qe=0.42 [dotted (black)], and qe=0.43 [dashed (red)]. Negative potential shock wave profile for qe=0.35 [solid (blue)], qe=0.36 [dotted (black)], and qe=0.37 [dashed (red)].
Figure 3:

Opposite polarity shock wave profile Φ plotted against χ for different values of nonextensivity of electrons qe with δd=0.76, δp=0.16, and fixed values of other parameters as in Figure 1. Positive potential shock wave profile for qe=0.41 [solid (blue)], qe=0.42 [dotted (black)], and qe=0.43 [dashed (red)]. Negative potential shock wave profile for qe=0.35 [solid (blue)], qe=0.36 [dotted (black)], and qe=0.37 [dashed (red)].

Figure 4: Opposite polarity shock wave profile Φ plotted against χ for different values of dust concentration δd with qe=0.4, δp=0.16, and fixed values of other parameters as in Figure 1. Positive potential shock wave profile for δd=0.7 [solid (blue)], δd=0.71 [dotted (black)], and δd=0.72 [dashed (red)]. Negative potential shock wave profile for δd=0.8 [solid (blue)], δd=0.81 [dotted (black)], and δd=0.82 [dashed (red)].
Figure 4:

Opposite polarity shock wave profile Φ plotted against χ for different values of dust concentration δd with qe=0.4, δp=0.16, and fixed values of other parameters as in Figure 1. Positive potential shock wave profile for δd=0.7 [solid (blue)], δd=0.71 [dotted (black)], and δd=0.72 [dashed (red)]. Negative potential shock wave profile for δd=0.8 [solid (blue)], δd=0.81 [dotted (black)], and δd=0.82 [dashed (red)].

Figure 5: Shock wave profile Φ plotted against χ for different values of rotational frerquency Ω0, cyclotron frequency ωci, and viscosity η with fixed values of the other parameters as in Figure 1. For positive potential shock profile: δd=0.7 and for negative potential shock profile: δd=0.8 at ωci=0.3, Ω0=0.1, η=0.1 [solid (blue)], ωci=0.4, Ω0=0.1, η=0.1 [dotted (black)], ωci=0.3, Ω0=0.2, η=0.1 [dashed (red)], ωci=0.3, Ω0=0, η=0.1 [longdashed (red)] and ωci=0.3, Ω0=0.1, η=0.15 [dotdashed (green)].
Figure 5:

Shock wave profile Φ plotted against χ for different values of rotational frerquency Ω0, cyclotron frequency ωci, and viscosity η with fixed values of the other parameters as in Figure 1. For positive potential shock profile: δd=0.7 and for negative potential shock profile: δd=0.8 at ωci=0.3, Ω0=0.1, η=0.1 [solid (blue)], ωci=0.4, Ω0=0.1, η=0.1 [dotted (black)], ωci=0.3, Ω0=0.2, η=0.1 [dashed (red)], ωci=0.3, Ω0=0, η=0.1 [longdashed (red)] and ωci=0.3, Ω0=0.1, η=0.15 [dotdashed (green)].

Figure 6: Shock wave profile Φ plotted against χ for different values of positron concentration δp and nonextensivity of positrons qp with fixed values of the other parameters as in Figure 1. For positive potential shock profile: δd=0.7 and for negative potential shock profile: δd=0.8 at qp=0.5, δp=0.15 [solid (blue)], qp=0.7, δp=0.15 [dotted (black)], and qp=0.5, δp=0.16 [dashed (red)] with δd=0.7.
Figure 6:

Shock wave profile Φ plotted against χ for different values of positron concentration δp and nonextensivity of positrons qp with fixed values of the other parameters as in Figure 1. For positive potential shock profile: δd=0.7 and for negative potential shock profile: δd=0.8 at qp=0.5, δp=0.15 [solid (blue)], qp=0.7, δp=0.15 [dotted (black)], and qp=0.5, δp=0.16 [dashed (red)] with δd=0.7.

Figure 7: 3D profile Φ vs. χ and qe for different values of δd and other parameters as in Figure 1: (a) δd=0.7 and (b) δd=0.72.
Figure 7:

3D profile Φ vs. χ and qe for different values of δd and other parameters as in Figure 1: (a) δd=0.7 and (b) δd=0.72.

Figure 8: 3D profile Φ vs. χ and δd for different values of qe and the other parameters same as in Figure 1: (a) qe=0.4 and (b) qe=0.42.
Figure 8:

3D profile Φ vs. χ and δd for different values of qe and the other parameters same as in Figure 1: (a) qe=0.4 and (b) qe=0.42.

5 Conclusions

We have studied DIA shock waves in an e-p-i-d magnetized and rotating plasma with electrons and positrons featuring the q-nonextensive distribution. Using a reductive perturbation technique, the ZKB equation was derived to study the DIA shock wave from its solution. The dependence of the shock structures on the various physical parameters was examined numerically. The critical values of nonextensivity of electrons (qe) and dust concentration (δd) were determined to check the existence with polarity of DIA shock structures. In the absence of dust concentration, only positive polarity shock structures are observed. The amplitude and width of DIA shock waves decrease with increase in nonextensivity of electrons qe, nonextensivity of positrons qp, and the positron concentration δp, while they increase with increase in the rotational frequency Ω0, δd, and η. Since shock waves dissipate their energy and accelerate the electrons and ions present in space and astrophysical regions, the findings of our study may be useful to understand the nonlinear electrostatic excitations and the mechanism of acceleration of charged particles in different environments of space/astrophysical dusty plasmas (e.g. interstellar region).

Acknowledgement

This work was supported by the University Grants Commission, New Delhi, India, under DRS-II (SAP) No. F 530/17/DRS-II/2015 (SAP-I).

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Received: 2017-11-05
Accepted: 2017-12-24
Published Online: 2018-01-30
Published in Print: 2018-02-23

©2018 Walter de Gruyter GmbH, Berlin/Boston

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