Solitons and shocks in e-p-i magneto-rotating plasma using Cairns distribution

Chaos, Solitons and Fractals 107 (2018) 13–17

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Solitons and shocks in e-p-i magneto-rotating plasma using Cairns distribution M. Yaqub Khan, Javed Iqbal∗ Department of Mathematics, Riphah International University, Islamabad, Pakistan

a r t i c l e

i n f o

Article history: Received 24 September 2017 Accepted 13 December 2017

Keywords: Solitons and shocks HPM Tanh-method

a b s t r a c t Solitons and shocks formation are studied in a magnetized rotating electron-ion-positron plasma using Cairns distribution. We derive an admitted solitary wave solution KdV equation and an admitted travelling wave solution KdVB equation. We apply HPM technique on derived KdV equation and tanh-method on derived KdVB equation. It is observed that γ = Th /TP , the ratio of electron temperature to positron temperature, and α = n0P /n0h , the ratio of number density of positrons to electrons, affect both the soliton width and amplitude. It is also found that γ = Te /TP , α = n0P /n0h , kinematic viscosity and angular frequency affects the structure of shocks. We have compared our results with publish papers and conclude our results are good. This work may be helpful in order to study the rotating flows of magnetized plasma. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction The study of nonlinear waves has become of great interest in plasma sciences for laboratory and space research. Several authors have investigated the formation of solitons via both reductive perturbation technique and pseudo potential approach in two and three component unmagnetized and magnetized plasmas [1–3]. In plasma, the magnetic field has an important role in the soliton dynamics. The compressive and rarefactive solitons in the magnetized plasma are due to the dispersive effects caused by the external magnetic field. Many authors investigated the trapping effect on electrostatic solitary waves. [4–6]. There are many problems in tokamak physics and in astrophysics where rapidly rotating plasmas are found. To understand the physics of these problems suitable non-inertial frames are used. In the pioneering investigations of Chandrasekhar [7] it was pointed that the force generated from rotation has significant role in plasmas. The rotating flow of plasma has been seen in space plasmas and in geophysical fluids. Many authors showed that the Coriolis force has effect on wave propagation [8–9]. The study of rotating plasmas has become a subject of extensive research due to their applicability in fusion devices and astrophysics [10−11]. The Coriolis force, which is important physical parameters in the plasma system and is generated due to rotation, has significance effect on nonlinear structures of waves. Mushtaq investigate ion acoustic (IA) solitons in rotating magneto-plasmas

∗

Corresponding author.

https://doi.org/10.1016/j.chaos.2017.12.009 0960-0779/© 2017 Elsevier Ltd. All rights reserved.

[12]. He considered that electrons are following Maxwell Boltzmann distribution and conclude that the external magnetic field and magnitude of rotational frequency did not affect directly the soliton amplitude. Mushtaq and Shah [13] studied the propagating two-dimensional IA solitary waves in a relativistic, rotating magnetized electron-positron-ion (e-p-i) plasma. Due to high energy particles the plasmas become nonthermal and such superathermal particles have a significant importance in pulsar magnetosphere. The velocity distribution functions effects the nonlinear structures of waves of plasma. The oscillatory IA shock waves in e-p-i plasma with nonthermal kappa distributed electrons and positrons is investigated by Hussain et al. [14]. Akhtar et al. [15] pointed the effects of ion temperature on modulation instability in e-p-i plasma. Dremin [16] studied the unexpected properties of interaction of high energy protons. Ferdousi et al. [17] derived KdV Burger equation and discussed briefly the effects of electrons and positrons non-extensively and effects of kinematic viscosity on the properties of the IA shocks in non-extensive e-p-i plasma. Cairns et al. [18] demonstrated that the nonthermal distribution of energetic particles can be used to explain IA solitary structures. Many authors used this nonthermal distribution in explaining the features of nonlinear structures of space and astrophysical plasmas [19]. Cairns distribution is used to investigate the dust acoustic solitons. [20]. Cairns distribution is used to study freak waves in a plasma [21]. During the last decades, a great interest is shown in interpreting solitons and shocks in e-i plasma and in e-p-i plasmas with non Maxwellian particles [22]. Most of these studies did not consider the propagation of nonlinear IA structures in three compo-

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nent e-p-i magneto-plasma. Also in these investigations the effects of kappa distributed function and ion kinematic viscosity were discussed. Therefore, the purpose of the present work is to study the formation of the solitons and shocks in rotating magneto plasma composed of inertial ions, Cairns distributed electrons and positrons and taking into account the kinematic viscosity of the ion fluid. Nonlinear evolution equations (NLEEs) are widely used to describe many phenomena in plasma physics, optical fibers, biology etc. So, to find the solutions for NLEEs is of great interest for the understanding of most nonlinear physical phenomena. Homotopy perturbation method (HPM) [23] and tanh-method [24] are widely used in finding the solitary wave solution and travelling wave solution of these NLEEs. Guo et al. [25] studied the solitary and shock waves in e-p-i plasma. But in their study they used the kappa distribution. Hussain [26] studied the effect of spectral index parameter on soliton in magneto-rotating plasma. But in his study he used the kappa distribution and effect of positrons on solitary wave structure is not considered. Hussain et al. [14] studied the dissipative shocks in rotating multicomponent magneto-plasma. But in their study they used kappa distribution. According to our best knowledge no one has studied the solitary wave structures in rotating e-p-i magnetoplasma using Cairns distribution. Cairns distribution can serve as a useful model for the family of non-Maxwellian space plasmas. It has been used in large number of papers. Therefore, to represent as astrophysical plasma, it could be of interest to consider an ep-i plasma having Cairns distributed electrons. It is believed that the presence of Cairns distributed electrons may change the nature of IA structures and allow the existence of structures very like those observed by the Viking and Freja satellites. Moreover it has been found from both theoretical analysis and experimental observation that presence of Cairns distributed electrons modifies the basic features of IA solitary waves [27]. So, it is better to study the solitons and shocks profiles in rotating e-p-i magneto-plasma using Cairns distribution. The organization of paper is as follows: in Section 2 model equations and derivation of KdV equation is presented. Results and discussion is in Section 3, while Section 4 is devoted to conclusion.

In component form (1) and (2) yield

⎫ ∂t ni + ∂x (ni vix ) + ∂z (ni viz ) = 0 ⎪ ⎬ (∂t + vix ∂x + viz ∂z )vix = −∂x φ + viy eff (∂t + vix ∂x + viz ∂z )viy = −vix eff + 20 sinθ viz ⎪ ⎭ (∂t + vix ∂x + viz ∂z )viz = −∂z φ − 20 sinθ viy Here eff = ωci + 20 cosθ The normalized Poisson equation for this system is [30]



 ∂x2 + ∂z2 φ = nh − n p − ni

We consider e-p-i plasma embedded in a uniform magnetic field B0 = B0 zˆ. We assume that magnetized plasma is consisting of cold dynamical ions, non-Maxwellian hot electrons and positrons. The energetic electrons and positrons are assumed to follow the Cairns distribution function. Due to small Coriolis force the plasma is rotating around an axis in xz plane. Also, the plasma is rotating with the angular frequency . In this rotating frame, the continuity equation and momentum equations can be written as

∂t ni + ∇ .(ni vi ) = 0

(1)

(∂t + vi .∇ )vi = −∇ φ + ωci (vi × zˆ) + 2(vi × )

(2)

The quasi-neutrality requires that nio = neo = n0 . Here ni is the ion number density normalized by n0 , vi is the ion fluid velocity  normalized by the Cs = Te /mi , φ = eϕ /Te is the normalized electrostatic wave potential. ωci = eB0 /mi c is the ion gyro-frequency. ∇ andt are the space coordinates normalized by the Debye length λDe = Te /4π n0 e2 . We consider the rotation to be slow in order to neglect the quadratic and higher terms in  = 0 sinθ xˆ + 0 cosθ zˆ.

(4)

The normalized form of number density of electrons and positrons is





nh = 1 − βφ + βφ 2 exp (φ ) = 1 + α1 φ +



φ2 2

+ ...



nP = 1 + βγ φ + βγ 2 φ 2 exp (−γ φ ) = 1 − γ α1 φ +

(5)

γ 2φ2

+ ...

2

(6) Using (5) and (6) in (4) gives



(∂ + ∂ )φ = [α1 (1 + γ )φ + 2 x

2 z

1−γ2 2



φ2

+ . . .] − ni

(7)

Here β = 4α /(1 + 3α ), α1 = 1 − β , φ = eϕ /Te , γ = Th /TP , α = n0P /n0h . For derivation of KdV equation, we consider χ = ε 1/2 lx x + ε 1/2 lz z − ε 1/2 λt , τ = ε 3/2 λt where lx and lz are direction cosines such that lx2 + lz2 = 1 and λ is phase velocity of IA waves. To proceed we assume

⎫

ni = 1 + ni1 + ε 2 ni2 + . . . ⎪ ⎪ ⎬ vix = ε 2 vix1 + ε 3 vix2 + . . . ⎪ 3 5 viy = ε 2 viy1 + ε 2 viy2 + . . . ⎪ viz = viz1 + ε 2 viz2 + . . . ⎪ ⎪ ⎭ φ = φ 1 + ε 2 φ2 + . . .

(8)

Substituting (8) in (3) and in (7) and collecting the lowest order of ɛ yields:

λ

2. Model equations and derivation of KdV equation

(3)

lx λ ∂ φ = viy1 , ∂ v = vix1 , eff χ 1 eff χ iy1 −λ∂χ viz1 + lz ∂χ φ1 = −2viy1 0 sinθ , ni1 = α1 (1 + γ )φ1

ni1 = lz

viz1 ,

Note that the equation lx ∂χ φ1 = viy1 , is not in correct form eff in [14]. Also we have used the assumption of small angle θ and 0 /ωci < 1. Using lowest order of set of these equations, λ is obtained as follows



λ

2

1 2lz lx 0 sinθ = l2 + α1 (1 + γ ) z eff



The next higher order terms in ɛ give a set of equations:

⎫

−λ∂χ ni2 + ∂τ ni1 + lx ∂χ vix1 + lz ∂χ viz2 + lz ∂χ (viz1 ni1 ) = 0 ⎪ ⎪ ⎪ −λ∂χ vix1 + lx ∂ φ2 − viy2 eff = 0 ⎬ −λ∂χ viy1 + vix1 eff − 2viz2 0 sinθ = 0 (9) −λ∂χ viz2 + ∂τ viz1 + viz1 lz ∂χ viz1 + lz ∂χ φ2 + 2viy2 0 sinθ = 0⎪ ⎪

∂χ2 φ1 −

α1 (1−γ 2 ) 2

(φ1 )2 − α1 (1 + γ )φ2 + ni2 = 0

⎪ ⎭

It should be noted that the second equation of this set is not in correct form in [26]. Using this set equations and after some calculations we obtain nonlinear KdV equation.

∂τ φ + Aφ∂χ φ + B∂χ3 φ = 0 where

φ = φ1 ,

(10)

A = λα1 (1 + γ ) + (λ/2 α1 )(γ − 1 )

2 λ (1 + l2x (1 − llxz 20 sinθ 2 α1 (1+γ ) eff eff

))

and

B=

M.Y. Khan, J. Iqbal / Chaos, Solitons and Fractals 107 (2018) 13–17

Now we solve this equation by Homotopy Perturbation method. In Eq. (10) L(φ ) = φτ and N (v ) = Aφφχ + Bφχ χ χ We construct a homotopy (1 − q )[L(v ) − L(φ0 )] + q (L(v ) + N (v ) = 0

i.e. (1 − q )[vt − φ0τ ] + q



vτ + Avvχ + Bvχ χ χ = 0

v 0 τ − φ0 τ = 0 , v 0 ( χ , 0 ) = φ 0 v 1 τ + φ0 τ + A v 0 v 0 + B v 0 χ χ χ = 0 , v 1 ( χ , 0 ) = 0

… … … … … … … … … … … … … … … … … … … … …. … We consider initial approximation φ0 = a sech2 ( χb ), then from above equations we get:

v0 (χ , τ ) = a sech2

χ  b

χ 



χ 

ab A − 12B sech

χ 

φ (χ , τ ) = a sech2

 −3

+2ab

b

+ τ 8ab−3 B sech2



2

ab A − 12B sech

4

χ  b

α1 (1−γ 2 )

χ 

χ 

⎪ ⎭

= φ1 ,

b

b

tanh

tanh

χ 

A = λα1 (1 + γ ) + (λ/2 α1 )(γ − 1 ),

B=

)) and

ηi 0 2

dφ dφ d3 φ d2 φ + Aφ + B 3 −C 2 = 0 dξ dξ dξ dξ Integrating, we get

(11)

b

−V φ + A

φ2 2

For derivation of KdVB equation we take the following set of equations:



(12)

The normalized kinematic viscosity of ions is ηi = μi ω pi /mi niCs2 In component form Eq. (12) yields

⎫ ∂t ni + ∂x (ni vix ) + ∂z (ni viz ) = 0 ⎪  2  ⎪ ⎪ (∂t + vix ∂x + viz ∂z )vix = −∂x φ + viy eff + ηi ∂x + ∂z2 vix  ⎪ ⎬ (∂t + vix ∂x + viz ∂z )viy = −vix eff + 20 sinθ viz +ηi ∂x2 + ∂z2 viy ⎪ (∂t + vix ∂x + viz ∂z )viz = −∂z φ − 20 sinθ viy + ηi ∂x2 + ∂z2 viz ⎪ ⎪ ⎪ ⎭ 1 −γ 2 )φ 2 ( 2 2 (∂x + ∂z )φ = [α1 (1 + γ )φ + + . . . ] − n i 2 (13) For derivation of KdVB equation we consider χ = ε 1/2 lx x +

ε 1/2 lz z − ε 1/2 λt , τ = ε 3/2 λt . To proceed we consider

⎫

+B

d2 φ dφ −C = c1 dξ dξ 2

(17)

We now introduce the new independent variable Z = tanhξ and W (Z ) = φ (ξ ) Using this variable Z in Eq. (17), we get

2.1. Model equations and derivation of KdVB equation

∂t ni + ∇ .(ni vi ) = 0   (∂t + vi .∇ )vi = −∇ φ + ωci vi × zˆ + 2(vi × ) + ηi ∇ 2 vi ∇ 2 φ = ne − nP − ni

(16)

Now we solve this equation by tanh-method. For this we define ξ = χ − Vτ Using this transformation in Eq. (16), we get

−V

It should be noted  that Eq. (11) is same as used by [28–29], where a = 3Au and b = 4uB .

ni = 1 + ε ni1 + ε 2 ni2 + . . .⎪ ⎪ vix = ε 2 vix1 + ε 3 vix2 + . . . ⎪ ⎪ ⎬ 3 5 viy = ε 2 viy1 + ε 2 viy2 + . . . viz = ε viz1 + ε 2 viz2 + . . . ⎪ ⎪ ⎪ ⎪ φ = ε φ 1 + ε 2 φ2 + . . . ⎭ 1/2 ηi = ε ηi 0

(φ1 )2 − α1 (1 + γ )φ2 + ni2 = 0

2

(15)

Using this set and after some calculations we obtain nonlinear KdVB equation.

C=

b

φ ( χ , τ ) = v0 + v1 

∂χ2 φ1 −

2 λ (1 + l2x (1 − llxz 20 sinθ 2 α1 (1+γ ) eff eff

………………………………………………………… By HPM

χ 

⎫

−λ∂χ ni2 + ∂τ ni1 + lx ∂χ vix1 + lz ∂χ viz2 + lz ∂χ (viz1 ni1 ) = 0⎪ ⎪ ⎪ −λ∂χ vix1 + lx ∂χ φ2 − viy2 iy2 eff = 0 ⎪ ⎪ ⎬ λ∂χ viy1 + vix1 eff − 2viz2 iy2 0 sinθ = 0 −λ∂χ viz2 + ∂τ viz1 + viz1 lz ∂χ viz1 + lz ∂χ φ2 + 2viy2 0 sinθ ⎪ ⎪ ⎪ − − ηi0 ∂χ2 viz1 = 0 ⎪

where

tanh

b

viz1 ,

∂τ φ + Aφ∂χ φ + B∂χ3 φ − C ∂χ2 φ = 0

v1 (χ , τ ) = τ 8ab−3 Bs ech2 tanh b b

  2  −3 4 χ +2ab

lx λ ∂ φ = viy1 , ∂ v = vix1 , eff χ 1 eff χ iy1 −λ∂χ viz1 + lz ∂χ φ1 = −2viy1 0 sinθ and ni1 = α1 (1 + γ )φ1

ni1 = lz

The next higher order terms in ɛ give a set of equations:

Substituting v = v0 + v1 + v2 + . . . in above equation and equating like powers of q, we obtain

q0 : q1 :

λ

15

B ( 1 − Z 2 )2

d 2W dW − {2BZ (1 − Z 2 ) + C (1 − Z 2 )} dZ dZ 2 A 2 + W − V W = c1 2

(18)

Consider the solution in series form

W = a0 + a1 Z1 + a2 Z2

(19)

Using (19) in (18), we obtain the set of equations:

⎫

2a2 B − C a1 + 12 Aa20 − V a0 − c1 = 0 ⎪ ⎪ ⎪ ⎬ 2a1 B + 2C a2 − Aa0 a1 + V a1 = 0 1 2 C a1 − 8a2 B + 2 Aa1 + Aa0 a2 − V a2 = 0 ⎪ ⎪ 2a1 B + 2C a2 + Aa1 a2 = 0 ⎪ ⎭ 1 2 6a2 B + 2 Aa2 = 0

(20)

Solving these equations, we obtain

12C 12B 1 , a2 = − , (V + 12B), a1 = − A 5A A 100B2 − C 2 + 60BC V = 25AB

a0 =

Finally the solution of KdVB equation is

φ (ξ ) = a0 + a1t anh(ξ ) + a22t anh2 (ξ ) (14)

Substituting (14) in (13) and collecting the lowest order of ɛ yields:

(21)

3. Results and discussion In this section, we will study the effects of different plasma parameters on the profile of the solitons and shocks in rotating multicomponent magneto plasma. Firstly, we will investigate the nature of the solitary structures. In the non-conservative plasma,

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Fig. 1. Effect of different parameters on soliton profile lx = 0.8, lz = 0.6, θ = 50 , α = 0.2, γ = 10, 0 = 0.6, β = 0.4, B0 = 2 G. Fig. 4. Effect of different parameters on shock profile lx = 0.8, θ = 50 , α = 0.45, γ = 10, 0 = 0.6, B0 = 2 G, η0 = 0.5.

Fig. 2. Effect of different parameters on soliton profile lx = 0.8, lz = 0.6, θ = 50 , α = 0.1, γ = 10, 0 = 0.6, β = 0.4, B0 = 2 G.

Fig. 5. Effect of different parameters on shock profile lx = 0.8, θ = 50 , α = 0.75, γ = 10, 0 = 0.6, B0 = 2 G, η0 = 0.5.

Fig. 3. Effect of different parameters on soliton profile lx = 0.8, lz = 0.6, θ = 50 , α = 0.2, γ = 11, 0 = 0.6, β = 0.4, B0 = 2 G.

the dissipative soliton can be generated due to the counter balance between the nonlinearity and dispersion. The effects of variation of Cairns distributed normalized parameters on the propagation of solitons are demonstrated in Fig. 1. It is seen that due to Cairns distributed positrons and electrons the plasma system is more favorable to the propagation of solitary waves. In Fig. 2 we see that by increasing α1 = 1 − β , the width and amplitude of the soliton shows a change as compared to Fig. 1. It means that by decreasing the α = n0P /n0h , the soliton width increases and soliton amplitude decreases. Our result is in agreement with ref.30. In Fig. 3 we observe that the soliton width and amplitude are sen-

sitive to γ = Th /TP . We found that the soliton width and amplitude decreases as compared to Fig. 1, when we increase the ratio of electron temperature to positron temperature. Our result is congruent with result of ref.30. The effects of variation of Cairns distributed normalized parameters on the propagation of shocks are shown in Fig. 4. It should be noted that the viscosity parameter η is contained only in (12). For η → 0, the dissipative term in (16) disappears and leads to the formation of solitary pulses only. However, the dissipative term in our model is taken into account, which means our system will lose the energy and shock like structures will arise. Thus, we conclude that the dissipative coefficient affects the profile of the nonlinear waves. Actually kinematic viscosity generates the dissipation in a plasma system. In Fig. 5, we observe that variation in the α , ratio of number density of protons to number density of electrons, has effect on shock profile. If we increase the number density of protons then the strength of shocks also increases as compared to Fig 0.4. Our result is same as proposed by [14]. In Fig. 6 we found that by decreasing γ = Th /TP , the strength of shock decreases as compare to Fig. 4. Our result is congruent with result of ref 0.14. In Fig. 7 we have shown the effect of kinematic viscosity η on shock profile. By increasing η, the shock amplitude is seen to be increased as compared to Fig. 4. It is noticed that the strength of the shock increases by increasing the kinematic viscosity of the ions. In Fig. 8 it is observed the strength of the shock enhances if we increase the value of the eff as compared to Fig. 4. This result and the result of [14] are in agreement.

M.Y. Khan, J. Iqbal / Chaos, Solitons and Fractals 107 (2018) 13–17

17

4. Conclusion

Fig. 6. Effect of different parameters on shock profile lx = 0.8, θ = 50 , α = 0.45, γ = 9, 0 = 0.6, B0 = 2 G, η0 = 0.5.

We have presented the theory of the IA solitons and shocks in the magnetized e-p-i rotating plasma using Cairns distribution. We include the effects of Coriolis force in our plasma system. We have applied the reductive perturbation technique in order to study the problem of formation of solitons and shock waves in rotating magneto-plasma with Cairns distribution. KdV and KdVB equations are derived and their solution are presented by HPM and tanhmethod. It is noticed that the α = n0P /n0h and γ = Th /TP affect the soliton profile. By decreasing the α , the soliton width increases and the soliton amplitude decreases but by increasing the γ , the soliton width and amplitude decreases. It is also found that ratio of number density of protons to number density of electrons, ratio of electron temperature to proton temperature, kinematic viscosity and angular frequency has effect on shock like structures. An increase in α , η and eff , leads to increase in strength of the shock structure. While a decrease in γ means a decrease in strength of shocks. This work may be helpful in order to study the rotating flows of magnetized plasma which exist in pulsar magnetosphere. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

Fig. 7. Effect of different parameters on shock profile lx = 0.8, θ = 50 , α = 0.45, γ = 10, 0 = 0.6, B0 = 2 G, η0 = 0.7.

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

Fig. 8. Effect of different parameters on shock profile lx = 0.8, θ = 50 , α = 0.45, γ = 10, 0 = 0.8, B0 = 2 G, η0 = 0.5.

Shukla PK, Yu MY. J Math Phys 1978;19:2506. Ikezi H, Taylor RJ, Baker DR. Phys Rev Lett 1970;25:11. Popel SI, Vladimirov SV, Shukla PK. Phys Plasmas 1995;2:716. Mushtaq AA, Shah HA. Phys Plasmas 2006;13:012303. Abbasi H, Pajouh HH. Phys Plasmas 2007;14:012307. Moslem WM, El-Taibany WF, El-Shewy EK, El-Shamy EF. Phys Plasmas 2005;12:052318. Chandrasekhar S. Mon Not R Astron Soc 1953;113:667. Das GC, Nag A. Phys Plasmas 2007;14:083705. Moslem WM, Sabry R, Abdelsalam UB, Kourakisand I, Shukla PK. New J Phys 2009;11:033028. Lehnert B. Astrophys. J 1954;119:647. Hide R. Philos. Trans R Soc A 1954;259:615. Mushtaq A. J. Phys A 2010;43:315501. Mushtaq A, Shah HA. Phys Plasmas 2005;12:072306. Hussain S, Akhtar N, Hasnain H. Astrophys Space Sci 2015;360:25. Akhtar N, Mahmood S, Siddiqui S. Plasma Phys Control Fusion 2014;56:095027. Dermin MI. Phys Uspekhi 2017;60(4):333–44. Ferdousi M, Yasmin S, Ashraf S, Mamun AA. Chin Phys Lett 2015;32:015201. Cairns RA, Mamun AA, Bingham R, Bostrm R, Dendy RO, Nairn CMC, Shukla PK. Geophys Res Lett 1995a;22:2709. Asaduzzaman M, Mamun AA. Phys Rev E 2012;86:016409. Kalita BC, Kalita R. J Plasma Phys 2016;82:905820201. El-Tantawy SA, El-Awady EI, Schlickeiser R. Astrophys Space Sci 2015;360:49. Sultana S, Kourakis I, Saini NS, Hellberg MA. Phys Plasmas 2010;17:032310. Ganji DD, Rafei M. Phys Lett A 2006;356:131–7. Malfliet W. Am J Phys 1992;60:650. Guo S, Liquan M, Ling HY, Chenchen M, Youfa S. Plasma Sources Sci Technol 2016;25:055006. Hussain S. Chin Phys Lett 2012;29:065202. Cairns RA, Bingham R, Dendy RO, Nairn CMC, Shukla PK, Mamun AA. J Phys IV Fr. 1995;5:C6 b. Iqbal J, Khan MY. Phys Plasmas 2017;24:042506. Iqbal J, Khan MY. Phys Plasmas 2017;24:082514. Jilani K, Arshad MM, Khan T. Astrophys Space Sci 2013. doi:10.1007/ s10509- 013- 1637- 5.