ARTICLE IN PRESS

0032-0633/$ - se

doi:10.1016/j.ps

�CorrespondE-mail addr

Planetary and Space Science 56 (2008) 510–518

www.elsevier.com/locate/pss

Numerical study of dust-ion-acoustic solitary waves in aninhomogeneous plasma

Yu Zhang, Wei-Hong Yang, J.X. Ma�, De-Long Xiao, You-Jun Hu

CAS Key Laboratory of Basic Plasma Physics, Department of Modern Physics, University of Science and Technology of China, Hefei,

Anhui 230026, People’s Republic of China

Received 2 August 2007; received in revised form 19 October 2007; accepted 29 October 2007

Available online 12 November 2007

Abstract

The hydrodynamic equations describing the propagation of nonlinear dust-ion-acoustic solitary waves in an inhomogeneous dusty

plasma are solved numerically, taking into consideration dust charging process. Equilibrium plasma densities and dust charge are

assumed to be spatially nonuniform, but dust number density is taken to be uniform. It is shown that wave evolutions depend sensitively

on the plasma density distributions and electron-to-ion density ratios. Numerical results show that an initial solitary pulse evolves into a

dispersive pulse with an oscillatory tail when the wave propagates from high- into low-density region and whereas the pulse can keep its

shape when it propagates in the opposite direction. The dust charging process results in the damping of the solitary pulses.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Simulation; Inhomogeneous plasma; Dust-ion-acoustic soliton; Dust charging

1. Introduction

The physics of complex (dusty) plasmas have beenvigorously explored during the past decade. Charged dustgrains exist in diverse environments such as planetaryrings, cometary tails, interplanetary medium and inter-stellar clouds, the lower part of the Earth’s ionosphere(Whipple et al., 1985; Angelis et al., 1988), as well as inlaboratory low-temperature plasmas (Barkan et al., 1995;Merlino et al., 1998). Due to the (predominantly) electronand ion fluxes hitting on the grain surface, the dustparticles are usually negatively charged.

Recently, many authors (Ma and Liu, 1997; Popel et al.,2003; Shukla, 2003; Xiao et al., 2005) have theoreticallyinvestigated various aspects of nonlinear acoustic wavepropagation in three-component plasmas, consisting ofelectrons, ions and dust grains. There can be different typesof acoustic modes for different time scales, in the form ofshocks, solitons and periodic wave trains. One such mode,an ultra low-frequency electrostatic mode called the dust-

e front matter r 2007 Elsevier Ltd. All rights reserved.

s.2007.10.004

ing author. Tel.: +86551 3601160; fax: +86 551 3601164.

ess: jxma@ustc.edu.cn (J.X. Ma).

acoustic (DA) wave was first theoretically predicted by Raoet al. (1990) by including the dust collective dynamics. Onthe contrary, at higher frequencies, Shukla and Silin (1992)studied the dust-ion-acoustic (DIA) waves involving onlyion and electron dynamics. Both these waves have beenexperimentally observed (Barkan et al., 1996; Prabhakaraand Thana, 1996; Nakamura et al., 1999).Coherent and dissipative structures of the DIA waves

have been one of the subjects of interest in recent years(Popel et al., 2003). If dissipative effects such as fluidviscosity and nonadiabatic dust charging are neglected,solitons can emerge as a balance between nonlinearity anddispersion. The soliton structure was observed along themain tail of the Halley comet (Voelzke and Matsuura,1998) and Saturn’s rings (Kotsarenko et al., 1998). Thenonlinear DIA solitons have been extensively analyzed bymany authors (Nakamura and Sarma, 2001; Mahmoodand Saleem, 2003; Popel et al., 2003; Li et al., 2004). Mostexisting studies concentrated mainly on ideal homogeneousdusty plasmas, whereas one encounters plasma inhomo-geneities in laboratory or in space in which the localenvironment changes as the wave propagates. It is thus ofinterest to investigate how this nonuniformity affects the

ARTICLE IN PRESSY. Zhang et al. / Planetary and Space Science 56 (2008) 510–518 511

soliton behavior, such as its evolution in the case ofdifferent plasma density distributions or along differentdirections with respect to the density gradient. Particularlydoes its profile suffer a significant deformation in theprogress?

The dust charging process involves the collection ofelectrons and ions by the dust grains through microscopicelectron and ion currents to the grain surface, because ofthe potential difference between the grain surface and itsambient plasma. Fluctuations of plasma parameters canvary these currents and therefore alter the dust-chargenumber (Popel et al., 2003). However, most pervious worksin dusty plasmas (Rao et al., 1990; Shukla and Silin, 1992;Merlino, 1997; Amin et al., 1998) considered fixed graincharges. When the dust-charge variation is taken intoaccount, the waves become damped (Melandso et al., 1993;Varma et al., 1993) due to the anomalous dissipationoriginating from the charging process. The effects of elasticand inelastic collisions of the plasma with the dust as wellas Landau damping can bring on dissipation mechanismsand result in nonlinear wave structures (Popel et al., 1996,2005).

So far, most theories on nonlinear DIA waves werebased on the Korteweg–deVries Burgers (KdVB) equationusing the reductive perturbation technique (RPT) (Washi-mi and Taniuti, 1966), introducing the stretched coordi-nates and then expanding the dependent variables intopower series. However, the wave propagation may not bedirectly observed in the stretched coordinates. Further-more, the RPT only contains the terms in the lowest orders,restricting its applicability to the weakly nonlinear DIAwaves. Some others (Popel et al., 2003; Ma and Liu, 1997;Mahmood and Saleem, 2003) utilized Sagdeev potentialapproach for large amplitude solitons, but the method isinaccessible for inhomogeneous cases.

In this paper, we study the DIA soliton propagation ininhomogeneous plasmas, taking into account the dust-charge variations. The plasma is assumed inhomogeneouswith spatially varying equilibrium ion and electrondensities as well as the dust charge, wherein the dustnumber density is assumed uniform. By numericallysolving the hydrodynamic equations, we investigate thecharacteristics of the soliton propagating against and alongthe density gradient. The results are then compared for thecases of different density distributions and differentelectron-to-ion density ratios. The effects of the dustcharging process in the different parameter regimes arealso investigated.

2. Basic equations and numerical method

Consider a one-dimensional dusty plasma consisting ofelectrons, ions, and negatively charged dust grains,including the grain charge variations. Assume all theequilibrium quantities vary with spatial coordinate x, withthe exception of the dust number density nd, which isassumed to be a constant. Five parameters will be involved

in the hydrodynamic equations, i.e., the density na(a ¼ e, i),the dust-charge number Zd, the electrostatic potential f,and the ion velocity vi. For simplicity, we introduce a set ofnormalized quantities

X ¼ x=lD; lD ¼ ½Te=4pe2ni0ðX 0Þ�1=2,

T ¼ opit; opi ¼ ½4pe2ni0ðX 0Þ=mi�1=2,

Na ¼ na=na0ðX 0Þ,

V i ¼ vi=cs; cs ¼ ðT e=miÞ1=2,

F ¼ ef=Te,

z ¼ e2Zd=rT e,

where opi and lD are the ion oscillation frequency andeffective Debye length evaluated at the left boundary, e isthe electronic charge, Te and mi are the electron tempera-ture and ion mass, respectively, r is the dust radius, and csis the ion-acoustic speed. In the above, the density na0(X0)represents its equilibrium value at the left boundaryX ¼ X0.

2.1. The equilibrium state

The equilibrium state is assumed nonuniform along theX direction with a given density profile Na0 ¼ Na0(X)(a ¼ e, i), and there is neither external electric field normacroscopic particle flow, F0 ¼ 0 and Va0 ¼ 0. Theelectron and ion temperatures Te and Ti are constants.The dust particles are assumed to be extremely massive,immobile point charges (vd0 ¼ 0 and nd0 ¼ constant),forming a stationary negative background. However, thecharge number on the grains z0(X) is space-dependent andself-consistently determined by the charging equation. Inthe dimensionless form, the quasi-neutrality conditionreads

d0Ne0ðX Þ þ ð1� d0Þz0ðX Þ=z0ðX 0Þ ¼ N i0ðX Þ, (1)

where d0 ¼ ne0(X0)/ni0(X0) represents the electron-to-iondensity ration at X ¼ X0 and correspondingly d ¼ ne0(X)/ni0(X) ¼ d0Ne0(X)/Ni0(X) will denote the local density ratio.Here and in the following, the subscript zero denotes theequilibrium quantities.In the orbital-motion-limited (OML) theory of dust

charging (Cui and Goree, 1994), the microscopic electronand ion currents collected by the dust grains are expressedas

I e0 ¼ �epr2vene0 expð�z0Þ, (2)

I i0 ¼ epr2vini0ð1þ sz0Þ, (3)

where s ¼ Te/Ti, and va ¼ ð8Ta=pmaÞ1=2 is the mean speed

of the a species. In the equilibrium state, the microscopiccharging currents balance (Ie0+Ii0 ¼ 0), thus we obtain

d0ðve=viÞNe0 expð�z0Þ ¼ N i0ð1þ sz0Þ. (4)

ARTICLE IN PRESSY. Zhang et al. / Planetary and Space Science 56 (2008) 510–518512

From Eqs. (1) and (4), z0(X) and Ne0(X) can be solvednumerically for a given ion density distribution Ni0(X).

2.2. The hydrodynamic equations for the wave propagation

Assume an arbitrary amplitude DIA soliton propagatesalong the X direction. On the time scale of the ion motion,the electron density is given by Boltzmann relation

Ne ¼ Ne0 expðFÞ. (5)

The ion dynamics are governed by the continuity andmomentum equations

qT N i þ qX ðN iV iÞ ¼ �GidN i, (6)

qT V i þ V iqX V i ¼ �qXF� GiV i, (7)

where Gid=nid/opi is the normalized capture rate of ions bythe dust grains, and Gi ¼ neffi =opi is the normalizedeffective frequency of the ion momentum-transfer colli-sions, which includes ion-neutral, ion-electron, ion-dustcharging, and ion-dust elastic collisions (Tsytovich andAngelis, 2004).

The heavy dust grains cannot dynamically respond tothe ion-acoustic waves. They only act as a negativebackground. On the other hand, the grain charging occurson a faster time scale that is comparable to the ion plasmafrequency, depending on the dust and plasma parameters.Thus the dust charge can vary with and influence the ion-acoustic waves. The dust charge qd is determined by dqd/dt ¼ Ie+Ii, which in the dimensionless form becomes (Xiaoet al., 2006)

qT z ¼ JeNe expð�zÞ � J iN ið1þ szÞ, (8)

where Ja ¼ e2prvana0ðX 0Þ=T eopi is the dimensionless char-ging flux of the a species. The system is closed by thePoisson equation

q2XF ¼ d0Ne þ ð1� d0Þz=z0ðX 0Þ �N i. (9)

2.3. Boundary conditions and initial values

Suppose the wave propagates in an infinite free space.However, the numerical calculations are done in finitespace with artificial numerical boundaries X ¼ X0 and X1.In order to prevent the reflection of the wave off thenumerical boundaries, we impose the free boundaryconditions at X ¼ X0 and X1 for the five dynamicquantities.

The initial values are adopted as follows. At T ¼ 0, aninitially soliton-like pulse

FðX Þ ¼ F0a sech2

ZdX 0=M

� �=w

� �(10)

for the electrostatic potential is assumed, where F0 is thepeak value of the initial pulse, and a(X), w(X), and M(X)are the spatially varying amplitude factor, width, andMach number of the pulse, respectively. It is assumed that

the initial amplitude is small such that the analyticalexpressions for a(X), w(X), and M(X) (Li et al., 2004) canbe used. Then the initial value for Ne is calculated directlyfrom Eq. (5). Since the initial soliton is weak, the initialvalues for the remaining three quantities can be drawnfrom F(X). From the weak DIA soliton theory, therelationships among the electrostatic potential, the ionvelocity, and the ion density are given by (Xiao et al., 2006)

V iðX Þ ¼ FðX Þ=VfðX Þ and

N iðX Þ ¼ N i0ðX Þ½1þ V iðX Þ=VfðX Þ�, ð11Þ

where

Vf ¼N i0 þ ð1� d0Þb1=z0ðX 0Þ

d0Ne0 þ ð1� d0Þb1=z0ðX 0Þ

� �1=2(12)

is the phase velocity of the linear DIA waves whichdepends on the equilibrium parameters, and b1 ¼ (1+sz0)/(1+s+sz0) characterizes the dust-charge perturbation(Tsytovich and Morfill, 2004). Finally, the initial valuefor the dust-charge number z is obtained from the Poissonequation (9).

2.4. Numerical method

To solve the set of hydrodynamic equations numerically,we rewrite Eqs. (6)–(8) in the matrix form

qU

qTþ

qF ðUÞ

qX¼ S, (13)

where U=[Ni, Vi, z] is a column matrix, F ðUÞ ¼

½N iV i;V2i =2þ F; 0� is a matrix function of U, and S ¼

½�GidN i;�GiV i; JeNe expð�zÞ � J iN ið1þ szÞ� is the sourceterm. The accurate 4th-order Runge–Kutta scheme isapplied consecutively to determine the time evolution ofU. The quantities at (n+1)th time step Unþ1

j is obtained bythat at nth time step Un

j

LðUjÞ ¼ �1

2DXðF jþ1 � F j�1Þ þ Sj,

K0j ¼ DT LðUnj Þ,

K1j ¼ DT L Unj þ

1

2K0j

� �,

K2j ¼ DT L Unj þ

1

2K1j

� �,

K3j ¼ DT LðUnj þ K2jÞ,

Unþ1j ¼ Un

j þ1

6ðK0j þ 2K1j þ 2K2j þ K3jÞ,

where the subscript j stands for the jth space grid. Thenthe successive over-relaxation (SOR) iteration method(Nicholls and Honig, 1991) is utilized to solve the PoissonEq. (9) for F, and consequently Ne is obtained fromEq. (5). In this way the time evolution and spatialdistribution of the five dynamic variables can be obtained.

ARTICLE IN PRESSY. Zhang et al. / Planetary and Space Science 56 (2008) 510–518 513

3. Simulation results and discussions

In the numerical simulation, the computational domainis �150pXp100 with free boundaries. We use a uniformspatial mesh of 1250 grid nodes, which are equally spacedwith spatial step DX ¼ 0.2 such that the details of thesoliton profiles in propagation can be observed. The timestep is chosen to be DT ¼ 0.2DX ¼ 0.04, satisfying theCFL condition (Leveque, 2002) for stable numericalsolutions. The typical laboratory dusty plasma parametersTe ¼ 2.0 eV, Ti ¼ 0.1 eV (s ¼ Te/Ti ¼ 20), ni0(0) ¼109 cm�3 and r ¼ 10�4 cm are used in the simulation. Herewe have ignored the collision terms.

It should be noted that the numerical simulation is basedon the fully nonlinear hydrodynamic equations. Thus, it

Fig. 1. (Color online) Time evolution of a small-amplitude ion-acoustic

soliton in a uniform two-component plasma without dust. The initial

soliton profile is taken to be F(X) ¼ 0.05 sech2[(X+100)/10.95].

Fig. 2. (Color online) (a) The variations of the equilibrium quantities with re

density profile. Dotted line: z0/z0(X0), dashed line: Ni0, and solid line: Ne0. (b)

can not only be used to study the weak soliton propagationas the RPT does, but also be applicable to cases ofarbitrary-amplitude solitons and pulses with other bound-ary conditions.

3.1. DIA soliton in hyperbolically decreasing-density

plasmas

In order to check the numerical code, we first present theresult of an initial small-amplitude soliton propagation inhomogenous two-component plasma. In this case, theequilibrium plasma density is Ne0,i0(X) ¼ 1 and the initialsoliton profile is taken to be F(X) ¼ 0.05sech2[(X+100)/10.95]. Fig. 1 shows the numerical result of the evolution ofF(X) with time. Though the time shown in the figure is upto T ¼ 120, the program runs for much longer time and thecalculated results remain stable. It is shown that the solitonkeeps its profile in the propagation with a speed of 1.0167,which is in agreement with the analytical prediction of asmall-amplitude K–dV soliton.We then apply the numerical code to investigate the DIA

soliton propagation in the inhomegeneous dusty plasmas.A hyperbolic tangential decreasing profile is consideredhere for the equilibrium ion density Ni0(X) ¼ [1�tanh(X/L)]/2, where L ¼ 50 is the density scale length. It is alsoassumed that d0 ¼ 0.8. Fig. 2 plots the variations of theequilibrium quantities, including z0/z0(X0), Ni0, Ne0, andthe dust-to-ion charge density ratio nd0z0/ni0 versus thedistance X. Since the dust density nd0 is assumed to be aconstant, the dust-charge number decreases with X.However the ratio nd0z0/ni0 increases as Ni0 and Ne0

decrease, suggesting that more percentage of negativecharges resides on the dust grains as the plasma densitiesbecome lower.

spect to distance for the case of the hyperbolic tangential decreasing ion

The ratio of the dust-to-ion charge density nd0z0/ni0 versus X.

ARTICLE IN PRESSY. Zhang et al. / Planetary and Space Science 56 (2008) 510–518514

For the initial soliton profile given by Eq. (10) withF0 ¼ 0.05, Fig. 3 displays the time evolution of theelectrostatic potential F, ion velocity Vi, and ion densityperturbation Ni�Ni0 with respect to X. It can be seen

Fig. 3. (Color online) Time evolution of an initial solitary pulse in the

nonuniform dusty plasma with Ni0(X) ¼ [1�tanh(X/50)]/2, d0 ¼ 0.8, and

F0 ¼ 0.05.

Fig. 4. Peak position of the pulse with respect to time.

clearly that, as the wave propagates down the densitygradient, both the amplitude and the width increase, whichis rather different from keeping constant as in thehomogeneous case. Furthermore, a weakly oscillatory taildevelops as the time progresses due to the dispersion effect.Fig. 4 shows the time-of-flight curve of the peak position ofthe solitary pulse in the propagation. It shows that thecurve remains linear when T is below 50, indicating that thepulse shape remains almost unchanged at this stage. WhenT is greater than 60, the curve bent upward, indicating theincrease in the pulse velocity, which corresponds to thewaveform deformation from the soliton shape as shown inlate stages in Fig. 3. It is interesting to note that theobserved trend of the speed as a function of time is similarto the one reported by Li et al. (2004) when the solitonpropagates from high to low-density regions. Since thepulse velocity is proportional to the phase velocity ofthe DIA wave and the latter increases with the increase ofthe percentage of the dust-charge density, the wave speedincreases with X as is obvious from Fig. 2(b).

3.2. Effect of different density distributions

As an comparison of the DIA solitary pulse propagationin plasmas with different density distributions, Fig. 5 plotsthe electrostatic potential profiles at two instants for thecases of exponentially decreasing Ni0(X) ¼ exp[�(X+100)/100] (dashed curves) and the hyperbolically decreasingNi0(X) ¼ [1�tanh(X/50)]/2 (solid curves) density distribu-tions. The initial soliton profile in the exponentiallydecreasing density case is calculated with the correspondingequilibrium quantity distributions and the parameters arethe same as in Fig. 3. It turns out that these two cases havenearly the same initial soliton profiles. It is found that atearly stages as in Fig. 5(a), the pulse profiles do not deviatesignificantly. However, at late stages as in Fig. 5(b), thepulse grows to a much larger amplitude in the case ofthe hyperbolic density distribution than that in the case

ARTICLE IN PRESS

Fig. 5. (Color online) Comparison of the electrostatic potential profiles in

the dusty plasma with different density distributions at (a) T ¼ 40 and (b)

T ¼ 80. Solid line: Ni0(X) ¼ [1�tanh(X/50)]/2, and dashed line:

Ni0(X) ¼ exp[�(X+100)/100]. Other parameters are the same as in Fig. 3.

Fig. 6. (Color online) Time evolution of the electrostatic potential profile

as the wave propagates into the higher density region. The equilibrium ion

density distribution is Ni0(X) ¼ [1+tanh(X/50)]/2+1 and the other

parameters are the same as in Fig. 3.

Fig. 7. (Color online) Time evolution of the electrostatic potential in the

inhomogeneous plasma with d0 ¼ 0.2. The other parameters are the same

as in Fig. 3.

Y. Zhang et al. / Planetary and Space Science 56 (2008) 510–518 515

of the exponential density distribution. Moreover, thehyperbolic density distribution case shows a more pro-nounced oscillatory tail. In both cases, the pulse speed doesnot change significantly.

3.3. Effect of propagation direction

In subsection 3.1 and 3.2, the solitons propagate in thedirection opposite to the density gradient. Here we assumethe equilibrium ion density to be Ni0(X) ¼ [1+tanh(X/50)]/2+1 such that the solitons propagate in the direction alongthe density gradient. The initial soliton profile is calculatedby Eq. (10) with the parameters the same as in Fig. 3. Fig. 6shows the time evolution of the electrostatic potentialprofile. It is shown that the wave amplitude and width areboth decreasing but the shape remains almost unchanged,without an oscillatory tail. It is also observed that the wavespeed is slower in this case than that in the abovesituations. Since here the electron-to-ion density ratio d

increases with X, the wave speed decreases in thepropagation.

3.4. Effect of electron-to-ion density ratio

Different electron-to-ion density ratio results in differentdust-charge density, and influences the distributions of theequilibrium quantities and the nonlinear wave propaga-tion. Here we assume d0 ¼ ne0(X0)/ni0(X0) ¼ 0.2, implyingthat the percentage of the dust-charge density is muchhigher than that in previous subsections. The equilibriumion density distribution is hyperbolic-tangentially decreas-ing and the other parameters are the same as in Fig. 3. Theinitial pulse shape is calculated by Eq. (10) for thecorresponding density distribution. The time evolution of

ARTICLE IN PRESS

Fig. 9. (Color online) Comparison of the electrostatic potential profiles at

T ¼ 80 for the cases without (solid line) and with (other lines) the dust

charge variation and for different ion densities at X ¼ X0. Other

parameters are the same as in Fig. 3.

Y. Zhang et al. / Planetary and Space Science 56 (2008) 510–518516

electrostatic potential is shown in Fig. 7. In the simulation,the initial soliton pulse begins to form a small weaklyoscillatory tail when T ¼ 17, and afterwards it developsinto an intense oscillation due to the dominant dispersioneffect. As the wave propagates down the density gradient,the oscillation enhances. The main soliton-like pulse beginsto enter the right boundary at about T ¼ 30 and has nearlypassed through at T ¼ 48. The small trailing pulsesincrease in amplitude and width, passing through the rightboundary successively as time elapses. However, thestrongly dispersive wave does not suffer obvious dissipa-tion as it propagates. In addition, the wave speed is muchgreater in this case than before because of the smaller valueof d.

In order to see quantitatively the strong dispersion effectin low d0 case, we compare, in Fig. 8, the simulation resultsof d0 ¼ 0.2 and 0.8 at T ¼ 60. It is obvious that thedispersive oscillations are much stronger in d0 ¼ 0.2 case.At this moment, the main pulse in d0 ¼ 0.2 case has alreadypassed the right boundary and the second pulse has grownin amplitude to a value as large as that of the main pulse ind0 ¼ 0.8 case.

To explain the result, we note that the dispersion arisesfrom the term on the left-hand side of the Poisson equationq2F/qX2

6¼0 (Xiao et al., 2006). This means that it is becauseof charge separation (deviation from quasi-neutrality). Theeffect of the Boltzmann distributed electrons is toneutralize the electric field of the charge separation in theion acoustic wave, i.e., the electrons tend to keep the quasi-neutrality and thus reduce the dispersion. The decrease ofthe electron-to-ion density ratio for the fixed ion densitymeans the lack of the neutralizing electrons. On the otherhand, the charge variation of the immobile dust grainscannot respond to the ion-acoustic wave as quickly as theelectrons do. Therefore the effect of dispersion enhances,and as a result, the oscillatory tails appear as the wavepropagates.

Fig. 8. (Color online) Comparison of the electrostatic potential profiles at

T ¼ 60 for the cases of d0 ¼ 0.8 (solid line) and d0 ¼ 0.2 (dashed line).

Other parameters are the same as in Fig. 3.

Fig. 10. (Color online) Comparison of the electrostatic potential profiles

at T ¼ 60 with (a) d0 ¼ 0.8 and (b) d0 ¼ 0.6. Other parameters are the

same as in Fig. 3. Solid line: fixed dust charge, and dashed line: variable

dust charge.

ARTICLE IN PRESSY. Zhang et al. / Planetary and Space Science 56 (2008) 510–518 517

3.5. Effect of the dust charging

To investigate the effect of dust charging process andexam its effect on wave damping, we choose severalequilibrium ion densities (ni0(X0) ¼ 109, 1010, and1011 cm�3) in the simulation, and compare the results withthat in the absence of the dust-charge variation. Fig. 9shows the electrostatic potential profile at T ¼ 80 for thesecases, with the ion density distribution and the parametersthe same as in Fig. 3. As compared with the fixed dust-charge case, the waves become damped due to thedissipation from the charging process, and higher plasmadensity results in stronger damping. We recall that thedust-charge variation leads to a damping of the ion-acoustic wave, and the damping rate is proportional to thecharging current |Ie0| ¼ Ii0 as well as the dust density nd0(Ma and Yu, 1994). Since the charging current isproportional to the ion density, the higher the ion densityis, the stronger the wave damping will be.

On the other hand, for the same equilibrium ion densityni0(X0) ¼ 109 cm�3, the solutions at T ¼ 60 for differentinitial electron-to-ion density ratio d0 are presented inFig. 10. It is found the electrostatic potential of d0 ¼ 0.6suffers stronger damping than that of d0 ¼ 0.8 when thedust charge variation is considered. Further studies showthat, this trend will go on if d0 continues to decrease. Sincend0Zd0/ni0 ¼ 1�d0Ne0/Ni0, for the other parameters un-changed, the decrease of d0 means the increase of the dustdensity. Thus, for the same reason as above, the wavedamping is stronger in the d0 ¼ 0.6 case than in thed0 ¼ 0.8 case.

4. Conclusion

To summarize, the properties of the nonlinear DIAsoliton propagation in inhomogeneous dusty plasmas withself-consistent grain charge variations are numericallyinvestigated. The characteristics of a solitary pulsepropagating in the plasma with hyperbolic-tangent iondensity distribution are presented and compared with otherion density distributions and different values of d0.Moreover, the effect of dust charging process for the casesof different plasma densities and d0 has been studied.

It is shown that both the amplitude and the width of thesolitary pulse increase as the wave propagates from high tolow-density region, whereas they decrease when the wavepropagates in the opposite direction. The wave speedincreases as the local electron-to-ion density ratio ddecreases. For small values of d0, the dispersion effectarising from the charge separation is so dominant that thewave evolves into a highly dispersive one with a strongoscillatory tail. Besides, the effect of dust charge variationleads to the damping of the pulse. High plasma densitiesand small values of d0 result in strong damping. Though aweak amplitude of the initial solitary pulse is chosen in thesimulation, the numerical code can also be applicable tosolitary waves of arbitrary amplitude since the fully

nonlinear hydrodynamic equations have been consideredin the simulation.

Acknowledgment

This work is supported by the National Natural ScienceFoundation of China (Grant nos. 10475075, 40336052).

References

Amin, M.R., Morfill, G.E., Shukla, P.K., 1998. Modulational instability

of dust-acoustic and dust-ion-acoustic waves. Phys. Rev. E 58,

6517–6523.

Angelis, U., Formisano, V., Giordano, M., 1988. Ion plasma wave in

dusty plasmas: Halley’s comet. J. Plasma Phys. 40, 399–406.

Barkan, A., Merlino, R.L., Angelo, N.D., 1995. Laboratory observation

of the dust-acoustic wave mode. Phys. Plasmas 2, 3563–3565.

Barkan, A., Merlino, R.L., Angelo, N.D., 1996. Experiments on ion-

acoustic waves in dusty plasmas. Planet. Space Sci. 44, 239–242.

Cui, C., Goree, J., 1994. Fluctuations of the charge on a dust grain in a

plasma. IEEE Trans. Plasma Sci. 22, 151–158.

Kotsarenko, N.Ya., Koshevaya, S.V., Stewart, G.A., Maravilla, D., 1998.

Electrostatic spatially limited solitons in a magnetized dusty plasma.

Planet. Space Sci. 46, 429–433.

Leveque, R.J., 2002. Finite Volume Methods for Hyperbolic Problems.

Cambridge University Press, Cambridge, UK, 68pp.

Li, Y.F., Ma, J.X., Li, J.J., 2004. Dust-ion-acoustic soliton in an

inhomogeneous collisional plasma. Phys. Plasmas 11, 1366–1371.

Mahmood, S., Saleem, H., 2003. Dust acoustic solitary wave in the

presence of dust streaming. Phys. Plasmas 10, 47–52.

Ma, J.X., Liu, J.Y., 1997. Dust-acoustic soliton in a dusty plasma. Phys.

Plasmas 4, 253–255.

Ma, J.X., Yu, M.Y., 1994. Self-consistent theory of ion acoustic waves in a

dusty plasma. Phys. Plasmas 1, 3520–3522.

Melandso, F., Aslahsen, T., Havnes, O., 1993. A new damping effect for

the dust-acoustic wave. Planet. Space Sci. 41, 321–325.

Merlino, R.L., 1997. Current-driven dust ion-acoustic instability in a

collisional dusty plasma. IEEE Trans. Plasma Sci. 25, 60–65.

Merlino, R.L., Barkan, A., Thompson, C., Angelo, N.D., 1998.

Laboratory studies of waves and instabilities in dusty plasmas. Phys.

Plasmas 5, 1607–1614.

Nakamura, Y., Sarma, A., 2001. Observation of ion-acoustic solitary

waves in a dusty plasma. Phys. Plasmas 8, 3921–3926.

Nakamura, Y., Bailung, H., Shukla, P.K., 1999. Observation of

ion-acoustic shocks in a dusty plasma. Phys. Rev. Lett. 83,

1602–1605.

Nicholls, A., Honig, B., 1991. A rapid finite difference algorithm, utilizing

successive over-relaxation to solve the Poisson–Boltzmann equation.

J. Comput. Chem. 12, 435–445.

Popel, S.I., Yu, M.Y., Tsytovich, V.N., 1996. Shock waves in

plasmas containing variable-charge impurities. Phys. Plasmas 3,

4313–4315.

Popel, S.I., Golub’, A.P., Losseva, T.V., Ivlev, A.V., Khrapak, S.A.,

Morfill, G., 2003. Weakly dissipative dust-ion-acoustic solitons. Phys.

Rev. E 67, 056402.

Popel, S.I., Losseva, T.V., Merlino, R.L., Andreev, S.N., Golub’, A.P.,

2005. Dissipative processes and dust ion-acoustic shocks in a Q

machine device. Phys. Plasmas 12, 054501.

Prabhakara, H.R., Thana, V.L., 1996. Trapping of dust and dust acoustic

waves in laboratory plasmas. Phys. Plasmas 3, 3176–3181.

Rao, N.N., Shukla, P.K., Yu, M.Y., 1990. Dust-acoustic waves in dusty

plasmas. Planet. Space Sci. 38, 543–546.

Shukla, P.K., 2003. Nonlinear waves and structures in dusty plasmas.

Phys. Plasmas 10, 1619–1627.

Shukla, P.K., Silin, V.P., 1992. Dust ion-acoustic wave. Phys. Scr. 45, 508.

ARTICLE IN PRESSY. Zhang et al. / Planetary and Space Science 56 (2008) 510–518518

Tsytovich, V.N., Angelis, U., 2004. Kinetic theory of dusty plasmas. V.

The hydrodynamic equations. Phys. Plasmas 11, 496–506.

Tsytovich, V.N., Morfill, G.E., 2004. Non-linear collective phenomena in

dusty plasmas. Plasma Phys. Control. Fusion 46, B527–B540.

Varma, R.K., Shukla, P.K., Krishan, V., 1993. Electrostatic oscillations in

the presence of grain-charge perturbations in dusty plasmas. Phys.

Rev. E 47, 3612–3616.

Voelzke, M.R., Matsuura, O.T., 1998. Morphological analysis of the

plasma structures of comet P/Halley. Planet. Space Sci. 46, 835–841.

Washimi, H., Taniuti, T., 1966. Propagation of ion-acoustic solitary waves

of small amplitude. Phys. Rev. Lett. 17, 996–998.

Whipple, E.C., Northrop, T.G., Mendis, D.A., 1985. The electrostatics of

a dusty plasma. J. Geophys. Res. 90, 7405–7413.

Xiao, D.L., Ma, J.X., Li, Y.F., 2005. Dust-acoustic shock waves: effect of

plasma density gradient. Phys. Plasmas 12, 052314.

Xiao, D.L., Ma, J.X., Li, Y.F., Xia, Y.H., Yu, M.Y., 2006. Evolution of

nonlinear dust-ion-acoustic waves in an inhomogeneous plasma. Phys.

Plasmas 13, 052308.