IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 4, APRIL 2013 1005

Effects of Two-Temperature Ions and ExternalMagnetic Field on Dust-Acoustic Solitary

Waves in a Dusty PlasmaEsmaeil Eslami and Rasool Baraz

Abstract— In this paper, a reductive perturbation theory isused to derive a Korteweg–de Vries equation for a magnetizeddusty plasma consisting of cold dust grains, Boltzmann distrib-uted electrons, and Boltzmann distributed ions at two differenttemperatures. The renormalization method is used to obtainstationary solutions of these coupled equations. The effects of two-temperature ions and external magnetic field on the properties ofsolitary waves are studied. Such a study showed that the presenceof the hot and cold ions modifies the nature of dust-acousticsolitary structures and admits mainly solitary waves with negativepotential. However, for a small amount of electron density, theplasma may allow rarefactive dust-acoustic solitary waves to existin such a dusty plasma system.

Index Terms— Dust plasma, external magnetic field, Korteweg–de Vries (KdV), solitary wave.

I. INTRODUCTION

DUSTY plasma has been studied extensively recently. Itis different from the usual electron–ion plasma. The

dusty plasma consists of electrons, ions, and charged dustparticles. Such plasmas occur in laboratory, astrophysical andspace environments, such as interstellar medium, planetaryrings, cometary tails, the earth’s environment, etc. [1]–[4].In the laboratory, the dust particles appear as impurities andcan significantly influence the behavior of the surroundingplasma [5]. Dusty plasmas also play an important role inother interesting fields such as low temperature physics, radiofrequency plasma discharge, coating and etching of thin films,plasma crystal, etc. Dust particles usually charge negatively toa large value because of the ion current and electron currenton their surface [6]. However, the presence of positivelycharged dust particles has also been observed in a laboratory,in the anode region of a dc glow discharge [7], and indifferent regions of space, i.e., upper mesosphere [8], cometarytails [3], [9], and [10], Jupiter’s magnetosphere [11]. Usually,the dust grains are of micrometer or submicrometer size. Themasses of the dust particles are very large, so they haveno consequence on high-frequency oscillations except on thedamping factor [12].

Manuscript received October 24, 2012; revised February 4, 2013; acceptedFebruary 11, 2013. Date of current version April 6, 2013.

The authors are with the Department of Physics, Iran University of Sci-ence and Technology, Tehran 16846-13114, Iran (e-mail: eeslami@iust.ac.ir;shahangshad@gmail.com).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPS.2013.2248099

In recent years, there has been continuing interest in thepropagation of linear and nonlinear electromagnetic wavesin dusty plasmas [13]–[15]. Study of nonlinear wave phe-nomena in dusty plasmas began with the work of Bliokhand Yarashenko [16]. Early, Rao et al. [17] predicted thecollective processes of nonlinear dust-acoustic waves in spaceand laboratory environments, which were confirmed by thelaboratory experiment [18], [19]. In their study, dust particleswere considered as constant charge or variable charge. Allthese studies can be treated in both unmagnetized and magne-tized dusty plasmas. Kakati and Goswami [20] investigatedthe nonlinear properties of solitary wave structure in anunmagnetized dusty plasma consisting of nonisothermal ions,isothermal electrons, and variable-charged dust grains. Theyshowed that variations of dust charge and reflected ions modifythe amplitude and width of solitary waves. Xie et al. [21]studied the effect of adiabatic dust charge variation on dust-acoustic solitary waves in an unmagnetized dusty plasma.El-Labany et al. [22] investigated the dust-acoustic solitarywaves and double layers in an unmagnetized dusty plasmaconsidering the effects of two-temperature trapped ions anddust charge variation. They found that the presence of two-temperature ions plays an important role in the coexistence ofboth compressive and rarefactive solitons as well as the doublelayers. Their recent works investigated nonlinear propertiesof the dust-acoustic solitary waves in a magnetized dustyplasma consisting of negatively variable-charged dust particles,vortex-like distributed ions, and two-temperature isothermalelectrons [23]. They showed that only rarefactive solitonwave permits us to propagate. Choi et al. [24] studied theeffect of ion thermal pressure in an unmagnetized plasmawith immobile dust grains. Lin and Lin [25] have analyt-ically investigated the nonlinear dust waves in a magne-tized dusty plasma with many different dust grains. Later,Lin et al. [26] have analytically investigated the nonlineardust-acoustic waves in a 2-D dust plasma with dust chargevariation in an unmagnetized dusty plasma. Mamun [27],[28] theoretically investigated dust-acoustic solitary structuresboth magnetized and unmagnetized with immobile dust grains.He showed that the presence of the second component ofions modifies the nature of dust-acoustic solitary structuresand allows rarefactive dust-acoustic solitary waves to existin a magnetized dusty plasma. Also, he showed that in anunmagnetized dusty plasma system consisting of electrons,ions, and negative as well as positive dust, this positive dust

0093-3813/$31.00 © 2013 IEEE

1006 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 4, APRIL 2013

component causes the coexistence of positive and negativesolitary potential structures. Moslem [29] studied obliquelydust-acoustic solitary waves in a collisional, magnetized dustyplasma having cold dust grain, isothermal electrons, two-temperature isothermal ions, and stationary neutrals. The non-linear properties of dust-acoustic waves propagating obliquelyto an external magnetic field in a dusty plasma comprisingof Boltzmannian electrons, two-temperature trapped ions, andcharged dust grains are investigated by Bagchi et al. [30]. Theyfound that the Mach speed of the dust-acoustic solitary waveincreases with the dust fluid temperature and the directioncosines of the wave vector, but decreases with the low-to-hightemperature ratio for the trapped ions. The work of Asaduz-zaman and Mamun [31] deals with a dusty plasma, whichis composed of negatively charged cold mobile dust fluids,inertialess Boltzmann electrons, and ions with two distinctivetemperatures in an unmagnetized plasma. They found thatcoexistence of both compressive and rarefactive solitons maybe propagated beyond the Korteweg–de Vries (KdV) limit. Theeffects of two-temperature nonthermally distributed ions in anunmagnetized dusty plasma system consisting of negativelycharged mobile dust fluid and Boltzmann distributed electronswere discussed by Tasnim et al. [32]. They found that the dustyplasma system under consideration supports finite amplitudeshock waves, whose amplitude depends on the ratio of lowerto higher temperature of ions and the ratio of ion to electrontemperature as well as relative ion number densities.

Recently, Masud and Mamun [33] considered the effect oftwo-temperature Maxwellian ions and obtained the Zakharov–Kuznetsov equation. But, this paper is limited to rarefactivesoliton waves only. Thus, to obtain a more generalized work ondusty plasma, we have derived the KdV equation and analyzedboth compressive and rarefactive solitary waves in this paper.This paper has been organized as follows.

In Section II, we present the basic equations govern-ing the dynamics of our plasma. In Section III using thereductive perturbation method, the KdV equation, which gov-erns the evolution of the first-order perturbed potential, hasbeen derived. The renormalization method is used to obtainstationary solutions of these coupled equations. Section IVis devoted to the discussion of numerical analysis of solitonwaves, and then we summarize the main conclusions.

II. MATHEMATICAL FORMALISM

We consider a dusty plasma, which consists of threecomponents, i.e., extremely massive, micronsized, negativelycharged, cold dust fluid, inertialess free electrons withBoltzmann distribution, and Boltzmann distributed ions in thepresence of an external static magnetic field ( �B = B0z). In theequilibrium situation, the charge neutrality condition becomesZini0 = ne0+Zd0nd0, where ne0 and nd0 are the unperturbedelectron and dust number densities, respectively. Zi, and Zd0

are the unperturbed number of charges on the ions and thenumber of charges residing on the dust particles, respectively.Here, we consider a singly ionized plasma system for whichZi = 1. An equilibrium condition for two unperturbed low(nil0) and high (nih0) temperature ion densities is satisfied byni0 = nil0 + nih0.

The dust particles are governed by normalized continuity,momentum, and Poisson equations for a nondimensional case

∂nd

∂t+ �∇ · (nd�ud) = 0 (1)

∂�ud

∂t+

(�ud · �∇

)�ud =

e

md

�∇φ (2)

∇2φ = 4πe (Zdnd + ne − ni) (3)

where nd, �ud, and md refer to the dust number density, fluidvelocity, and mass of the dust grain, respectively. φ is theelectrostatic wave potential and e is the electron charge.

The space coordinate x, time t, velocities, and electrostaticpotential φ are normalized by the Debye length λDe =(Teff/4πnd0Zd0e

2)1/2

, the inverse of dust plasma frequency

ω−1pd =

(md/4πnd0Z

2d0e

2)1/2

, the dust-acoustic speed Cd =(Zd0Teff/md)

1/2, and Teff/e, respectively. The effective tem-perature is given by

Teff =(

TilTihTe

μlTihTe + μhTilTe + νTilTih

)

where Til (or/and Tih) and Te are the low (or/and high)temperature ion and temperature electron, respectively. μl

(or/and μh) and ν are normalized low (or/and high) ion andelectron number densities by Zd0nd0 with the intention that

μl =δ1

δ1 + δ2 − 1, μh =

δ2

δ1 + δ2 − 1, ν =

1δ1 + δ2 − 1

where δ1 = nil0/ne0 and δ2 = nih0/ne0.Therefore, we obtain the set of normalized basic equations

∂nd

∂t+

∂ (nduxd)∂x

+∂ (nduyd)

∂y+

∂ (nduzd)∂z

= 0 (4)

∂uxd

∂t+ uxd

∂uxd

∂x+ uyd

∂uxd

∂y+ uzd

∂uxd

∂z

=∂φ

∂x− ωcduyd (5)

∂uyd

∂t+ uxd

∂uyd

∂x+ uyd

∂uyd

∂y+ uzd

∂uyd

∂z

=∂φ

∂y+ ωcduxd (6)

∂uzd

∂t+ uxd

∂uzd

∂x+ uyd

∂uzd

∂y+ uzd

∂uzd

∂z=

∂φ

∂z(7)

∂2φ

∂x2+

∂2φ

∂y2+

∂2φ

∂z2= nd + ne − nih − nil. (8)

The variables uxd, uyd, and uzd are the dust fluid velocities inthe x, y, and z directions normalized to the dust-acoustic speedCd, respectively. t is the time variable. nd is the dust particlenumber density normalized by nd0, and φ is normalized byTeff/e. Here, the normalized dust cyclotron frequency is defineas ωcd = (zd0eB0/md) /ωpd.

In our analysis, we assume that the electrons and ions haveBoltzmann distribution. Thus, we can express normalized formof ne and ni as

ne = νe−ϑeφ

nil = μle−ϑlφ, nih = μhe−ϑhφ

where ϑe = Til/Te, ϑl = Teff/Til, and ϑh = Teff/Tih.

ESLAMI AND BARAZ: EFFECTS OF TWO-TEMPERATURE IONS AND EXTERNAL MAGNETIC FIELD 1007

III. KDV EQUATION

To study the properties of soliton dynamics in the Boltzmancomplex plasma, we, following Washimi and Taniuti [34],employ the reductive perturbation technique to the basic equa-tions. Accordingly, the time-space coordinates are stretchedthrough the following relations:

ξ = ε12 (lxx + lyy + lzz − v0t) , τ = ε

32 t (9)

where ε (0 < ε < 1) is a small parameter characterizingthe strength of nonlinearity of dust-acoustic waves and ν0

is the wave phase velocity normalized to Cd. lx, ly, and lzare directional cosines of the wave vector in the x, y, andz directions, respectively, so that l2x + l2y + l2z = 1. We cannow expand all the physical quantities about their equilibriumvalues in power of ε including terms of order ε3/2 (a measureof the nonisothermality), which means that we let

nd = 1 + εn(1)d + ε2n

(2)d + · · · (10)

uxd = u(0)xd + ε3/2u

(1)xd + ε2u

(2)xd + · · · (11)

uyd = u(0)yd + ε3/2u

(1)yd + ε2u

(2)yd + · · · (12)

uzd = u(0)zd + εu

(1)zd + ε2u

(2)zd + · · · (13)

φ = εφ(1) + ε2φ(2) + · · · (14)

where u(0)xd , u

(0)yd , and u

(0)zd are assumed to be zero.

Substituting (10)–(14) into the set of basic normalizedequations (4)–(8) and then collecting terms of different powerof ε, we obtain in the lowest order for continuity, momentum,and Poisson equations

n(1)d = −φ(1) (15)

n(1)d =

lzν0

u(1)zd (16)

u(1)yd =

lxωcd

∂φ(1)

∂ξ, u

(1)xd = − ly

ωcd

∂φ(1)

∂ξ, u

(1)zd = − lx

ν0φ(1)

(17)

ly∂φ(1)

∂ξ+ ωcdu

(1)xd = 0. (18)

To the next order in ε by using (15)–(18) and eliminating u(1)xd

and u(1)yd , we obtain

u(2)xd =

lxν0

ω2cd

∂φ(1)

∂ξ, u

(2)d = − lyν0

ω2cd

∂φ(1)

∂ξ(19)

n(2)d =

∂2φ(1)

∂ξ2− φ(2) − K

2

(φ(1)

)2

(20)

∂n(1)d

∂τ− ν0

∂n(2)d

∂ξ+ lx

∂u(2)xd

∂ξ+ ly

∂u(2)yd

∂ξ

+lz∂

(u

(2)zd + n

(1)d u

(1)zd

)

∂ξ= 0 (21)

ν0∂u

(2)zd

∂ξ+

∂u(1)zd

∂τ+ lzu

(1)zd

∂u(1)zd

∂ξ− lz

∂φ(2)

∂ξ= 0 (22)

where

K =

(νϑ2

e − μl − μhϑ2

(μl + νϑe + μhϑ)2

)and θ ≡ θh

θl=

Til

Tih.

(a)

(b)

Fig. 1. Nonlinear coefficient A as a function of θ for (a) θe = 0.2, μl = 0.2,lz = 0.5, and different values of μh and ν and (b) θe = 0.2, μh = 0.95,lz = 0.5, and different values of μl and ν.

By substituting (19)–(22) into (15)–(18), one can eliminate∂n(2)

dx/∂ξ, ∂n(2)dz/∂ξ, and ∂φ(2)/∂ξ and the KdV equation is

obtained

∂φ(1)

∂τ− Aφ(1) ∂φ(1)

∂ξ+ B

∂3φ(1)

∂ξ3= 0 (23)

where

A =lz2

(3 + K) and B =lz2

(1 +

1 − l2zω2

cd

).

To consider solitonic solution related to a frame whichmoves with velocity u, we define a new variable χ = ξ − uτ ,with the following appropriate boundary conditions: 1) φ(1) →0; 2) ∂φ(1)/∂χ → 0; and 3) ∂2φ(1)/∂χ2 → 0 at |χ| → ∞.

Following some simple algebra, the well-known stationarysolitary wave solution of the modified-KdV equation (23) canbe written as

φ(1) = −φ(1)m sech2

(χ

σ

)(24)

where φ(1)m = 3u/A and σ =

√4B/u are the amplitude

and the width of acoustic soliton solution, respectively. Thissolution also holds for n

(1)d if we replace φ

(1)m by −φ

(1)m .

As u > 0, it is clear from (19)–(24) that if K > −3, thereexist solitary waves with negative potential or compressivesolitary waves (solitary waves with density hump) and ifK < −3, there exist solitary waves with positive potentialor rarefactive solitary waves (solitary waves with density dip).

IV. NUMERICAL RESULTS AND DISCUSSION

The coefficient of nonlinear term A for (23) as a function ofθ for different parameters is shown in Fig. 1. The parameter Aindicates the inverse square root of the linear dust density massper unit length. It seems that the sign of Astrongly dependson the values of ν, μl, μh, and θe. As is well known: 1) if thecoefficient of nonlinear term Ais negative, there are rarefactivesolitary waves and 2) if Ais positive, the compressive solitarywaves exist in this system. The value of A becomes large asTil � Tih, while we see that the acoustic pulse with small

1008 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 4, APRIL 2013

Fig. 2. Relative plot of φ(1) versus χ for negative solitons. Parameter valuesare as follows: μl = 0.2, μh = 0.9, θe = 0.2, θ = 0.9, ωcd = 0.1,ν = 0.1,and u = 1 where (�) lz = 0.1, (◦) lz = 0.2, (�) lz = 0.3, and(∇) lz = 0.4.

Fig. 3. Relative plot of φ(1) versus χ for negative solitons. Parameter valuesare as follows: μl = 0.2, μh = 0.9, θe = 0.2, θ = 0.9, ωcd = 0.1,ν = 0.1, and u = 1 where (�) lz = 0.5, (◦) lz = 0.6, (�) lz = 0.7, and(∇) lz = 0.8.

amplitude grows with increasing θ (Til → Tih). The last casecorresponds to the higher amplitude of acoustic soliton wave.

In this paper, we have investigated dust ion-acoustic solitarywave in the magnetized dusty plasma consisting of ionswith two-temperature, and charged dust grain. For nonlineardust ion-acoustic solitary wave, the reductive perturbationtheory is used to reduce the basic set of equation (4)–(8)to the KdV equation (23). For better accuracy, the higherorder nonlinear and dispersion terms have to be included.Moreover, the numerical results are applied to investigatesome nonlinear characteristics of the dust-acoustic solitarywaves. The results of our analysis for both the amplitudes andwidths of the nonlinear dust ion-acoustic solitary waves arepresented in Figs. 2–7. Here, we consider the effects of two-temperature ions and external magnetic field, on the featuresof solitary waves. The parameters, which we have chosen inour numerical calculations, are relevant to different regionsof space, viz., cometary tails [9], [10], mesosphere [10],

Fig. 4. Relative plot of φ(1) versus χ for negative solitons. Parameter valuesare as follows: μl = 0.2, μh = 0.9, θe = 0.2, θ = 0.9, lz = 0.5, ν = 0.1,and u = 1 where (�) ωcd = 0.05, (◦) ωcd = 0.1, (�) ωcd = 0.5, and (∇)ωcd = 1.

Fig. 5. Relative plot of φ(1) versus χ for negative solitons. Parameter valuesare as follows: μl = 0.2, μh = 0.9, θe = 0.2, ωcd = 0.1, lz = 0.5,ν = 0.1, and u = 1 where (�) θ = 0.1, (◦) θ = 0.2, (�) θ = 0.3, and (∇)θ = 0.4.

Jupiter’s magnetosphere [11], etc. Figs. 2 and 3 are drawnwhen μl = 0.2, μh = 0.9, θe = 0.2, θ = 0.9, ωcd = 0.1,ν = 0.1, and u = 1 for the different values of directionalcosine lz(=0.1−0.8). The amplitude decreases with lz , whilethe width of the solitary structures increases. The effect ofan external magnetic field is presented in Fig. 4 for the setof parameters μl = 0.2, μh = 0.9, θe = 0.2, θ = 0.9,lz = 0.5, ν = 0.1, and u = 1. It is seen that the magnitudeof the external magnetic field has no effect on the amplitudeof the solitary waves. However, it does have an effect onthe widths of these compressive solitary waves. It is shownthat as we increase the magnitude of the external magneticfield, the widths of these solitary structures decrease, i.e., theexternal magnetic field makes the solitary structures morespiky. Figs. 5 and 6 demonstrate the variation of φ(1) fordifferent values of θ = Til/Tih. The physical properties thatwe use in the numerical analysis are as follows: μl = 0.2,μh = 0.9, θe = 0.2, lz = 0.5, ωcd = 0.1, ν = 0.1, and

ESLAMI AND BARAZ: EFFECTS OF TWO-TEMPERATURE IONS AND EXTERNAL MAGNETIC FIELD 1009

Fig. 6. Relative plot of φ(1) versus χ for negative solitons. Parameter valuesare as follows: μl = 0.2, μh = 0.9, θe = 0.2, ωcd = 0.1, lz = 0.5,ν = 0.1, and u = 1 where (�) θ = 0.5, (◦) θ = 0.6, (�) θ = 0.7, and (∇)θ = 0.8.

Fig. 7. Relative plot of φ(1) versus χ for positive solitons. Parameter valuesare as follows: μh = 0.95, θe = 0.2, θ = 0.01, ωcd = 0.1, lz = 0.5, andu = 1 where (�) (μl = 0.09, ν = 0.04), (◦) (μl = 0.1, ν = 0.05), (�)(μl = 0.11, ν = 0.06), and (∇) (μl = 0.12,ν = 0.07).

u = 1. The presence of two different types of temperatureTil and Tih ions modifies significantly the amplitude of thecompressive solitary waves. It is obvious that the amplitudeof φ(1) decreases as θ increases from 0.1 up 0.4. However,with further increasing of θ(= 0.5 − 0.8), little change indust-acoustic solitary waves is found.

All the aforementioned figures represent a compressivesolitary wave. However, it has been found from this numericalanalysis that for typical dusty plasma parameters, especiallyin the low number of electron ν and low temperature ions μl

densities, we have the existence of solitary waves with positivepotential (rarefactive solitons) for ν < 0.07 and μl < 0.12(Fig. 7). The other particular set of plasma parameters inthis figure are μh = 0.95, θe = 0.2, θ = 0.01, lz = 0.5,ωcd = 0.1, and u = 1. The profiles of the ion-acousticsolitary waves show that a dusty plasma with a lower electrondensity, the amplitude of the wave decreases whereas width

increases along with the decrease in the ion number with lowtemperature, respectively.

In summary, we investigated the effects of two-temperatureions and external magnetic field on the nonlinear propaga-tion of dust-acoustic waves in magnetized dusty plasmas. Itwas found that in the presence of negatively charged dustgrains and two-temperature ions, the system supports mainlycompressive soliton waves. The amplitude and width of thepulse wave strictly depend on plasma parameters. It was alsofound that as we decrease the background electron numberdensity, the rarefactive solitary wave may be propagated. Theresults of our investigation could be helpful to understandingsome impressive features of solitary waves associated with iondistribution which may occur in astrophysical objects suchas Saturn’s E-ring, interstellar molecular clouds, and insideionopause of Halley’s comet.

REFERENCES

[1] M. Horanyi and D. A. Mendis, “The effects of electrostatic chargingon the dust distribution at Halley’s Comet,” Astrophys. J., vol. 307,pp. 800–807, Aug. 1986.

[2] T. G. Northrop, “Dusty plasmas,” Phys. Scripta, vol. 45, no. 5, p. 475,1992.

[3] D. A. Mendis and M. Rosenberg, “Cosmic dusty plasmas,” Annu. Rev.Astron. Astrophys., vol. 32, no. 1, pp. 419–463, 1994.

[4] F. Verheest, “Waves and instabilities in dusty space plasmas,” Space Sci.Rev., vol. 77, nos. 3–4, pp. 267–302, 1996.

[5] S. Benkadda, V. N. Tsytovich, and A. Verga, “Self organization in dustyplasma,” Comments Plasmas Phys. Cont. Fusion, vol. 16, no. 2, p. 321,1995.

[6] C. K. Goertz, “Dusty plasmas in the solar system,” Rev. Geophys.,vol. 27, no. 2, pp. 271–292, 1989.

[7] A. A. Samarian, O. S. Vaulina, A. P. Nefedov, V. E. Fortov, B. W. James,and O. F. Petrov, “Positively charged particles in dusty plasmas,” Phys.Rev. E, vol. 64, no. 5, pp. 056407-1–056407-8, 2001.

[8] O. Havnes, J. Trøim, T. Blix, W. Mortensen, L. Næsheim, E. Thrane,and T. Tønnesen, “First detection of charged dust particles in the Earth’smesosphere,” J. Geophys. Res., vol. 101, no. A5, p. 1039, 1996.

[9] M. Horányi, “Charged dust dynamics in the solar system,” Annu. Rev.Astron. Astrophys., vol. 34, pp. 383–418, Sep. 1996.

[10] V. W. Chow, D. A. Mendis, and M. Rosenberg, “Role of grain size andparticle velocity distribution in secondary electron emission in spaceplasmas,” J. Geophys. Res., vol. 98, no. A11, pp. 19065–19076, 1993.

[11] M. Horányi, G. E. Morfill, and E. Grün, “Mechanism for the accelerationand ejection of dust grains from Jupiter’s magnetosphere,” Nature,vol. 363, pp. 144–146, May 1993.

[12] U. De Angelis, R. Bingham, and V. N. Tsytovich, “Ion plasma waves industing plasmas,” J. Plasma Phys., vol. 40, no. 3, pp. 399–406, 1989.

[13] S. K. El-Labany and W. M. Moslem, “Higher-order contributions to dust-acoustic waves in a magnetized dusty plasmas,” Phys. Scripta, vol. 65,no. 5, p. 416, 2002.

[14] P. K. Shukla, “Nonlinear waves and structures in dusty plasmas,” Phys.Plasmas, vol. 10, no. 5, pp. 1619–1627, 2003.

[15] P. K. Shukla and B. Eliasson, “Colloquium: Fundamentals of dust-plasma interactions,” Rev. Mod. Phys., vol. 81, no. 1, pp. 25–44, 2009.

[16] P. V. Bliokh and V. V. Yarashenko, “Electrostatic waves in Saturn’srings,” Sov. Astron., vol. 29, no. 3, p. 330, 1985.

[17] N. N. Rao, P. K. Shukla, and M. Y. Yu, “Dust-acoustic waves in dustyplasmas,” Planet. Space Sci., vol. 38, no. 4, pp. 543–546, 1990.

[18] A. Barkan, R. L. Merlino, and N. D’Angelo, “Laboratory observationof the dust-acoustic wave mode,” Phys. Plasmas, vol. 2, no. 10,pp. 3563–3565, 1995.

[19] R. L. Merlino, A. Barkan, C. Thompson, and N. D’Angelo, “Laboratorystudies of waves and instabilities in dusty plasmas,” Phys. Plasmas,vol. 5, no. 5, pp. 1607–1614, 1998.

[20] M. Kakati and K. S. Goswami, “Solitary wave structures in presence ofnonisothermal ions in a dusty plasma,” Phys. Plasmas, vol. 5, no. 12,pp. 4508–4510, 1998.

[21] B. S. Xie, K. F. He, and Z. Q. Huang, “Effect of adiabatic variation ofdust charges on dust-acoustic solitary waves,” Phys. Lett. A, vol. 247,no. 6, pp. 403–409, 1998.

1010 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 4, APRIL 2013

[22] S. K. El-Labany, W. F. El-Taibany, A. A. Mamum, and W. M. Moslem,“Dust acoustic solitary waves and double layers in a dusty plasmawith two-temperature trapped ions,” Phys. Plasmas, vol. 11, no. 3,pp. 926–933, 2004.

[23] S. K. El-Labany, E. F. El-Shamy, and S. A. El-Warraki, “Effects oftwo-temperature electrons, external oblique magnetic field, and higher-order nonlinearity on dust acoustic solitary waves in a dusty plasma withvortex-like ion distribution,” Astrophys. Space Sci., vol. 315, nos. 1–4,pp. 287–295, 2008.

[24] C. R. Choi, C. M. Ryu, N. C. Lee, D. Y. Lee, and Y. Kim, “Ion thermalpressure effects on dust ion acoustic solitary waves in a dusty plasmaobliquely propagating to an external magnetic field,” Phys. Plasmas,vol. 12, no. 7, pp. 072301-1–072301-6, 2005.

[25] C. Lin and M. M. Lin, “Analytical study of the dust acoustic waves ina magnetized dusty plasma with many different dust grains,” Adv. Stud.Theory Phys., vol. 1, no. 12, pp. 563–570, 2007.

[26] C. Lin, W. G. Feng, and M. M. Lin, Communications in NonlinearScience and Numerical Simulation, vol. 13. New York, USA: Elsevier,2008.

[27] A. A. Mamun, “Compressive and rarefactive dust-acoustic solitary struc-tures in a magnetized two-ion-temperature dusty plasma,” Astrophys.Space Sci., vol. 260, pp. 507–514, May 1999.

[28] A. A. Mamun, “Coexistence of positive and negative solitary potentialstructures in dusty plasma,” Phys. Lett. A, vol. 372, no. 5, pp. 686–689,2008.

[29] W. Moslem, “Higher order solution of the dust ion acoustic solitonsin a warm dusty plasma with vortex-like electron distribution,” Chaos,Solitons Fractals, vol. 23, no. 2, pp. 581–588, 2005.

[30] S. Bagchi, A. P. Misra, K. R. Chowdhury, and A. R. Chowdhury, “Non-linear oscillations in a magnetized dusty plasma with two-temperaturetrapped ions,” Chaos, Solitons Fractals, vol. 40, no. 2, pp. 758-–765,Apr. 2009.

[31] M. Asaduzzaman and A. A. Mamun, “Compressive and rarefactive dust-acoustic Gardner solitons beyond the K-dV limit with two-temperatureions dusty plasma,” Astrophys. Space Sci., vol. 341, no. 2, pp. 535–542,2012.

[32] I. Tasnim, M. M. Masud, and A. A. Mamun, “Effects of nonther-mal ions of distinct temperatures on dust acoustic shock waves in adusty plasma,” Astrophys. Space Sci., vol. 343, no. 2, pp. 647–652,2013.

[33] M. M. Masud and A. A. Mamun, “Physics of non-Abelian matter inhigh energy hadronic and nuclear collisions,” JETP Lett., vol. 96, no. 8,pp. 554–557, 2012.

[34] H. Washimi and T. Taniuti, “Propagation of ion-acoustic solitary wavesof small amplitude,” Phys. Rev. Lett., vol. 17, no. 19, pp. 996–998,1966.

Esmaeil Eslami was born in Ghaemshar, Iran,in 1973. He received the Ph.D. degree in plasmaphysics from the University of Joseph Fourier,Grenoble, France, in 2005.

He is currently an Assistant Professor of physicswith the Iran University of Science and Technol-ogy, Tehran. His current research interests includetheoretical and experimental plasma physics whichinvolves collective processes in dusty plasmas, lowtemperature plasma, laser-produced plasma, andlaser-plasma interaction.

Rasool Baraz was born in Tehran, Iran, in 1982.He received the B.Sc. degree in physics from theUniversity of Semnan, Semnan, Iran, and the M.Sc.degree in plasma physics from the University ofSahand, Tabriz, Iran, in 2008 and 2010, respectively.

His current research interests include theoreticaland computational plasma physics which involvescollective processes in dusty plasmas.