Research Article Effect of Size Distribution on the Dust

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2013, Article ID 389365, 6 pageshttp://dx.doi.org/10.1155/2013/389365

Research ArticleEffect of Size Distribution on the Dust Acoustic Solitary Wavesin Dusty Plasma with Two Kinds of Nonthermal Ions

Samira Sharif Moghadam1 and Davoud Dorranian2

1 Physics Department, Science Faculty, North Tehran Branch, Islamic Azad University, Tehran, Iran2 Laser Lab., Plasma Physics Research Center, Science and Research Branch, Islamic Azad University, Tehran, Iran

Correspondence should be addressed to Samira Sharif Moghadam; [email protected]

Received 24 May 2013; Accepted 14 July 2013

Academic Editor: Mahmood Ghoranneviss

Copyright © 2013 S. Sharif Moghadam and D. Dorranian. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Effect of dust size,mass, and charge distributions on the nonlinear dust acoustic solitarywaves (DASWs) in a dusty plasma includingnegatively charged dust particles, electrons, and nonthermal ions has been studied analytically. Dust particles masses and electricalcharges are assumed to be proportional with dust size. Using reductive perturbation methods the Kadomtsev-Petviashvili (KP)equation is derived and its solitary answers are extracted.The coefficients of nonlinear term of KP equation are affected strongly bythe size of dust particles when the relative size (the ratio of the largest dust radius to smallest dust radius) is smaller than 2. Thesecoefficients are very sensitive to 𝛼, the nonthermal coefficient. According to the results, only rarefactive DASWs will generate insuch dusty plasma. Width of DASW increases with increasing the relative size and nonthermal coefficient, while their amplitudedecreases. The dust cyclotron frequency changes with relative size of dust particles.

1. Introduction

The study of dusty plasmas represents one of themost rapidlygrowing branches of plasma physics. A dusty plasma is anionized gas containing small particles of solid matter, whichacquire a large electric charge by collecting electrons andions from the plasma. Dust grains or particles are usuallyhighly charged. Charged dust components appear naturallyin space environments such as planetary rings, cometarysurroundings, interstellar clouds, and lower parts of earth’sionospheres [1–4]. Fusion reactors, film deposition reactors,and ions courses are other examples of dusty plasma systemsin laboratory. Since dust particles are relatively massive, thefrequency of DASW is typically few Hz and propagates ata speed of a few cm/s. This wave is spontaneously excitedin the plasma due to ion drift relative to the dust. In mostinvestigations, reductive perturbation method has been usedfor deriving the Korteweg de Vries (KdV) or modified KdVequations in one-dimensional studies [5–8] or Kadomstev-Petviashvili (KP) equation for two-dimensional cases [9, 10].

There aremany reports on the formation and propagationof DASW in dusty plasma. But far from ideal condition, thereare several phenomena which affect the DASW. Presence ofnonthermal particles and size distribution are two of them.A nonthermal plasma is in general any plasma which isnot in thermodynamic equilibrium, either because the iontemperature is different from the electron temperature, orbecause the velocity distribution of one of the species doesnot follow a Maxwell-Boltzmann distribution. The effects ofdust temperature and dust charge variation in a dusty plasmawith nonthermal ions have been investigated by Duan [11].Pakzad and Javidan have worked on the problem of variabledust charge in a dusty plasmawith two-temperature ions withnonthermal electrons [12, 13]. Dorranian and Sabetkar haveinvestigated the characteristic features of dust particles in adusty plasma with two ions at two different temperaturesand have observed the effect of nonthermal ions on theamplitude and width of DASW [14]. All of these theorieshave focused on dusty plasma containing only monosizeddust grains. In fact, the dust particles have many different

2 Advances in Materials Science and Engineering

species of size [15–17]. The dust size distribution can bedescribed by a power law distribution in space plasma andgiven by a Gaussian distribution in laboratory plasma [18].The effects of dust size distribution on the dust acousticwaves,solitons, or instabilities in a dusty plasma have been studiedby several researchers previously [15, 17]. The effect of dustsize distribution for two ion temperature dusty plasma inone dimension (KdV equation) was studied by Duan and Shi[19]. He et al. observed the effect of dust size distribution onthe nature of DASW in two dimensions (KP equation) [20].In this work the collective behavior of dust particles is notconsidered, and the effect of relative size is not taken intoaccount.

Continuing the former works, we have added the effectof dust size distribution to the model which contains twononthermal ions at two temperatures, including Maxwellianelectrons. In this paper, characteristic features of dust par-ticles with variable size, mass, and electric charge in a non-thermal two-temperature ions dusty plasma are investigated.The mass and charge of dust particles are assumed to beproportional with their size.The paper is organized as follow:following the introduction in Section 1, in Section 2, wepresent the model description and the theoretical aspectsof the model. The KP equation and its solitary answer arederived and described in this section. Section 3 is devoted toresults and discussion, and Section 4 includes the conclusion.

2. Model Description

2.1. Basic Equations. We consider the propagation of DASWin collisionless, unmagnetized dusty plasma consisting ofhighly negatively charged dust grains, electrons, and two-temperature nonthermal ions. The dust particles are muchlarger and extremely more massive than ions or electrons,but the sizes of dust grains are different. We shall assumethat there are 𝑁 number of different dust grains sizes whosemasses are 𝑚𝑑𝑗(𝑗 = 1, 2, . . . 𝑁). In this case we have differentspecies of dusts with different size, which is counted with𝑗. Plasma is assumed to be quasineutral. The state of chargeneutrality can be written as

𝑛𝑒0 +

𝑁

∑

𝑗=1

𝑛𝑑𝑗0𝑍𝑑𝑗0 − 𝑛𝑖𝑙0 − 𝑛𝑖ℎ0 = 0 (1)

in which 𝑛𝑒0, 𝑛𝑖𝑙0, and 𝑛𝑖ℎ0 are the unpertubrbed number den-sity of electrons, low-temperature ions, and high-temperatureions, respectively. 𝑍𝑑𝑗0 is the unperturbed number of chargesresiding on the 𝑗th dust grainmeasured in the unit of electroncharge. The dust grains are assumed to be much smallerthan electron Debye length 𝜆𝑑𝑒 as well as the intergraindistance. Thus, we can treat the dust grains as heavy pointmasses with constant negative charge [21]. The governing

equations including the normalized Poisson’s equation, con-tinuity equation, and the equation of motion describing thedynamics of the dust particles in the plasma are

𝜕2𝜙

𝜕𝑥2+

𝜕2𝜙

𝜕𝑦2= 𝑛𝑒 +

𝑛

∑

𝑗=1

𝑛𝑑𝑗𝑍𝑑𝑗 − 𝑛𝑖𝑙 − 𝑛𝑖ℎ, (2a)

𝜕𝑛𝑑𝑗

𝜕𝑡

+

𝜕 (𝑛𝑑𝑗𝑢𝑑𝑗)

𝜕𝑥

+

𝜕 (𝑛𝑑𝑗V𝑑𝑗)

𝜕𝑦

= 0, (2b)

𝜕𝑢𝑑𝑗

𝜕𝑡

+ 𝑢𝑑𝑗

𝜕𝑢𝑑𝑗

𝜕𝑥

+ V𝑑𝑗𝜕𝑢𝑑𝑗

𝜕𝑦

=

𝑧𝑑𝑗

𝑚𝑑𝑗

𝜕𝜙

𝜕𝑥

, (2c)

𝜕V𝑑𝑗𝜕𝑡

+ 𝑢𝑑𝑗

𝜕V𝑑𝑗𝜕𝑥

+ V𝑑𝑗𝜕V𝑑𝑗𝜕𝑦

=

𝑧𝑑𝑗

𝑚𝑑𝑗

𝜕𝜙

𝜕𝑦

. (2d)

In these equations, 𝜙 is the electrostatic potential; 𝑛𝑒, 𝑛𝑖𝑙, and𝑛𝑖ℎ are the number density of electrons, low-temperature ionsand high-temperature ions respectively. 𝑍𝑑𝑗 and 𝑛𝑑𝑗 are thecharge number and the density number of the 𝑗th species ofdust particles, respectively. 𝑒 is the basic charge magnitude.𝑢𝑑𝑗 and V𝑑𝑗 are the components of the velocity of 𝑗th speciesof dust particles in the 𝑥- and 𝑦-directions, respectively. Forthe case of dusty plasma with two kinds of ions at differenttemperatures, 𝑇eff, the effective temperature is

𝑇eff = [1

𝑁tot𝑧𝑑0(

𝑛𝑒0

𝑇𝑒

+

𝑛𝑖𝑙0

𝑇𝑖𝑙

+

𝑛𝑖ℎ0

𝑇𝑖ℎ

)]

−1

, (3)

in which 𝑇𝑒, 𝑇𝑖𝑙, and 𝑇𝑖ℎ are the plasma electron temperatureand the temperatures of plasma ions at lower and highertemperatures, respectively. 𝑁tot is the total number densityof dust grains which can be written as 𝑁tot = ∑

𝑁

𝑗=1𝑛𝑑0𝑗; 𝑧𝑑0

is determined by the equation 𝑧𝑑0𝑁tot = ∑𝑁

𝑗=1𝑛𝑑0𝑗𝑧𝑑0𝑗. 𝑎 is

the average radii of dust size given by 𝑎 = ∑𝑁

𝑗=1𝑛𝑑0𝑗𝑎𝑗/𝑁tot.

The dust density is normalized by 𝑁tot; 𝑧𝑑𝑜𝑗 is normalizedby 𝑧𝑑0. The space coordinates 𝑥 and 𝑦, time 𝑡, the compo-nents of dust velocity 𝑢𝑑𝑗, V𝑑𝑗, and the electrostatic potential𝜙 are normalized by the effective Debye lenght 𝜆𝐷𝑑 =

(𝑇eff/4𝜋𝑁tot𝑍𝑑0𝑒2)1/2, the inverse of effective dust plasma

frequency 𝜔−1𝑝𝑑

= (𝑚𝑑/4𝜋𝑁tot𝑍𝑑02

𝑒2)1/2, the effective dust

acoustic speed 𝐶𝑑 = (𝑧𝑑0𝑇eff/𝑚𝑑)1/2, and 𝑇eff/𝑒, respectively,

where 𝑚𝑑 is the average mass of dust grains determined bythe equation of 𝑚𝑑 = ∑

𝑁

𝑗=1𝑛𝑑0𝑗𝑚𝑑𝑗/𝑁tot; 𝑚𝑑𝑗 is normalized

by𝑚𝑑.Ions in the plasma are assumed to be nonthermal, so their

density can be described by the known Maxwell Boltzmanndistribution function. For the dimensionless density number

Advances in Materials Science and Engineering 3

of electrons, 𝑛𝑒, lower temperature ions, 𝑛𝑖𝑙, and highertemperature ions, 𝑛𝑖ℎ, we have

𝑛𝑒 =

1

𝛿1 + 𝛿2 − 1

exp (𝛽1𝑠𝜙) , (4a)

𝑛𝑖𝑙 =

𝛿1

𝛿1 + 𝛿2 − 1

(1 +

4𝛼

1 + 3𝛼

𝑠𝜙 +

4𝛼

1 + 3𝛼

(𝑠𝜙)2) exp (−𝑠𝜙) ,

(4b)

𝑛𝑖ℎ =

𝛿2

𝛿1 + 𝛿2 − 1

× (1 +

4𝛼

1 + 3𝛼

𝛽3𝑠𝜙 +

4𝛼

1 + 3𝛼

(𝛽3𝑠𝜙)2) exp (−𝛽3𝑠𝜙) ,

(4c)

in which 𝛽1 = 𝑇𝑖𝑙/𝑇𝑒, 𝛽2 = 𝑇𝑖ℎ/𝑇𝑒, 𝛽3 = 𝛽1/𝛽2, 𝛿1 = 𝑛𝑖𝑙0/𝑛𝑒0,𝛿2 = 𝑛𝑖ℎ0/𝑛𝑒0, and 𝑠 = 𝑇eff/𝑇𝑖𝑙 = 𝛿1 + 𝛿2 − 1/𝛿1 + 𝛿2𝛽3 + 𝛽1. 𝛼is the nonthermal parameter that determines the number offast (nonthermal) ions [22, 23].

2.2. The KP Equation. In this part by using the basicequations, the so called KP equation in the plasma withvariable dust particle electric charge is derived. KP equationis a fundamental equation, and its solution describes thecharacteristic of DASW in detail. In order to simplify theequations based on the reductive perturbation theory, it isuseful to introduce three independent variables [24]:

𝜉 = 𝜀 (𝑥 − 𝜆𝑡) , (5a)

𝜏 = 𝜀3𝑡, (5b)

𝜂 = 𝜀2𝑦, (5c)

in which 𝜀 is the dimensionless expansion parameter andcharacterizes the strength of the nonlinearity of the system.𝜆 is the phase velocity of the wave along the 𝑥 direction.Different parameters of the dust particles can be expandedin terms of 𝜀 as

𝑛𝑑𝑗 = 𝑛𝑑𝑗0 + 𝜀2𝑛𝑑𝑗1 + 𝜀

4𝑛𝑑𝑗2 + ⋅ ⋅ ⋅ , (6a)

𝑢𝑑𝑗 = 𝜀2𝑢𝑑𝑗1 + 𝜀

4𝑢𝑑𝑗2 + ⋅ ⋅ ⋅ , (6b)

V𝑑𝑗 = 𝜀3V𝑑𝑗1 + 𝜀

5V𝑑𝑗2 + ⋅ ⋅ ⋅ , (6c)

𝜙 = 𝜀2𝜙1 + 𝜀

4𝜙2 + ⋅ ⋅ ⋅ .

(6d)

By substituting (5a), (5b), (5c), (6a), (6b), (6c), and (6d) in(2a), (2b), (2c), and (2d) and collecting the coefficient of thelowest order of 𝜀, we have

𝑛𝑑𝑗1 = −

𝑛𝑑𝑗0𝑧𝑑𝑗

𝜆2𝑚𝑑𝑗

𝜙1, 𝑢𝑑𝑗1 = −

𝑧𝑑𝑗

𝜆𝑚𝑑𝑗

𝜙1,

𝜕V𝑑𝑗1𝜕𝜉

= −

𝑧𝑑𝑗

𝜆𝑚𝑑𝑗

𝜕𝜙1

𝜕𝜂

,

(7)

𝜆 =[

[

𝑁

∑

𝑗=1

𝑛𝑑𝑗0𝑧2

𝑑𝑗

𝑚𝑑𝑗

(𝛿1 + 𝛿2𝛽3 + 𝛽1) (1 + 3𝛼)

(1 + 3𝛼) 𝛽1 + (1 − 𝛼) (𝛿1 + 𝛿2𝛽3)

]

]

1/2

. (8)

Tending 𝛼 → 0 results are the same as the resultspresented by He et al. [20], and neglecting the effect ofsize distribution, equations are the same with the resultspublished by Dorranian and Sabetkar [14]. For the higherpower of 𝜀, we have

𝜕𝑛𝑑𝑗1

𝜕𝜏

− 𝜆

𝜕𝑛𝑑𝑗2

𝜕𝜉

+

𝜕 (𝑛𝑑𝑗0𝑢𝑑𝑗2)

𝜕𝜉

+

𝜕 (𝑛𝑑𝑗1𝑢𝑑𝑗1)

𝜕𝜉

+

𝜕 (𝑛𝑑𝑗0V𝑑𝑗1)

𝜕𝜂

= 0,

(9a)

𝜕𝑢𝑑𝑗1

𝜕𝜏

− 𝜆

𝜕𝑢𝑑𝑗2

𝜕𝜉

+ 𝑢𝑑𝑗1

𝜕𝑢𝑑𝑗1

𝜕𝜉

=

𝑧𝑑𝑗

𝑚𝑑𝑗

𝜕𝜙2

𝜕𝜉

(9b)

𝜕V𝑑𝑗1𝜕𝜏

− 𝜆


+ 𝑢𝑑𝑗1


=

𝑧𝑑𝑗

𝑚𝑑𝑗

𝜕𝜙2

𝜕𝜂

, (9c)

𝜕2𝜙1

𝜕𝜉2=

𝑁

∑

𝑗=1

𝑛𝑑𝑗2𝑧𝑑𝑗 +

(1 + 3𝛼) 𝛽1 + (1 − 𝛼) (𝛿1 + 𝛿2𝛽3)

(𝛿1 + 𝛿2𝛽3 + 𝛽1) (1 + 3𝛼)

𝜙2

−

𝑠2(𝛿1 + 𝛿2𝛽

2

3− 𝛽2

1)

2 (𝛿1 + 𝛿2 − 1)

𝜙2

1.

(9d)

Using (9a), (9b), (9c), and (9d) the KP equation for aunmagnetized dusty plasma with two ions can be derived as

𝜕

𝜕𝜉

[

𝜕𝜙1

𝜕𝜏

+ 𝐴𝜙1

𝜕𝜙1

𝜕𝜉

+ 𝐵

𝜕3𝜙1

𝜕𝜉3] + 𝐶

𝜕2𝜙1

𝜕𝜂2= 0, (10)

in which

𝐴 =

𝜆

2

(𝛿1 + 𝛿2𝛽2

3− 𝛽2

1) (𝛿1 + 𝛿2 − 1)

(𝛿1 + 𝛿2𝛽3 + 𝛽1)2

(𝛿1 + 𝛿2𝛽2

3− 𝛽2

1)

(𝛿1 + 𝛿2 − 1)

×

(𝛿1 + 𝛿2𝛽3 + 𝛽1) (1 + 3𝛼)

(1 + 3𝛼) 𝛽1 + (1 − 𝛼) (𝛿1 + 𝛿2𝛽3)

−

3

2𝜆3

𝑁

∑

𝑗=1

𝑛𝑑𝑗0𝑧3

𝑑𝑗

𝑚2

𝑑𝑗

(𝛿1 + 𝛿2𝛽3 + 𝛽1) (1 + 3𝛼)

(1 + 3𝛼) 𝛽1 + (1 − 𝛼) (𝛿1 + 𝛿2𝛽3)

,

(11)

𝐵 =

𝜆

2

(𝛿1 + 𝛿2𝛽3 + 𝛽1) (1 + 3𝛼)

(1 + 3𝛼) 𝛽1 + (1 − 𝛼) (𝛿1 + 𝛿2𝛽3)

, (12)

𝐶 =

𝜆

2

. (13)

For dust grains with radii 𝑎 in a given range [𝑎min, 𝑎max], thedifferential polynomial expressed distribution function is ofthe form [19]

𝑛 (𝑎) 𝑑𝑎 = 𝐾𝑎−𝛽𝑑𝑎. (14)

It is found that values of 𝛽 = 4, 5 for the F-ring ofSaturn, while for the G-ring values of 𝛽 = 7 or 6. For


cometary environments, we recall a value of 𝛽 = 3, 4. 𝐾 isa normalization constant, and 𝑎 is the radii of dust particle,which should satisfy the following equation [19]:

𝑛tot = ∫𝑎max

𝑎min

𝑛 (𝑎) 𝑑𝑎. (15)

Outside the limits 𝑎min < 𝑎 < 𝑎max, we have 𝑛𝑎 = 0. Inthe equations 𝑧𝑑𝑗 = 𝑘𝑧𝑎𝑗 and 𝑚𝑑𝑗 = 𝑘𝑚𝑎

3

𝑗, 𝑘𝑧 = 4𝜋𝜀0V0/𝑒

is the electrical potential of the surface of dust particles atequilibrium. 𝑘𝑚 = 4/3𝜋𝜌𝑑, in which 𝜌𝑑 is the mass density ofdust particles, 𝜌𝑑 is taken to be constant for all dust particles,V0 is the electric surface potential at equilibrium, and 𝜀0 isthe vacuum permittivity [25]. Integrating (8), (11), using theprevious definitions, we have

𝜆 = [

(𝛿1 + 𝛿2𝛽3 + 𝛽1) (1 + 3𝛼)

(1 + 3𝛼) 𝛽1 + (1 − 𝛼) (𝛿1 + 𝛿2𝛽3)

×

𝐾𝑘2

𝑧

𝑘𝑚𝛽

(𝑎−𝛽

min − 𝑎−𝛽

max)]

1/2

,

𝐴 =

𝜆

2

(𝛿1 + 𝛿2𝛽2

3− 𝛽2

1) (𝛿1 + 𝛿2 − 1)

(𝛿1 + 𝛿2𝛽3 + 𝛽1)2

(𝛿1 + 𝛿2𝛽2

3− 𝛽2

1)

(𝛿1 + 𝛿2 − 1)

×

(𝛿1 + 𝛿2𝛽3 + 𝛽1) (1 + 3𝛼)

(1 + 3𝛼) 𝛽1 + (1 − 𝛼) (𝛿1 + 𝛿2𝛽3)

+

3𝐾𝑘3

𝑧

2𝜆3𝑘2𝑚(𝛽 + 2)

(𝛿1 + 𝛿2𝛽3 + 𝛽1) (1 + 3𝛼)

(1 + 3𝛼) 𝛽1 + (1 − 𝛼) (𝛿1 + 𝛿2𝛽3)

× (𝑎−𝛽−2

max − 𝑎−𝛽−2

min ) .

(16)

The general steady state solution of (10) for 𝜙1 has the form[14]

𝜙1 = 𝜙𝑚sech2(

𝜒

𝑤

) , (17)

in which the new variable 𝜒 = 𝜉 + 𝜂 − 𝑢𝜏 is the transformedcoordinate relative to the frame which moves with thevelocity 𝑢 [14]; 𝜙𝑚 = 3(𝑢−𝐶)/𝐴 and𝑤 = 2√𝐵/(𝑢 − 𝐶) are theamplitude and width of soliton, respectively. The coefficientsof KP equation would change to the results presented in[14, 20] in case neglecting of nonthermal effects and sizedistribution effects, respectively.

3. Results and Discussion

Effects of ions temperature and density on the coefficientsof KP equation are very small. In other words, variations ofions temperature and density lead to negligible effects on thecoefficients of KP equation in this model.

To plot our data, 𝑎min is taken to be unity. In order toavoid the effects of ions density and temperature, 𝛽1 and 𝛽2are taken to be equal to 0.25, and 𝛿1 and 𝛿2 are both taken tobe 2.

×106

A

0

−2

−4

−6

−86

42

0 00.5

1

𝛼c

Figure 1: Variation of 𝐴 versus 𝑐 and 𝛼.

1500

1000

500

06

42

0 00.5

1

𝛼

B

42 0 5

c

Figure 2: Variation of 𝐵 versus 𝑐 and 𝛼.

For a typical dusty plasma [19], 𝑛𝑖0 ∼ 105–1010 cm−3,

𝑇𝑖 ∼ 𝑇𝑒 ∼ 0.1 eV, 𝑛𝑑0 ∼ 105 cm−3, 𝑄𝑑 ∼ (−10

4 e)–(−105 e),𝑚𝑑 ∼ 10

−15–10−12 𝑔 ∼ 109–1012m, 𝜌𝑑 ∼ 1 g/cm3, and

𝑎 ∼ 0.1–1 𝜇m, so it can be estimated that 𝑘𝑚 = 4, 𝑘𝑧 = 108,

and𝐾 = 10−12. Here 𝛽 is taken to be 6.

The variation of the nonlinear coefficient, 𝐴, with themaximum dust size ratio 𝑐 = 𝑎max/𝑎min and the nonthermalcoefficient 𝛼 is presented in Figure 1. 𝐴 increases withincreasing 𝑐. For all the ranges of 1 < 𝑐 < 5 and 0 < 𝛼 < 1, thenonlinear coefficient 𝐴 is negative, mean that in this range ofvariation only rarefactive solitons will be generated in dustyplasma, and it is not useful to continue our work for modifiedKP equation. This point is similar with the presented resultsin [20] in which the effect of size distribution is taken intoaccount. But in Dorranian and Sabetkar’s work [14], in thepresence of thermal effect without size distribution effect, justin the case of small 𝛼, rarefactive soliton will appear in thedusty plasma medium. As the function of 𝑐, 𝐴 varies fasterwhen 1 < 𝑐 < 2. When the magnitude of the maximum ratioof dust size is larger than 2,𝐴 is approximately constant.Withincreasing 𝛼 also 𝐴 increases. Variation of 𝐴 with 𝛼 is largerwhen 1 < 𝑐 < 2. With increasing 𝑐, the effect of 𝛼 decreases.Effect of 𝛼 is noticeable when 𝑐 is small.

Effects of 𝑐 and 𝛼 on 𝐵 are presented in Figure 2. Incontrast with 𝐴, effect of 𝛼 on 𝐵 is noticeable when 𝑐 is large.

Advances in Materials Science and Engineering 5

80

60

40

20

064

20 0

0.51

𝛼

C

c

Figure 3: Variation of 𝐶 versus 𝑐 and 𝛼.

For 𝑐 → 1, the coefficient 𝐵 is not changed with 𝛼. The sameresult can be seen for 𝑐. When 𝛼 is small, 𝐵 is not changedwith 𝑐.When nonthermality is high, increasing themaximumratio of dust size leads to increasing the coefficient 𝐵 of KPequation.

The same effect are occurred for the coefficient 𝐶. Effectof 𝑐 on 𝐶 is noticeable when 𝛼 tends to 1, and 𝐶 increases fastwith 𝛼 when 𝑐 tends to 5 as is shown in Figure 3.

Width of generated soliton in this model stronglydepends on 𝑐 and 𝛼.This dependence is shown in Figure 4. Inany case, solitons are broadened with increasing 𝑐 and 𝛼. Thebroadest soliton generates when 𝛼 and 𝑐 are maximum at thesame time.The same results have been obtained inDorranianand Sabetkar’s paper when the effect of size distribution wasnot taken into account [14]. The rate of increasing solitonwidth with 𝛼 is smaller in the presence of size distributioneffect.

Profiles of generated solitons in dusty plasma regardingthe requirements of our model are presented in Figures 5and 6. In Figure 5, the effect of 𝛼 is shown, when 𝑐 = 5.With increasing 𝛼, the width of generated solitons increaseswhile their amplitude decreases. This result is in goodagreement with the results of works in which the effect ofsize distribution is neglected [14]. With increasing 𝑐 also theamplitude of generated solitons decreases but their width isincreased.

𝛼 is the nonthermal coefficient which changes the distri-bution function of ions from Boltzman distribution. Increas-ing 𝛼 leads to increasing the energy of ions in the dustyplasmamedium or decreasing the dust particles energy.Withdecreasing the energy of dust particles, the amplitude ofsolitons decreases. Less energetic particles aremore localized,so the width of generated solitons increases with increasing𝛼.

𝑐 is the ratio of the maximum dust particle size tothe minimum dust particle size. With increasing 𝑐 thewidth of the size distribution function of dust particles isincreased, while their amplitude decreases. Increasing 𝑐 isactually increasing the dust species. In this case, we havelarger variety of dust particles accompanied in solitonicoscillation. Increasing the ratio of dust particles size leadsto increasing the perturbation in their uniform motion. Theresult will be reducing the amplitude of solitons. Different size

201510505

43

2 1 0 0.2 0.4 0.6 0.8 1

𝛼

w

c

Figure 4: Variation of width versus 𝑐 and 𝛼.

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

1

1.5

2×10−3

𝛼 = 0.4

𝛼 = 0.6

𝛼 = 1

𝜙

𝜒

Figure 5: The profile of soliton at different 𝛼s, when 𝑐 = 5.

particles moving with different velocities lead to generatingless localized oscillations.

4. Conclusion

In the present paper, nonlinear dust acoustic solitary waves(DASWs) in a cold unmagnetized dusty plasma containingnegatively charged dust particles, electrons, and nonthermalions are investigated. The size of dust particles is takento be variable, and the charge and mass of dust particlesare supposed to be proportional with their size. For thispurpose, a reasonable normalization of the hydrodynamicand Poisson’s equations is used to derive the Kadomstev-Petviashvili (KP) equation for our dusty plasma system.

In this model DASWs are found to be rarefactive forall magnitudes of the maximum ratio of dust size 𝑐 and𝛼. In this case, soliton nature is under the influence ofsize distribution rather than nonthermal ions, since in thepresence of nonthermal ions without taking into account theeffect of size distribution in both rarefactive and compressivesolitons have been observed [14]. But, the results of themodelin which the effect of size distribution is taken into account isonly generation of rarefactive solitons is reported [26].

For 1 < 𝑐 < 5, amplitude of DASW varies noticeablywith 𝑐 rather than temperature and density of two species ofdusts particles. With increasing 𝑐, the amplitude of DASWdecreases sharply, and for 𝑐 > 2 it is almost independent of𝑐. With decreasing the nonthermal coefficient, the amplitude


−10 −8 −6 −4 −2 0 2 4 6 8 10−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2×10−5

c = 2

c = 3

c = 4

𝜙

𝜒

Figure 6: The profile of soliton at different 𝑐s, when 𝛼 = 0.2.

of DASW increases. The same results have been reportedby Dorranian and Sabetkar [14]. Decreasing the ratio ofmaximum size of dust to its minimum magnitude leads togeneration of more localized solitons, that is, decreasing thewidth of DASW. The most energetic, localized solitons arewhen 𝑐 = 1 that is, there is not any size, mass, or chargedistribution for dusts. Since in this case the dusty plasmamedium is uniform, it leads to formation of less perturbationin the medium.

References

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[5] R. Bharuthram and P. K. Shukla, “Large amplitude ion-acousticsolitons in a dusty plasma,” Planetary and Space Science, vol. 40,no. 7, pp. 973–977, 1992.

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[7] M. Wadati, “Wave propagation in nonlinear lattice. I,” Journalof the Physical Society of Japan, vol. 38, no. 3, pp. 673–680, 1975.

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[10] H. R. Pakzad and K. Javidan, “Solitary waves in dusty plasmaswith variable dust charge and two temperature ions,” Chaos,Solitons and Fractals, vol. 42, no. 5, pp. 2904–2913, 2009.

[11] W. S. Duan, “The effects of dust temperature and the dust chargevariation in a dusty plasma with vortex-like ion distribution,”Chinese Physics, vol. 12, no. 5, p. 479, 2003.



[14] D. Dorranian and A. Sabetkar, “Dust acoustic solitary waves ina dusty plasma with two kinds of nonthermal ions at differenttemperatures,” Physics of Plasmas, vol. 19, no. 1, Article ID013702, 6 pages, 2012.

[15] S. K. El-Labany, F. M. Safi, and W. M. Moslem, “Higher-orderZakharov-Kuznetsov equation for dust-acoustic solitary waveswith dust size distribution,” Planetary and Space Science, vol. 55,no. 14, pp. 2192–2202, 2007.

[16] B. Smith, T. Hyde, L. Matthews, J. Reay, M. Cook, and J.Schmoke, “Phase transitions in a dusty plasmawith two distinctparticle sizes,” Advances in Space Research, vol. 41, no. 9, pp.1510–1513, 2008.

[17] M. M. Lin and W. S. Duan, “The effect of dust size distributionfor the dust acoustic waves in a cold dusty plasma withnonthermal ions,” Physica Scripta, vol. 76, no. 4, p. 320, 2007.

[18] W.-S. Duan, “Weakly two-dimensional dust acoustic waves,”Physics of Plasmas, vol. 8, no. 8, pp. 3583–3586, 2001.

[19] W.-S. Duan and Y.-R. Shi, “The effect of dust size distributionfor two ion temperature dusty plasmas,” Chaos, Solitons andFractals, vol. 18, no. 2, pp. 321–328, 2003.

[20] G.-J. He, W.-S. Duan, and D.-X. Tian, “Effects of dust sizedistribution on dust acoustic waves in two-dimensional unmag-netized dusty plasma,” Physics of Plasmas, vol. 15, no. 4, ArticleID 043702, 6 pages, 2008.

[21] K. N. Ostrikov and M. Y. Yu, “Ion-acoustic surface waves on adielectric-dusty plasma interface,” IEEE Transactions on PlasmaScience, vol. 26, no. 1, pp. 100–103, 1998.

[22] R. A. Cairns, A. A. Mamun, R. Bingham et al., “Electro-static solitary structures in non-thermal plasmas,” GeophysicalResearch Letters, vol. 22, no. 20, pp. 2709–2712, 2011.

[23] A. A. Mamun, “Effects of ion temperature on electrostaticsolitary structures in nonthermal plasmas,” Physical Review E,vol. 55, no. 2, pp. 1852–1857, 1997.

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Research Article Effect of Size Distribution on the Dust