Large-amplitude dust-ion acoustic solitary waves in a dusty plasma with nonthermal electrons

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1 Astrophys Space Sci (2012) 341: DOI /s ORIGINAL ARTICLE Large-amplitude dust-ion acoustic solitary waves in a dusty plasma with nonthermal electrons S.K. El-Labany W.F. El-Taibany M.M. El-Fayoumy Received: 13 February 2012 / Accepted: 12 April 2012 / Published online: 3 May 2012 Springer Science+Business Media B.V Abstract Propagation regimes of large-amplitude dust-ion acoustic solitary wave in a dusty plasma with nonthermal electrons are analyzed by employing the Sagdeev potential technique. Two domains of the Mach numbers are defined depending on the nonthermal and plasma parameters. The two types of soliton solution are found to be exited corresponding to certain values of the nonthermal parameter. Numerical solutions are presented that illustrate the dependence of soliton characteristics on practically interesting plasma and nonthermal parameters. The findings of this investigation could be useful in understanding the detected solitary waves in space plasma in the presence of nonthermal electrons such as electrostatic solitary structures observed in Saturn s E-ring. Keywords Dust-ion acoustic waves Sagdeev potential Nonthermal electrons Solitons S.K. El-Labany W.F. El-Taibany ( ) M.M. El-Fayoumy Department of Physics, Faculty of Science, Damietta Branch, Mansoura University, Damietta El-Gedida, P.O , Egypt eltaibany@hotmail.com S.K. El-Labany skellabany@hotmail.com M.M. El-Fayoumy manal_200724@yahoo.com W.F. El-Taibany eltaibany@kku.edu.sa W.F. El-Taibany Department of Physics, College of Science for Girls in Abha, King Khalid University, Abha, P.O. 960, Kingdom of Saudi Arabia 1 Introduction The study of dusty plasmas is a very exciting area of research that has been developing very rapidly in the last decade due to the widespread occurrence in both the laboratory and space environments. Examples include cometary comae and tails, planetary rings, the interstellar medium, the lower ionosphere, plasma processing devices, limiter regions of fusion plasmas, etc. (Baumjohann and Treumann 1997; Shukla and Mamun 2002; Tsytovich et al. 2008; Shukla and Eliasson 2009). The consideration of charged dust grains in plasma doesn t only modify the existing plasma wave spectra, but also introduces a number of new novel eigenmodes, such as dust-acoustic (DA) waves (Rao et al. 1990), dust-ion acoustic (DIA) (Shukla and Silin 1992), dust lattice, etc. Among the modified dusty modes discussed in the literature, DIA wave (Shukla and Silin 1992) has received a wide attention due to some space observations (Shukla and Mamun 2002; Shukla and Eliasson 2009) as well as experimental confirmation in several low temperature dusty plasma devices (Barkan et al. 1996; Merlino et al. 1998; Rahman et al. 2011). In the DIA waves, the restoring force comes from the pressures of inertialess electrons, whereas the ion mass provides the inertia similar to the usual ion-acoustic (IA) waves in an electron-ion plasma (Shukla and Silin 1992; Shukla and Mamun 2002). Thus, the DIA waves are characterized by a phase speed much larger (smaller) than the ion (electron) thermal speed and the mode occurs on time scale much shorter than the dust plasma period. Hence, charged dust grains are regarded as immobile, and their effects appear through the modification of the equilibrium quasineutrality. Shukla and Silin (1992) were the first to report theoretically the existence of DIA waves. They showed that due

2 528 Astrophys Space Sci (2012) 341: to the conservation of equilibrium charge density n eo + Z d n do = n io and the strong inequality n eo n io (where n so is the particle number density of the species with s = e(i)d for electrons(ions) dust grains), a dusty plasma with negatively charged static dust grains supports DIA waves. The dispersion relation of the linear DIA waves is (Shukla and Mamun 2002) ω 2 = (n io /n eo )k 2 C 2 ia /(1 + k2 λ 2 De ) where C ia = T e /m i is the IA speed (with T e and m i being the electron temperature and the ion mass, respectively), k is the wave number and λ De = T e /(4πn eo e 2 ) is the electron Debye radius. For a long wavelength limit (namely kλ De 1), the dispersion relation for the DIA waves reads ω = kc ia nio /n eo. This form of spectrum is a modified form of the usual IA waves. On the other hand, adopting a fluid theory, Merlino (1997) analyzed the DIA wave instability. He showed that for constant dust charge and under certain conditions the relative drift between the electrons and ions is sufficient to excite the DIA wave instability. He pointed out that the mode becomes less difficult to excite as the relative concentration of negatively charged dust increases. Of particular interest is the case that electrons follow nonthermal distribution function which is turning out to be a characteristic feature of space plasma. Specifically, these distribution is a common feature of the auroral zone (Hal et al. 1991). The observation of energetic or nonthermal electrons has also been noted in the upper ionosphere by Freja and Viking satellites (Bostrom 1992; Dovner et al. 1994). In this case, Cairns et al. (1995) proposed the nonthermal distribution of electrons to study the IA solitary structures observed by the Freja satellite. They have shown that the presence of nonthermal electrons allows the existence of both positive and negative density perturbations. They offer a considerable increase in richness and varieties of wave motion that exist in plasma and further significantly influence the conditions required for the formation of these waves. Mamun and Shukla (2002) had discussed a plasma system consists of ion fluid, Boltzmann electrons, and charge fluctuating immobile dust particles to illustrate the effect of charge variation on the small-amplitude DIA waves. On the other side, the higher-order nonlinearity and dispersion to nonlinear acoustic waves are estimated incorporating nonthermal electrons using the reductive perturbation method (RPM) (El-Shewy 2007). Including nonthermal electrons in their study of largeamplitude DA waves, Tribeche and Boumezoued (2008) found that the nonlinear localized potential structure shrinks when the electrons deviate from isothermality. Also, the dust particles are locally expelled and pushed out the region of the soliton localization as the electrons evolve far away from their thermodynamic equilibrium. Later on, Berbri and Tribeche (2009) extended the model of Mamun and Shukla (2002) by including nonthermal electron and they derived Kortweg de-vries (KdV) Burger equation for describing small-amplitude DIA waves in a dusty plasma. In addition, Mamun and Shukla (2009) have investigated the effects of the nonthermal electrons as well as the polarity of the net dust-charge number density on nonplanar (both cylindrical and spherical) DIA solitary waves by employing the RPM. It is found that the basic features of the DIA waves are significantly modified by the effects of nonthermal electron distribution, polarity of net dust-charge number density, and nonplanar geometry. Recently, employing a RPM, Alinejad (2010) has showed that the amplitude of the nonlinear DIA rarefactive solitary waves and shock structures are reduced due to the presence of the nonthermal electron component in a dusty plasma system comparison with the isothermal case. El-Labany et al. (2010) have investigated the properties of nonlinear IA solitary waves in a warm magnetoplasma with positive-negative ions and nonthermal electrons. On the other hand, Sagdeev (1966) first investigated the features of the arbitrary amplitude nonlinear wave in a simple unmagnetized isothermal plasma conceiving electrons and cold ions. He studied the nonlinear wave in the form of an integral energy equation analog to a classical particle oscillation in a potential well derived from the basic equations governing the plasma dynamics and have been found later on to be a keen interest to study the nonlinear waves in plasmas. The Sagdeev potential could be analyzed to predict the existence domain of various features of localized solitons in various configurations of plasmas. The advantages of the method in finding the solitary waves or double layers stemming from the nonlinear waves are found useful in investigating arbitrary amplitude wave propagation (Das et al. 1998; El-Labany and El-Taibany 2003; Sabry et al. 2009; Verheest and Pillay 2010). Very recently, employing the Sagdeev potential analysis, Alinejad (2011a) has analyzed the large-amplitude DIA waves with two-temperature trapped electrons. He found that only compressive DIA solitons are observed. Moreover, the possible existence domain of the large-amplitude DIA waves is estimated including the effect of arbitrary dust charge variable, Alinejad (2011b, 2012). Mayout and Tribeche (2011) have considered another nonthermal distribution; superthermal, for ions in their theoretical model for large-amplitude DA waves applied for Saturn s F-ring. Up to the best knowledge of the authors, no research study concerns with the existence domain of the largeamplitude DIA solitary waves in a dusty plasma including nonthermal electrons. However, many authors concerned with small-amplitude analysis (Mamun and Shukla 2002; El-Shewy 2007; Berbri and Tribeche 2009; Alinejad 2010, 2011a, 2011b, 2012; Rahman et al. 2011) or truncated Sagdeev potential for lower order of the electrostatic potential. Therefore, it is of practical interest to examine the effect

3 Astrophys Space Sci (2012) 341: of nonthermal electrons on the propagation regime (s) of the large-amplitude DIA waves in the presence of nonthermal electrons. This paper is organized as follows: the basic equations governing the dusty plasma system under consideration are given in Sect. 2. The new Sagdeev pseudopotential of the current system is derived in Sect. 3. The Mach number regime in which DIA solitary waves would propagate is derived analytically and discussed in Sect. 4. Small-amplitude expansion for Sagdeev pseudopotential discussed in Sect. 5. Finally, the conclusion of the analysis is stated in Sect Basic equations We consider a dusty plasma system which consists of nonthermal electrons, warm ion and cold negatively charged heavy dust fluid. For simplicity, we assume the dust grains have a constant surface charge. The normalized basic equations of this model are given by, for dust, n d t v d t for ions, + x (v dn d ) = 0, (1) + v d v d x μ ϕ x = 0 (2) n i t v i t for electrons, + x (n iv i ) = 0 (3) + v i v i x + 3σn i n i x + ϕ = 0, (4) x n e = ( 1 βϕ + βϕ 2) exp(ϕ) (5) where, β = 4α/(1 + 3α) and α is a parameter determining the ratio of nonthermal electrons present in our plasma system. The parameter β cannot exceed a certain range (0 β 1.3) (El-Taibany et al. 2010), beyond which α is negative which is not acceptable. When β 0, Eq. (5)gives the Boltzmann distribution of electrons (Cairns et al. 1995) and Poisson s equation, 2 ϕ x 2 = δn e n i + (1 δ)n d (6) Here we used the following normalization: the number densities n j,(j = i for ions, e for electrons and d for dusts) are normalized by n jo. The space coordinate x, time t, velocities and electrostatic potential ϕ are normalized by the Debye length λ Di = T e /(4πn io e 2 ), the inverse dusty plasma frequency ωpi 1 = m i /(4πe 2 n io ), C ia and T e /e, respectively. Also, the following definitions are introduced σ = T i /T e, δ = n eo /n io and μ = Z d m i /m d (the mass to charge ratio for ions to dust species). Z d is the residing number of charges on the dust surface. To study the properties of stationary and large-amplitude DIA waves, we transform the above set of Eqs. (1) (6), to a stationary frame (Kakati and Goswami 1998), η = x Mt where M = V s /C ia is the normalized Mach number with respect to C ia. Employing this transformation to Eqs. (1) and (2), then solving the resultant equations in terms of n d, we get n d = 1/ 1 + 2μϕ M 2 (7) Employing the same procedure to Eqs. (3) and (4), we obtain the number density of the ions as, n i = 1 2 {[ (M + 3σ) 2 2ϕ ] 1 2 3σ ± [ (M 3σ) 2 2ϕ ] 1 2 } (8) It is noted here that choosing the positive/negative sign in Eq. (8) is based on which one verifies n i ϕ=0 = n io = 1. Therefore, for M> 3σ(< 3σ), the negative (positive) sign has to be used as shown in Fig. 1. In other words, the positive (negative) sign refers to subsonic (supersonic) species. To examine the existence of solitary waves, we choose the parameter ranges corresponding to space dusty plasma situation. In Saturn s E-ring Bharuthram and Shukla 1992; Mendis and Rosenberg 1994; Mamun and Shukla 2002; Berbri and Tribeche 2009; Alinejad 2010; Merlino and Goree 2004; Wahlund et al. 2009). Accordingly, we will use the following physical parameter ranges: β = 0 1, Z d = 10 3, μ = , σ = , δ = The dependence of the ion number density, n i, on the electrostatic potential, ϕ is illustrated in Fig. 1 with the possibilities of M> 3σ(< 3σ) in panel a (b). 3 Sagdeev potential and solitary waves Using the values of densities given by Eqs. (5), (7) and (8), their coupling in Poisson s equation can be written as ( ) 1 dϕ 2 + V(ϕ)= 0, (9) 2 dx where V(ϕ), looks like the potential energy of a particle with unit mass in classical mechanics and is called the Sagdeev potential or pseudopotential and in this case it takes the following form: V(ϕ)= δ { 1 + 3β ( 1 + 3β 3βϕ + βϕ 2) exp(ϕ) } [ (1 δ)m 2 ]( 1 + 2μϕ ) μ M 2 1

4 530 Astrophys Space Sci (2012) 341: Fig. 1 n i against ϕ for M< 3σ (M > 3σ ) in panel a (panel b). In panel a (b), we have to select positive (negative) sign in Eq. (8) to satisfy the equilibrium value of n i at ϕ = 0. The selected parameters are μ = 0.001, β = 0.1, σ = ; M = 1(0.5) in panel a (b) 1 6 {[ (M + 3σ) 2 2ϕ ] 3 2 3σ [ (M 3σ) 2 2ϕ ] 3 2 (M + 3σ) 3 + (M 3σ) 3} (10) 4 Existence domains for solitary waves This kind of energy-type equation (9), has been analyzed in classical mechanics and it leads to the following conditions for the existence of solitary waves: The Sagdeev potentials have satisfied V(ϕ) ϕ=0 = 0 and dv (ϕ)/dϕ ϕ=0 = 0 and the third condition for the solitary waves to exist is d 2 V(ϕ)/dϕ 2 ϕ=0 < 0. It is obvious from Eq. (10) that V(ϕ) ϕ=0 = 0 and dv (ϕ)/dϕ ϕ=0 = 0, which confirm the neutrality of this system are satisfied. On the other hand, the third condition, d 2 V(ϕ)/dϕ 2 ϕ=0 < 0, leads to F(M)= δ(β 1) + μ(1 δ) 1 M 2 + M 2 3σ < 0. (11) Fig. 2 (a) A generic graph to show the possible existence domains in the current system. The dependence of M 1 (M 3 ) root on δ changes with (β,σ,μ) variations is illustrated in panel b (c) respectively; the curve references are as follows: the solid black curve is for (0.1, 0.2, 0.001), the red dashed curve is for (0.6, 0.2, 0.001), the blue dotted curve is for (0.1, 0.5, 0.001), though the green dot dashed curve is for (0.1, 0.2, ) Figure 2a shows a generic graph for the existence domains in the current system. Here F(M) is plotted against M for arbitrary physical parameters. There are two possible Mach number regimes for DIA wave propagation which are nom-

5 Astrophys Space Sci (2012) 341: inated as regime A and regime B where, } M 1 <M<M 2, (regimea) and M>M 3, (regimeb) (12) where M 1 and M 3 can be calculated by setting F(M)= 0 and their expressions are given by B 1 + B1 2 M 1 = 4A 1C 1, 2A 1 M 2 = 3σ, (13) B 1 B1 2 M 3 = 4A 1C 1, 2A 1 with A 1 = δ(β 1), B 1 = μ(1 δ) + 1 3σδ(β 1), C 1 = 3σμ(1 δ). It is noted here the inclusion of the dynamical dust particles and the ion temperature effects is responsible for creating the low-velocity regime, regime A. This velocity domain disappears in the previous studies (Berbri and Tribeche 2009; Alinejad 2010) where they ignored these effects. In Figs. 2b and c, we have illustrated the dependence of the M 1 and M 3 roots on the system physical parameters; δ and the set (β,σ,μ). They show that as β or σ increases both M 1 and M 3 increase. However they decrease as δ increases. M 1 increases as μ though M 3 doesn t be affected by μ changes. Since M 2 is a function of σ only, therefore it is directly proportional to σ and it doesn t be affected by the other parameters variations. It is remarked here that increasing M 1 results as reduction of propagation regime A, though increasing M 3 leads to creating the second propagation regime, B, at higher values of the Mach number. In other words, the increment of β and σ parameters results in reducing the propagation regime A, and also transfer regime B to a higher Mach number domain. The increment of the parameter μ does the same action to regime A, but it doesn t affect on regime B. In addition, increasing δ widen regime A, and it transfer regime B to be created at lower Mach number values. Now, it is obvious that the propagation regimes are strongly affected by the electron nonthermal parameter, β, and the existence of the heavy dust species, μ, and also ionto-electron temperature and electron-to-ion density ratios, σ and δ. On the other hand, the dependence of V(ϕ) on system parameter variations is illustrated through Fig. 3. The intersection value of V(ϕ) with ϕ-axis, ϕ max, refers to the amplitude of the produced soliton. On the other hand, the minimum value of V(ϕ), V min, equals half of the width of the Fig. 3 Sagdeev pseudopotential V(ϕ)is plotted against ϕ. In panel a, β = 0(solid curve) andβ = 0.21 (dashed curve) with M = 1.85, μ = 0.001, σ = and δ = 0.5. In panel b, β = 0.22 (solid curve), β = 0.25 (dashed curve), β = 0.3 (dotted curve) withm = 1.85, σ = , μ = 0.001, and δ = 0.5. In panel c, β = 0.31 (solid curve) and β = 0.32 (dashed curve), with M = 1.85, σ = and δ = 0.5 produced soliton. Figure 3a illustrates that the Sagdeev potential is created only for negative ϕ-range for either isothermal case (β = 0) or for very low value of β Therefore, the produced soliton is of rarefactive type as shown in Fig. 3a. Whereas, by increasing the value of β, there is a certain domain for β values, 0.22 β 0.3, at which the Sagdeev potential released the existence of both positive and negative ϕ-range as shown in Fig. 3b. Therefore, the pro-

6 532 Astrophys Space Sci (2012) 341: duced soliton in this β-domain is of compressive and rarefactive, but it is obvious that the rarefactive soliton is larger than the compressive soliton. Thereafter, by increasing β, β 0.31, the rarefactive solitary waves disappear and only the compressive one is created as shown in Fig. 3c. Moreover, it is noted here that increasing β results in reducing the amplitude and in shrinking the width of both rarefactive and compressive solitons. The predicted soliton solutions corresponding to different domains of β and corresponding to the three cases mentioned in Fig. 3 are investigated in Fig Small-amplitude expansion Here considering the small-amplitude theory ( ϕ 1), to add a clear comparison with the recent studies (Berbri and Tribeche 2009; Alinejad 2010; Akpabio et al. 2010; Shalaby et al. 2010; Mamun and Islam 2011), one may expand the Sagdeev potential up to the third order of ϕ. This yields ( ) 1 dϕ 2 + V(ϕ)= 0, (14) 2 dξ where the Sagdeev potential is given by V(ϕ) 1 2 F(M)ϕ2 + Dϕ 3, (15) with D = δ 6 + μ2 (δ 1) 2M 4 + M2 + σ 2(M 2 3σ) 3. The soliton solution of Eq. (14) isgivenby ϕ = ϕ s sech 2 (ξ/e), (16) where, ϕ s = F(M)/(2D) and E = 2/ F(M) are the amplitude and the width, respectively. The small-amplitude soliton solutions, presented through Eq. (16) and corresponded to the KdV solution, are shown in Fig. 5. Itisremarked here that the condition (11), is necessary here for the existence of small-amplitude soliton with real (i.e. non zero positive) width. On the other hand, the nature of the solitary waves, i.e., whether the system will support compressive or rarefactive solitary waves, depends on the sign of D. IfD is negative (positive), a compressive (rarefactive) soliton exists. Considering the case of small-amplitude DIA waves in a plasma comprising of Boltzmann electrons, cold ion fluid and variable charge stationary dust grains and setting σ = 0 and μ 1, we obtain the phase velocity of these smallamplitude soliton = 1/ δ(1 β) which agrees exactly with that derived recently by Berbri and Tribeche (2009), Alinejad (2010), Akpabio et al. (2010), Shalaby et al. (2010), Mamun and Islam (2011). This value can be derived from the expression of M 3 presented in Eq. (13) since for this Fig. 4 The dependence of the large-amplitude soliton solution, corresponding to Eqs. (9) and(10), on the physical parameter variations is illustrated. The soliton solution presented in panel (a) is plotted for the same parameters as of Fig. 3a. In panel b, the used parameters are β = 0.24, M = 1.85, σ = , μ = 0.001, and δ = 0.5. In panel c, the selected ones are β = 0.32, M = 1.85, σ = , μ = 0.001, and δ = 0.5 limiting case only one velocity is permitted. On the other side, the coefficient D appeared in Eq. (15) agrees exactly with the nonlinear coefficient of the KdV equation derived by Akpabio et al. (2010) with σ = 0. However, Berbri and Tribeche (2009) proved that there exist compressive and rar-

7 Astrophys Space Sci (2012) 341: Fig. 5 The dependence of the small-amplitude soliton solutions, corresponding to Eq. (15) on the physical parameter variations is investigated. In panel a, β = 0(solid curve)andβ = 0.24 (dashed curve) with M = 1.85, μ = 0.001, σ = and δ = 0.5. In panel b, M = 1.7 (solid curve) andm = 1.75 (dashed curve) withβ = 0.1, σ = , μ = and δ = 0.5 efactive DIA solitary waves, Alinejad (2010) found only rarefactive one in his plasma model. The comparison between Sagdeev potential analysis and small-amplitude analysis has been addressed recently (El-Taibany and Wadati 2007). Figure 5 illustrates the dependence of small-amplitude soliton on physical parameter variations. It shows that the rarefactive soliton amplitude decreases as β increases though its width becomes wider. In addition, increasing the wave velocity results in increasing the amplitude of the compressive soliton and reducing its width. 6 Conclusion In this paper, the possible propagation regimes of a largeamplitude DIA wave in an unmagnetized dusty plasma with warm ions, cold dust fluid and nonthermal electrons are investigated. An exact expression of the Sagdeev potential is obtained in terms of ϕ appropriating to describe largeamplitude solitons. On the other hand, employing the necessary conditions for solitary wave existence, on the Sagdeev potential delimits the existence Mach number domain for DIA wave propagation. Two possible regimes, A and B, are released. The new Mach number domain regime, regime A, is created due to including the effects of the ion temperature and the dynamic of the dust grains which are ignored in recent studies (Berbri and Tribeche 2009; Alinejad 2010). The produced soliton solutions and also the Sagdeev potential are illustrated against the system physical parameter variations. It is found that there is a certain domain of β values at which both the compressive and rarefactive solitons are created. On the other hand, for very low β values only rarefactive solitons are found, though for the higher β values, the compressive soliton is observed only. Therefore, this model supports the creation of both the two soliton types. Based on the numerical investigation presented here, it can be concluded that the propagation Mach number domains, the Sagdeev potential as well as the basic characteristics of the produced solitons are strongly affected by the electron nonthermality and the presence of the dynamical dust particles. Moreover, the small- amplitude limit analysis of the Sagdeev potential is provided showing the existence of both compressive and rarefactive soliton in the present model. At the end, these findings would be helpful in estimating the velocity domain appropriate for observing DIA waves and also understanding the nonlinear features of these detected solitary waves revealed where the dust grain dynamics, the ion temperature and nonthermal electrons are present in Saturn s E-ring. References Akpabio, L.E., Ikot, A.N., Udoimuk, A.B.: Lat. Am. J. Phys. Educ. 4, 724 (2010) Alinejad, H.: Astrophys. Space Sci. 327, 131 (2010) Alinejad, H.: Astrophys. Space Sci. 334, 325 (2011a) Alinejad, H.: Astrophys. Space Sci. 334, 331 (2011b) Alinejad, H.: Astrophys. Space Sci. 337, 223 (2012) Barkan, A., D Angelo, N., Merlino, R.L.: Planet. Space Sci. 44, 239 (1996) Baumjohann, W., Treumann, R.A.: Basic Space Plasma Physics. Imperial College Press, London (1997) Berbri, A., Tribeche, M.: Phys. Plasmas 16, (2009) Bharuthram, R., Shukla, P.K.: Plant Space Sci. 40, 973 (1992) Bostrom, R.: IEEE Trans. Plasma Sci. 20, 756 (1992) Cairns, R.A., Mamun, A.A., Bingham, R., Dendy, R.O., Bostrom, R., Nairn, C.M.C., Shukla, P.K.: Geophys. Res. Lett. 22, 2709 (1995) Das, G.C., Tagare, S.G., Sarma, A.: Planet. Space Sci. 46, 417 (1998) Dovner, P.O., Eriksson, A.I., Boston, R., Holback, B.: Geophys. Res. Lett. 20, 1827 (1994) El-Labany, S.K., El-Taibany, W.F.: Phys. Plasmas 10, 4685 (2003) El-Labany, S.K., Sabry, R., El-Taibany, W.F., Elghmaz, E.A.: Phys. Plasmas 17, (2010) El-Shewy, E.K.: Chaos Solitons Fractals 34, 628 (2007) El-Taibany, W.F., Wadati, M.: Phys. Plasmas 14, (2007)

8 534 Astrophys Space Sci (2012) 341: El-Taibany, W.F., Mushtaq, A., Moslem, W.M., Wadati, M.: Phys. Plasmas 17, (2010) Hal, D.S., Chaloner, C.P., Bryant, D.A., Lepine, D.R., Tritakis, V.P.: J. Geophys. Res. 96, 7869 (1991) Kakati, M., Goswami, K.S.: Phys. Plasmas 5, 4508 (1998) Mamun, A.A., Shukla, P.K.: IEEE Trans. Plasma Sci. 30, 720 (2002) Mamun, A.A., Shukla, P.K.: Phys. Rev. E 80, (2009) Mamun, A.A., Islam, S.: J. Geophys. Res. 116, A12323 (2011). doi: /2011ja Mayout, S., Tribeche, M.: Astrophys. Space Sci. 335, 443 (2011) Mendis, D.A., Rosenberg, M.: Annu. Rev. Astron. Astrophys. 32, 418 (1994) Merlino, R.L.: IEEE Trans. Plasma Sci. 25, 60 (1997) Merlino, R.L., Goree, J.: Phys. Today 57(7), 32 (2004) Merlino, R.L., Barkan, A., Thompson, C., D Angelo, N.: Phys. Plasmas 5, 1607 (1998) Rahman, O., Mamun, A.A., Ashrafi, K.S.: Astrophys. Space Sci. 335, 425 (2011) Rao, N.N., Shukla, P.K., Yu, M.Y.: Planet. Space Sci. 38, 543 (1990) Sabry, R., Moslem, W.M., Shukla, P.K.: Phys. Plasmas 16, (2009) Sagdeev, R.Z.: In: Leontovich, M.A. (ed.) Reviews of Plasma Physics, vol. 4, p. 23. Consultants Bureau, New York (1966) Shalaby, M., El-Labany, S.K., El-Shamy, E.F., Khaled, M.A.: Astrophys. Space Sci. 326, 273 (2010) Shukla, P.K., Eliasson, B.B.: Rev. Mod. Phys. 81, 25 (2009) Shukla, P.K., Mamun, A.A.: Introduction to Dusty Plasma Physics. IOP, Bristol (2002) Shukla, P.K., Silin, V.P.: Phys. Scr. 45, 508 (1992) Tribeche, M., Boumezoued, G.: Phys. Plasmas 15, (2008) Tsytovich, V.N., Morfill, G.E., Vladimirov, S.V., Thomas, H.M.: In: Elementary Physics of Complex Plasmas. Lect. Notes Phys., vol Springer, Berlin/Heidelberg (2008) Verheest, F., Pillay, S.R.: Astrophys. Space Sci. 326, 151 (2010) Wahlund, J.-E., André, M., Eriksson, A.I.E., Lundberg, M., Morooka, M.W., Shafiq, M., Averkamp, T.F., Gurnett, D.A., Hospodarsky, G.B., Kurth, W.S., Jacobsen, K.S., Pedersen, A., Farrell, W., Ratynskaia, S., Piskunov, N.: Planet Space Sci. 57, 1795 (2009)