Nonlinear dust acoustic waves in inhomogeneous four-component dustyplasma with opposite charge polarity dust grainsW. F. El-Taibany Citation: Phys. Plasmas 20, 093701 (2013); doi: 10.1063/1.4820939 View online: http://dx.doi.org/10.1063/1.4820939 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v20/i9 Published by the AIP Publishing LLC. Additional information on Phys. PlasmasJournal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors

Nonlinear dust acoustic waves in inhomogeneous four-componentdusty plasma with opposite charge polarity dust grains

W. F. El-Taibanya)

Department of Physics, Faculty of Science, Damietta University, New Damietta P. O. 34517, Egypt

(Received 21 March 2013; accepted 16 August 2013; published online 10 September 2013)

The reductive perturbation technique is employed to investigate the propagation properties of

nonlinear dust acoustic (DA) waves in a four-component inhomogeneous dusty plasma (4CIDP). The

4CIDP consists of both positive- and negative-charge dust grains, characterized by different mass,

temperature, and density, in addition to a background of Maxwellian electrons and ions. The

inhomogeneity caused by nonuniform equilibrium values of particle densities, fluid velocities, and

electrostatic potential leads to a significant modification to the nature of nonlinear DA solitary waves.

It is found that this model reveals two DA wave velocities, one slow, ks, and the other is fast, kf. The

nonlinear wave evolution is governed by a modified Kortweg-de Vries equation, whose coefficients

are space dependent. Both the two soliton types; compressive and rarefactive are allowed

corresponding to ks. However, only compressive soliton is created corresponding to kf. The

numerical investigations illustrate the dependence of the soliton amplitude, width, and velocity on the

plasma inhomogeneities in each case. The relevance of these theoretical results with 4CIDPs

observed in a multi-component plasma configurations in the polar mesosphere is discussed. VC 2013AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4820939]

I. INTRODUCTION

Dusty plasmas (DPs) have received much care in the past

decades because of a number of observations in planetary

rings, cometary tails, the early Universe, supernovae, interstel-

lar cosmic matter, dense molecular clouds, the Martian atmos-

phere, Earth’s mesosphere, and laboratory experiments.1–4

Also, DPs have recently attracted the attention of industries

due to the contaminants formed during the plasma processing

(deposition, etching, etc.) of thin films. Experiments were con-

ducted in order to understand the origin and the evolution of

these contaminants. DP has further enhanced interest by the

experimental observation of dust acoustic (DA) waves5 and

the formation of Coulomb crystals in the laboratory.6

Theoretically, Rao et al.7 first predicted the existence of a

very low frequency mode known as the “dust acoustic mode.”Later, many authors have discussed various aspects of linear

and nonlinear wave propagations in DPs. Although most of

these analyses assume the plasmas to be homogenous, inho-

mogeneity exists widely in space and laboratory discharges.

The plasma inhomogeneity may stem from density8 and/or

temperature gradient.9 Singh and Rao10 presented the analysis

of linear and nonlinear DA wave propagations in inhomogene-

ous DPs, taking into account the gradients of the plasma

number density. It is found that the amplitude of the linear

DA waves is inversely proportional to the square root of the

equilibrium dust density N�1=2do , while the amplitude of the

nonlinear DA waves is proportional to N�1=4do . Considering

constant Ndo and spatially nonuniform dust charge, Li et al.11

proved the existence of compressive solitons whose velocities

show complicated behaviours. On the other hand, experiment

on observations of the dust density perturbations a neon dc

glow discharge in revealed that the phase velocity of the den-

sity perturbations varies considerably along the plasma

column.12

Dust grains are usually considered negatively charged as

they collect electrons from the background plasma.1,3 Whereas,

dust grains can acquire positive charge, either by photoemis-

sion under ultraviolet (UV) radiation or by thermionic emission

from grains heated by radiative sources.3,13–15 The presence of

positively charged dust particles has been observed in different

regions of space, such as cometary tails,2,16 Jupiter’s magneto-

sphere,17 the Earth’s polar mesosphere18 and in the Martian

atmosphere19 and also in some plasma experiment.20 Chow

et al.16 have shown that, due to the size effect on secondary

emission, insulating dust grains with different sizes in space

plasmas can have opposite polarity. Also, they have calculated

the equilibrium potential for insulating dust grains immersed in

both Maxwellian and generalized Lorentzian plasmas. It was

found that grains with diameters 0.01lm and 1 lm have oppo-

site polarity when the plasma temperature is in the range

25–48 eV for a Maxwellian plasma and in the range 7–17 eV

for a Lorentzian plasma. These values may be in the range of

the inferred values in different regions of planetary ring sys-

tems, comets, and in the interplanetary medium.2,16 El-Taibany

and Wadati21 have employed the Sagdeev potential analysis to

find out the existence domain of the DA solitary waves corre-

sponding to positive dust grain dynamics in nonthermal homo-

geneous DP.

Moreover, there is also direct evidence for the existence

of both positive and negative dusts in different regions of

space, viz. polar mesospheric summer echoes (PMSE),18,22

and a laboratory-produced DP.22 Recently, field experiments

on terrestrial dust devils, simulating significant meteorological

a)Present address: Department of Physics, College of Science for Girls in

Abha, King Khalid University, Abha P.O. 960, Kingdom of Saudi Arabia.

Electronic address: eltaibany@hotmail.com or eltaibany@kku.edu.sa

1070-664X/2013/20(9)/093701/7/$30.00 VC 2013 AIP Publishing LLC20, 093701-1

PHYSICS OF PLASMAS 20, 093701 (2013)

phenomena on Mars, showed some evidence of electric and

magnetic fields that are thought to be related to the simulta-

neous presence of positively and negatively charged dust

particles.23 The properties of DA solitons in a positive dust-

electron plasma have been analyzed by Wang et al.24

Homogeneous DP systems containing opposite polarity

grains have been investigated theoretically. Based on the

space observations,2,16 Mamun and Shukla25 have consid-

ered a very simple DP model, which assumes positive and

negative dust only, and they have theoretically investigated

the properties of linear and nonlinear electrostatic waves in

such DP model. The model of Mamun and Shukla25 is only

valid if a complete depletion of the background electrons

and ions is possible, and both positive and negative dust flu-

ids are cold. However, in most space DP systems, a complete

depletion of the background electrons and ions is not possi-

ble,26,27 and the positive dust component is of finite tempera-

ture. On the other hand, the nonlinear properties as well as

the propagation domain of DA waves in a four-component

DP containing both positive and negative dust grains are

investigated.28,29 Employing a reductive perturbation tech-

nique, Rahman et al.30 have calculated the condition under

which the DA shock waves would propagate in a dusty

plasma with dusts of opposite polarities. A comparison

between the modulational instability of the DA wave enve-

lopes in space and laboratory DP models is also presented.31

Since studying the effect of inhomogeneity in a DP model is

important due to its capability to illustrate some new features

that cannot be observed in the homogeneous model.8

Therefore, in the present work, a model for four-component

inhomogeneous DP (4CIDP) is presented. The nonlinear

properties of the DA waves will be discussed. The manu-

script is organized in the following fashion: in Sec. III, the

basic equations of the model are introduced and then carry-

ing out a nonlinear analysis of DA waves in the subsequent

section. In Sec. III, the reductive perturbation technique32 is

employed and the evolution equation, a modified Korteweg-

de Vries (MKdV) equation, is derived. The solution of the

MKdV equation is presented and numerical investigations

are provided along with discussion for the current findings.

Section IV is devoted for the conclusion.

II. THE MODEL

I consider a four-component collisionless, fully ionized,

unmagnetized warm inhomogeneuous DP consisting of elec-

trons (mass me; charge �e), ions (mass mi; charge þe), nega-

tively charged dust particles (mass m1, charge q1¼�Z1e)

and positively charged dust particles (mass m2, charge

q2¼þZ2e); the four species are henceforth denoted by the

indices e, i, 1, and 2, respectively. Adopting a one-

dimensional (1D) geometry, the normalized two-dust-fluid

model equations can be written as25–27,30

@n1

@tþ @ n1u1

@x¼ 0; (1)

@u1

@tþ u1

@u1

@xþ 3r1n1

@n1

@x¼ @/@x

; (2)

for negative dust,

@n2

@tþ @ n2u2

@x¼ 0; (3)

@u2

@tþ u2

@u2

@xþ 3r2n2

@n2

@x¼ �Z l

@/@x

; (4)

for positive dust, and Poisson’s equation is given by

@2/@x2¼ n1 � Z n2 þ ne � ni: (5)

Since I am interested in the low frequency, both electron and

ion species are assumed to be thermalized, i.e.,

neðxÞ ¼ NeoðxÞ expðb/Þ; (6)

niðxÞ ¼ NioðxÞ expð�/Þ: (7)

The normalized state variables in Eqs. (1)–(7) are: the j (¼1, 2)

dust fluid number density nj ¼ ~nj=N1oð0Þ, the fluid velocities

uj ¼ ~uj=Cd1, and the electrostatic potential / ¼ e ~/=Ti

(Boltzmann’s constant is omitted, where obvious). Here, the

tilde-marked parameters are the corresponding dimensional

ones. The reduced time and space variables are t ¼ ~t xpd1

and x ¼ ~x=kDd1. I have chosen to scale all quantities by

those of the negative dust species, where appropriate, so

xpd1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4pe2Z2

1 N1oð0Þ=m1

prepresents the plasma fre-

quency, Cd1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ1Ti=m1

pis the characteristic DA speed,

and kDd1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTi=½4pe2Z1N1oð0Þ�

pis the Debye length. I have

also defined the following parameters: the background elec-

tron and ion number density, NeoðxÞ ¼ ~neoðxÞ=½Z1N1oð0Þ�and NioðxÞ ¼ ~nioðxÞ=½Z1N1oð0Þ�, respectively; the tempera-

ture ratio(s) rj ¼ Tj=Z1Ti (with j¼ 1/2 for negative/positive

dust species, respectively). Zj is the number of charges resid-

ing on the j–dust species. Also the mass and (absolute)

charge ratios are l ¼ m1=m2 and Z¼Z2/Z1, respectively.

b ¼ Ti=Te.

At equilibrium, the charge neutrality reads

nioðxÞ þ Z N2oðxÞ ¼ neoðxÞ þ N1oðxÞ; (8)

where neoðxÞ¼NeoðxÞexpðbUoÞ and nioðxÞ¼NioðxÞexpð�UoÞ.Since the dust dynamic time, sd, is experimentally esti-

mated to be about 10�2 s (Refs. 3 and 5) and the dust charging

time, sch, for positive dust grains including four charging cur-

rents: photoemission, secondary electrons, as well as the elec-

tron and ion currents is calculated to be about 2� 10�5 s.21 In

addition, sch for negative dust grains is about 10�8–10�6 s

depending on the included charging currents.33 So, it is

clear that sch � sd in both cases which means the dust grain

reaches an equilibrium dust charge number very quickly and

this permits to consider constant dust charge in the

present model. In other words, the dust charge is always close

to the equilibrium value.34 So, I consider here constant dust

charge.

III. NONLINEAR DA WAVES

In this section, I study propagations of nonlinear DA

waves in a 4CIDP using the reductive perturbation method.32

093701-2 W. F. El-Taibany Phys. Plasmas 20, 093701 (2013)

In this analysis, I use a set of stretched variables, which is

appropriate for spatially inhomogeneous plasmas along with

the zeroth-order fluids velocities and field that are present in

the system due to density inhomogeneities. The suitable set

of stretched co-ordinates is

T ¼ e1=2

ðx

0

d nkðnÞ � t

� �and X ¼ e3=2x; (9)

where e is a small parameter, which measures the size of the

perturbation amplitude, and k is the normalized velocity of

the moving frame to be determined later self-consistently.

The plasma parameters Wj � ½n1; n2; ni; ne; u1; u2;/� can be

expanded as power series in e as

Wj ¼ Wjo þX11¼1

e1 wj1; (10)

where Wjo ¼ ½N1o;N2o;Nio;Neo;U1o;U2o;Uo�.Considering the spatial gradient only, I have the follow-

ing equation:

@Wjo

@T¼ @k@T¼ 0: (11)

At equilibrium, I find that

@N1oU1o

@X¼ 0;

@N2oU2o

@X¼ 0;

3r1

@N21o

@Xþ @U2

1o

@X� 2

@ Uo

@X¼ 0;

3r2

@N22o

@Xþ @U2

2o

@Xþ 2Zl

@ Uo

@X¼ 0:

9>>>>>>>>>>>=>>>>>>>>>>>;

(12)

Substituting Eqs. (9) and (10) into Eqs. (1)–(7), then collect-

ing terms of different powers of e, I see that the lowest order

leads to

n11 ¼ �N1oc1/1; u11 ¼ � ~k1c1/1;

n21 ¼ Z lN2oc2/1; u21 ¼ Z l~k2c2/1;

ne1 ¼ neoðxÞb/1; ni1 ¼ �nioðxÞ/1;

9>=>; (13)

with c1¼½~k2

1�3r1N21o��1; ~k1¼k�U1o; c2¼½~k

2

2�3r2N22o��1

,

and ~k2¼k�U2o.

By use of the Poisson’s equation, the linear velocity can

be determined from the following dispersion relation:

N1o c1 þ Z2lN2oc2 ¼ A; (14)

where A ¼ nioðxÞ þ bneoðxÞ. Equation (14) is a fourth-order

algebraic equation in k. For small values of the initial fluid

velocities compared to k, ðU1o and U2o � k; ~k1 � ~k2 ¼ kÞ, I

get two propagation DA wave velocities

kf ;s ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB 6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2 � Cpq

; (15)

where

B ¼ 3

2ðr1N2

1o þ r2N22oÞ þ

1

2AðN1o þ Z2N2olÞ;

C ¼ 3

AN1oN2oðr1N1olZ2 þ r2N2o þ 3r1r2N1oN2oÞ:

It is noted here that in Eq. (15), the positive (negative) sign

corresponds to the fast (slow) velocity, kf ðksÞ, respectively.

To get real and positive velocity values of kf and ks, the con-

dition B2 > C should be verified. On the other hand, putting

r1 ¼ r2 ¼ 0, i.e., for a cold 4CIDP system, I have only one

linear phase velocity, which is given by

ko ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðN1o þ Z2N2olÞ=A

p: (16)

It is obvious that the wave velocities, presented in Eqs. (15)

and (16), are strongly affected by the number density inho-

mogeneity of all plasma species, either in hot or cold 4CIDP

systems. Moreover, for three-component inhomogeneuous

DP (3CIDP), without positive dust grains, Eq. (15) becomes

k3comp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN1oð3r1N1o þ A�1Þ

p: (17)

Equation (17) coincides exactly with the relation derived in

Refs. 10 and 11 (by neglecting dust charge fluctuations) and

in Ref. 35 (by ignoring the electron nonthermal effect). It

remarked here that the limits of cold DP system, (16), or the

3CIDP lead to only one phase velocity ðk3comp / N1oÞ, how-

ever, this model results in finding two DA wave velocities,

(15) which are strongly influenced by the presence of the

second positive dust grains as well as the inhomogeneity

included in the proposed DP model. To highlight a physical

significance appreciation of the present results, I will numeri-

cally investigate the different characteristic properties of the

nonlinear DA waves in a typical DP.18,20,26 The equilibrium

plasma species densities are assumed8 to have the form

NioðxÞ¼Nioð0Þexp�ðaixÞ; NeoðxÞ¼Neoð0Þexp�ðaexÞ and

N10ðxÞ¼N1oð0Þexp�ða1xÞ. Here, ai¼�d lnNi0ðxÞ=dx,

ae¼�d lnNe0ðxÞ=dx, and ai¼�d lnN10ðxÞ=dx are the density

gradient scale length for ions, electrons and negative dust

species, respectively.8,10,11 The parameters those I have cho-

sen in the numerical calculations are relevant to different

regions of space, viz. cometary tails,2,16 to the PMSE,18,20,31

Jupiter’s magnetosphere,17 etc. I may now proceed by adopt-

ing a specific set of parameter values; Neoð0Þ¼ 0:3; Nioð0Þ¼0:5; N1oð0Þ¼ 1; r1¼ 10�5; r2¼ 10�4; z¼ 0:01; l¼ 100, and

b¼0.5. For this set of parameters, the condition B2 >C is

satisfied. Moreover, the neutrality condition, Eq. (8) should be

verified. Using this set of parameters, Eq. (8) is transformed to a

mathematical relation from which N2o(x) can be calculated. The

dependence of the controlling physical parameters on the behav-

iour of homogeneous DP has been already investigated.26,27,30

Therefore, the included inhomogeneity, especially the density

gradient will be focused in the following numerical investiga-

tions. Figure 1 shows the variations of ks and kf against x. It is

obvious that ks decreases [increases] as x [N2o(0)] increases.

Both of ks and kf increase as N2o(0) increases. Setting N2o(0)¼0

leads to only one DA velocity, which has a modified form of kf,

(15). This agrees with the analytical expression presented above.

Increasing of either ae or a1 results as reduction of ks and kf. As

ai increases, both of ks and kf increase. The variation of kf

093701-3 W. F. El-Taibany Phys. Plasmas 20, 093701 (2013)

against x strongly depends on the values of a1 and ai; for

a1 < 0:1 and ai > 0:8; kf increases as x increases.

Now, let us proceed to the next order of the perturbation

theory, I obtain the following set of nonlinear partial differ-

ential equations:

�~k1

@n12

@Tþ k

@

@XðN1ou11 þ n11U1oÞ

þ @ðn11u11Þ@T

þ N1o@u12

@T

� �¼ 0; (18)

�~k1

@u12

@Tþ u11

@u11

@Tþ 3r1 n11

@n11

@TþN1o

@n12

@Tþ @ðN1on11Þ

@X

� �

þ k@ðU1ou11Þ

@X� k

@/1

@Xþ @/2

@T

� �¼ 0; (19)

�~k2

@n22

@Tþ k

@

@XðN2ou21 þ n21U2oÞ

þ @ðn21u21Þ@T

þ N2o@u22

@T

� �¼ 0; (20)

�~k2

@u22

@Tþ u21

@u21

@Tþ 3r2 n21

@n21

@TþN2o

@n22

@Tþ @ðN2on21Þ

@X

� �

þ k@ðU2ou21Þ

@Xþ Z l k

@/1

@Xþ @/2

@T

� �¼ 0; (21)

1

k@2/1

@T2¼ n12 � Z n22 þ ðneobþ nioÞ/2 þ

1

2ðneob

2 � nioÞ/21:

(22)

Eliminating the second-order perturbed quantities from Eqs.

(18)–(22), I finally obtain a MKdV equation

@/1

@Xþ aðXÞ/1

@/1

@Tþ bðXÞ @

3/1

@T3þ cðXÞ/1 ¼ 0; (23)

where

dðXÞ ¼ �2k2ð~k2N2oZ2lc22 þ ~k1N1oc2

1Þ;

aðXÞ ¼ 1

df3½c3

1N1oð~k2

1 þ r1N21oÞ � Z3l2c3

2N2oð~k2

2 þ r2N22oÞ� þ neob

2 � niog;

bðXÞ ¼ �1=½k2dðXÞ�;

cðXÞ ¼ �kdðXÞ c1

~k1

@ðc1N1okÞ@X

þ N1o@½c1ð3r1N2

1o þ ~k1U1oÞ�@X

� �þc2lZ2 ~k2

@ðc2N2okÞ@X

þ N2o@½c2ð3r2N2

2o þ ~k2U2oÞ�@X

� �� �:

9>>>>>>>>>=>>>>>>>>>;(24)

Equation (23) is the general evolution equation adequate for describing nonlinear DA waves in 4CIDP, which have gradients in

particle number densities, velocities, and electrostatic potential. It is remarked that the fourth term’s coefficient; c, is responsible

for introducing the inhomogeneity effect. If c¼ 0, I get a KdV equation.26,27 On the other side, setting N2o¼U1o¼U2o¼ 0, the

coefficients presented in Eq (24) become

a3compðXÞ ¼ �ðk2 � 3r1N2

1oÞ2

2k3N1o

3N1oðk2 þ r1N21oÞ

ðk2 � 3r1N21oÞ

3þ neob

2 � nio

" #;

b3compðXÞ ¼ðk2 � 3r1N2

1oÞ2

2k5N1o

;

c3compðXÞ ¼k2 � 3r1N2

1o

2k1

N1o

@N1ok

k2 � 3r1N21o

!

@Xþ 3r1

k

@N2

1o

k2 � 3r1N21o

!

@X

2664

3775;

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;

(25)

that coincide with the results corresponding to a 3CIDP;

Singh and Rao,10 Li et al.11 (neglecting dust charging fluctu-

ation), and Singh and Bharuthram35 (omitting the nonther-

mal electron effect).

To get an analytical solution for Eq. (23), let us write its

solution as

/1ðX; TÞ ¼ HðX; TÞGðXÞ ¼ HðX; TÞexp�ð

cðXÞ dX; (26)

where G(X) is the amplitude factor, I get

@H

@Xþ �aðXÞH @H

@Tþ bðXÞ @

3H

@T3¼ 0; (27)

where �a ¼ a G. By the use of the following transformation:

T1 ¼ffiffiffi�a

b

rT and X1¼

ðXffiffiffiffiffi�a3

b

rdX: (28)

093701-4 W. F. El-Taibany Phys. Plasmas 20, 093701 (2013)

Equation (27) becomes

@H

@X1

þ H@H

@T1

þ @3H

@T31

¼ 0: (29)

Using the transformation g ¼ T1 � VX1, integrating twice and

using the boundary conditions; HðgÞ ! 0; @H =@g! 0;@2H =@g2 ! 0 as jgj ! 1, I finally get the soliton solution

H ¼ Hm sech2ðg=WÞ; (30)

where Hm¼ 3 V is the amplitude and W ¼ 2=ffiffiffiffiVp

is the

width. Combining Eqs. (26) and (30) leads to

/1ðX; TÞ ¼ Hm

"exp�

ðcðXÞ dX

!#sech2ðg=WÞ; (31)

the amplitude of the produced soliton in case of a cold

3CIDP, i.e., with N2o ¼ r1 ¼ 0; /1m3compo ¼ HmGðxÞjN2o¼r1¼0

¼ 3V½N1oðnioþ bneoÞ��1=4. The latter limit agrees exactly

with the result of Singh and Rao.10 On the other side, for a

cold 4CIDP, setting r1 ¼ r2 ’ 0 and k¼ ko, I have

GðXÞ ¼ ½ðN1o þ Z2N2olÞðnio þ bneoÞ��1=4;

it is clear that the amplitude factor is influenced by the exis-

tence of the second positive dust species. Figure 2 shows the

variation of the amplitude factor, G(X), against a1 and x. It

illustrates that G(X) increases as a1 or x increases. Based on

the properties of the KdV Eq. (27), I can deduce the main

characteristics of the produced soliton.36 Since the soliton

amplitude is inversely proportional to the nonlinear coeffi-

cient, �aðXÞ�1, its width is proportional to the square root of

the dispersion coefficient,ffiffiffiffiffiffiffiffiffiffib ðXÞ

p. Moreover, it is well-

known that the sign of the a(x) determines the type of the

produced soliton. Figure 3(a) shows that for small value of

l; l < 16, both of compressive and rarefactive solitons are

created corresponding to k ¼ ks. Moreover, Fig. 3(a) illus-

trates that for small l; l < 25, each amplitude curve has

local minimum, increasing l leads to a shift for this local

minimum to be appeared at lower values of x. On the other

hand, for higher values of l and by increasing l, the com-

pressive solitons disappear and only rarefactive one is

allowed. In Fig. 3(a), the transition form compressive soliton

to rarefactive one occurs at [a(X)]�1¼ 0; the corresponding

points appeared in Fig. 3(b) atffiffiffiffiffiffiffiffiffiffib ðXÞ

p¼ 0. At these transi-

tion points, no soliton solution is created. The results pre-

sented in Fig. 3 corresponding to k ¼ ks can be summarized

as: increasing x, leads to reduction of both of the compres-

sive soliton amplitude and its width. Whereas for rarefactive

soliton; l¼ 50, the width becomes wider with a smaller am-

plitude by increasing x, c.f. Figs. 3(c) and 3(d). In addition,

increasing ai (reducing a1) yields to smaller rarefactive soli-

ton amplitude with wider width. However, no effect is

observed for either the soliton amplitude or its width against

ae variation.

Figure 4 illustrates that only compressive soliton is

allowed corresponding to k ¼ kf . It is clear that increasing lleads to smaller amplitude and slimmer width. Increasing ai

(ae), the compressive soliton amplitude becomes larger with

smaller width, whereas both of the soliton amplitude and

width are reduced by a decrement of either a1 or x. It is noted

here that for a 3CIDP, Li et al.11 illustrated that their model

FIG. 2. G(X) is plotted against a1 and x for r1 ¼ r2 ¼ 0; Nioð0Þ¼ 0:5; Neoð0Þ ¼ 0:3; N1oð0Þ ¼ 1; ai ¼ 0:5; ae ¼ 1, and l¼ 100.

FIG. 1. The variations of ks ½kf � are plotted against x and different system

parameter changes for 4CIDP in panels (a), (c), (e), and (g) [(b), (d), (f), and

(h)] for Neoð0Þ ¼ 0:3; Nioð0Þ ¼ 0:5; N1oð0Þ ¼ 1; r1 ¼ 10�5; r2 ¼ 10�4;z ¼ 0:01; l ¼ 100, and b ¼ 0:5. In (a) and (b), against N2o with

ai ¼ 0:5; ae ¼ 1, and a1 ¼ 0:25. In (c) and (d), against a1 with ai ¼ 0:5 and

ae ¼ 1. In (e) and (f), against ai with a1 ¼ 0:25 and ae ¼ 1. In (g) and (h),

against ae with ai ¼ 0:5 and a1 ¼ 0:25.

093701-5 W. F. El-Taibany Phys. Plasmas 20, 093701 (2013)

supports only compressive soliton whose amplitude and

width increase by increasing x. The latter agrees with the

illustrations presented in Figs. 4(c) and 4(d).

IV. CONCLUSION

In summary, the nonlinear properties of the DA wave in

a 4CIDPs are investigated for the first time considering two

opposite polarity dust fluids. The considered 4CIDP is com-

posed of Maxwellian electrons and ions, in addition to both

positive and negative dust grain fluids, which differ in mass,

temperature, and density. The inhomogeneity effects due to

nonuniform equilibrium particle number densities, initial

streaming velocity, and electrostatic potential have been

included within the analytical study, though I focused on the

number densities inhomogeneity in the numerical

FIG. 4. For k ¼ kf ; ½aðXÞ��1fffiffiffiffiffiffiffiffiffiffibðXÞ

pg

is plotted against x in (a) and (c) {(b)

and (d)} with Ni(0)¼ 0.5, Ne(0)¼ 0.3,

and N1(0)¼ 1. In (a) and (b), for

ai ¼ 0:75; ae ¼ 1:5, and a1 ¼ 0:325,

solid curves belong to l¼ 60, red

dashed curves to l¼ 90, and blue dot-

ted curves to l¼ 120. In (c) and (d),

for l¼ 100, solid curves (reference

curves) belong to ai ¼ 0:75; ae ¼ 1:5;and a1 ¼ 0:325, red dashed curves to

ai¼ 0:75; ae ¼ 1; anda1 ¼ 0:325, blue

dotted curves to ai¼ 0:45; ae ¼ 1:5;anda1 ¼ 0:325, and green dotted-

dashed curves correspond to ai¼ 0:75;ae¼ 1:5; anda1 ¼ 0:125.

FIG. 3. For k ¼ ks; ½aðXÞ��1fffiffiffiffiffiffiffiffiffiffibðXÞ

pg

is plotted against x in (a) and (c)

{(b) and (d)} with Nið0Þ ¼ 0:5;Neð0Þ ¼ 0:3; and N1ð0Þ ¼ 1. In (a)

and (b), for ai ¼ 0:75; ae ¼ 1:5, and

a1 ¼ 0:5, solid curves belong to

l¼ 10, red dashed curves to l¼ 12,

blue dotted curves to l¼ 20, and green

dotted-dashed curves correspond to

l¼ 50. In (c) and (d), for l¼ 50, solid

curves (reference curves) belong to

ai ¼ 0:75; ae ¼ 1:5, and a1 ¼ 0:325,

blue dotted curves to ai ¼ 0:45; ae

¼ 1:5, and a1 ¼ 0:325, and green

dotted-dashed curves correspond to

ai ¼ 0:75; ae ¼ 1:5, and a1 ¼ 0:125.

093701-6 W. F. El-Taibany Phys. Plasmas 20, 093701 (2013)

investigations. The present model reveals two different DA

wave velocities; slow, ks, and fast, kf. It is found that ks

decreases as the inhomogeneity scale length parameter; x,

increases. On the other hand, the variation of kf against xdepends strongly on the inhomogeneity parameter ranges; a1

and a1 values. It is shown that the evolution equation appropri-

ate for describing these nonlinear DA waves is the MKdV Eq.

(23). The solution of the MKdV equation contains two parts,

one is the amplitude factor and the other has a sech2-bell shaped

soliton solution. The amplitude factor increases as either a1 or xincreases. Moreover, it is found that the two soliton types (com-

pressive and rarefactive) corresponding to ks though only com-

pressive soliton corresponds to kf. The present results agree

with the previous results presented in Refs. 10 and 11. The var-

iations of soliton amplitude and its width against species’ den-

sity gradient scale lengths, (ai, ae, and a1), and inhomogeneity

scale length, x, have been illustrated in Figs. 3 and 4.

By now, the different applications of the present model

should be clear to the reader. This problem is of relevance

in astrophysics, in relation with the recorded DA waves in

inhomogeneuous PMSE18,20,31 or with noctilucent clouds

(NLC),26,27,30 which have similar characteristics but are

observed at different altitudes of the Earth’s magnetosphere.

Finally, anticipating further extension of the present work,

including the role of the ambient magnetic field to study elec-

trostatic dust-cyclotron waves, may be an interesting direction.

1C. K. Goertz, Rev. Geophys. 27, 271, doi:10.1029/RG027i002p00271

(1989).2D. A. Mendis and M. Rosenberg, Annu. Rev. Astron. Astrophys. 32, 419

(1994); D. A. Mendis, Plasma Sources Sci. Technol. 11, A219 (2002).3P. K. Shukla and A. A. Mamun, Introduction to Dusty Plasma Physics(Institute of Physics Publishing Ltd., Bristol, 2002); S. V. Vladimirov and

K. Ostrikov, Physics and Applications of Complex Plasmas (Imperial

College Press, London, 2005).4S. V. Vladimirov and K. Ostrikov, Phys. Rep. 393, 175 (2004); V. E.

Fortov, A. V. Ivlev, S. A. Khrapak, A. G. Khrapak, and G. E. Morfill, ibid.421, 1 (2005).

5A. Barkan, R. L. Merlino, and N. D’Angelo, Phys. Plasmas 2, 3563

(1995).6G. E. Morfill and H. Thomas, J. Vac. Sci. Technol. A 14, 490 (1996).7N. N. Rao, P. K. Shukla, and M. Y. Yu, Planet. Space Sci. 38, 543 (1990).8W. F. El-Taibany, M. Wadati, and R. Sabry, Phys. Plasmas 14, 032304

(2007); W. F. El-Taibany and M. Wadati, ibid. 14, 042302 (2007).

9W. F. El-Taibany, W. M. Moslem, M. Wadati, and P. K. Shukla, Phys.

Lett. A 372, 4067 (2008).10S. V. Singh and N. N. Rao, Phys. Plasmas 5, 94 (1998).11Y.-f. Li, J. X. Ma, and D.-l. Xiao, Phys. Plasmas 11, 5108 (2004).12A. A. Samaryan, A. V. Chernyshev, O. F. Petrov, A. P. Nefedov, and V. E.

Fortov, JETP 92, 454 (2001).13N. Meyer-Vernet, Astron. Astrophys. 105, 98 (1982); E. C. Whipple, T. G.

Northrop, and D. A. Mendis, J. Geophys. Res. 90, 7405, doi:10.1029/

JA090iA08p07405 (1985).14V. W. Chow, D. A. Mendis, and M. Rosenberg, IEEE Trans. Plasma Sci.

22, 179 (1994); M. Rosenberg and D. A. Mendis, ibid. 23, 177 (1995);

M. Rosenberg, D. A. Mendis, and D. P. Sheehan, ibid. 27, 239 (1999).15B. Walch, M. Hor�anyi, and S. Robertson, Phys. Rev. Lett. 75, 838 (1995);

A. A. Sickafoose, J. E. Colwell, M. Hor�anyi, and S. Robertson, ibid. 84,

6034 (2000).16V. W. Chow, D. A. Mendis, and M. Rosenberg, J. Geophys. Res. 98,

19065, doi:10.1029/93JA02014 (1993).17M. Hor�anyi, G. E. Morfill, and E. Gr€un, Nature (London) 363, 144 (1993).18O. Havnes, J. Trøim, T. Blix, W. Mortensen, L. I. Naesheim, E. Thrane,

and T. Tønnesen, J. Geophys. Res. 101, 10839, doi:10.1029/96JA00003

(1996); O. Havnes, A. Torsten, and A. Brattli, Phys. Scr. T89, 133 (2001).19P. K. Shukla and M. Rosenberg, Phys. Scr. 73, 196 (2006).20N. D’Angelo, J. Phys. D: Appl. Phys. 37, 860 (2004).21W. F. El-Taibany and M. Wadati, Phys. Plasmas 14, 103703 (2007).22N. D’Angelo, Planet. Space Sci. 49, 1251 (2001); 50, 375 (2002).23W. M. Farrell, P. H. Smith, G. T. Delory, G. B. Hillard, J. R. Marshall, D.

Catling, M. Hecht, D. M. Tratt, N. Renno, M. D. Desch, S. A. Cummer,

J. G. Houser, and B. Johnson, J. Geophys. Res. 109, E03004, doi:10.1029/

2003JE002088 (2004).24Z. X. Wang, Y. Liu, J. Y. Liu, and X. Wang, Phys. Plasmas 12, 014505

(2005).25A. A. Mamun and P. K. Shukla, Geophys. Res. Lett. 29, 1870,

doi:10.1029/2002GL015219 (2002).26F. Sayed and A. A. Mamun, Phys. Plasmas 14, 014501 (2007).27A. A. Mamun, Phys. Rev. E 77, 026406 (2008).28P. H. Sakanaka and P. K. Shukla, Phys. Scr T84, 181 (2000); P. H.

Sakanaka and I. Spassovska, Braz. J. Phys. 33, 158 (2003); AIP Conf.

Proc. 784, 506 (2005).29F. Verheest, M. A. Hellberg, and I. Kourakis, Phys. Plasmas 15, 112309

(2008).30A. Rahman, A. A. Mamun, and S. M. K. Alam, Astrophys. Space Sci. 315,

243 (2008).31W. F. El-Taibany, I. Kourakis, and M. Wadati, Plasma Phys. Controlled

Fusion 50, 074003 (2008).32T. Taniuti and C. C. Wei, J. Phys. Soc. Jpn. 24, 941 (1968); N. Asano and

T. Taniuti, ibid. 27, 1059 (1969); N. Asano and H. Ono, ibid. 31, 1830

(1971).33S. K. El-Labany, W. F. El-Taibany, N. A. El-Bedwehy, and M. M. El-

Fayoumy, Eur. Phys. J. D 64, 375 (2011).34A. V. Ivelev and G. Morfill, Phys. Rev. E 63, 026412 (2001).35S. Singh and R. Bharuthram, IEEE Trans. Plasma Sci. 38, 852 (2010).36H. Alinejad and A. A. Mamun, Phys. Plasmas 18, 073706 (2011).

093701-7 W. F. El-Taibany Phys. Plasmas 20, 093701 (2013)