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Dust acoustic solitons and shocks in a charge varying dusty plasma
Abderrezak Berbri and Mouloud Tribeche
Faculty of Sciences-Physics, Theoretical Physics Laboratory, University of
Bab-Ezzouar, USTHB, B.P. 32, El Alia, Algiers 16111, Algeria.
Abstract
A theoretical investigation has been made to analyze the propagation of dust acoustic solitary
waves in an unmagnetized charge varying dusty plasma. The charge variation is found to
increase the pulse amplitude while its width decreases, i.e., the dust charge fluctuation makes
the solitary structure more spiky. On the other hand, it has been shown that the dust charge
fluctuations lead to a dissipation which is responsible for non collisional shock waves
formation the nature of which depends sensitively on the equilibrium dust density.
I- Introduction
The dust grains immersed in a plasma are highly charged, and the dust charge on the dust
grains also fluctuates due to the fluctuating electrostatic force.This is an important
characteristic of dusty plasma, which distinguishes it from a three component plasma. The
charging processes and grain charge fluctuations are also interesting phenomena and have
been recently investigated. In this paper, we study the properties of nonlinear dust acoustic
solitons and shocks by incorporating the effects of the dust charge fluctuation.
II- Basic equation
Let us consider an unmagnetized dusty plasma composed of Boltzmann distributed electrons,
ions, and variable charge dust grains of density ne, ni and nd, respectively. We assume for
simplicity that all the grains have the same charge, equal to qd =-eZd, with Zd positive for
negatively charged dust and negative for positively charged dust. All the dust grains are
assumed to be spheres of radius rd. In the presence of the dust grains, the electrons and ions
may be considered as point particles. Moreover, the electrons and ions are in a local
thermodynamic equilibrium and their densities are given by the Maxwell-Boltzmann
distribution
*
+
0exp /
ss s s
nn q T?/H
(1)
Let us begin to look at a very simple case, where each cold species is a beam of particles, each
particle of species s having the same speed at a given position. Thus, we choose [1]
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34th EPS Conference on Plasma Phys. Warsaw, 2 - 6 July 2007 ECA Vol.31F, P-5.056 (2007)
020
0
2
1
dd d
dd
vv qd
mv
h
h
ÃÔ
?/
Ä
Ð
ÅÖ
#Õ
(2)
where
1/ 2
020
0
2
1d
dd
dd
vv qd
mv
h
h
ÃÔ
Ä
?/Ð
Ä
ÅÖ
#Õ
Õ
(3)
Integrating the dust distribution function fd over all velocity space, we find
*+
0
1/2
0
1
1
dd
ddd
nv
Nnv
ce
???
/
# (4)
where
2
22
0
2de
dd
rT
emv
u
c
? (5)
and
0d
Qd
e
[
?
[
Ð (6)
Poisson’s equation can be expressed as
2
2
0
exp( ) exp( ) (1 ) d
d
d
Q
d
f
f
Q
dX
u
[?[/ /[// N
(7)
where f=ni0/ne0 and
u
=Ti/Te. The following normalized quantities: {=e
h
/Ti, X=x/
n
Dm,
Qd=eqd/rdTe are introduced and
n
Dm =(Ti/4
r
ne0e²)1/2.
Charge neutrality at equilibrium requires
*
+
2
00
1
de d de
0
/
f
rTQ e n n?/ (8)
The system is closed by the normalized charging equation
*+
1/ 2 exp( )exp( ) exp( ) 1
1
di
d
e
dQ m Q
KQf
dX m
u
uu
ez
Ç
Ã
?/[-/[/
ÈÙ
Ä
ÅÖ
ÈÙ/ÉÚ
d
Ô
Õ
(9)
with
K
given by
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34th EPS 2007; A.BERBRI et al. : Dust acoustic solitons and shocks in a charge varying dusty plasma 2 of 4
22
0
2
0
2ed
id
nre
Kmv
u
? (10)
Equation (9) is the additional dynamical equation that is coupled to the plasma equations
through the plasma currents. The dust electric charge becomes a dynamical variable which is
coupled self-consistently to the other dynamical variables such as density and potential.
Initially, in the absence of any perturbation ({=0), equation (9) yields
0
0
exp( )
1/
id
ed
mQ
fmQ
u
u
ÃÔ
?Ä
/
ÅÖ
Õ
(11)
In the following, the value of f is deduced from the above relation while the remaining other
parameters are given first.
Next, equations 6, 7 and 9 are integrated numerically. In figure 1, we study the effect of the
dust charge fluctuation on the soliton profiles. It is seen that the variation of the dust charge
produces a reduction of the dust acoustic soliton width as well as an increase of its amplitude.
In figure 2, it can be seen that as K decreases, the wave amplitude suffers the well known
anomalous damping. In figure 3, we study the effect of the parameter vd0 (the dust speed) on
the dust acoustic shock when the charge of the dust grain is variable. It is found that greater
values of vd0 favor the development of dissipative structures. In figure 4, it can be seen that
when r (the dust radius) increases, the shock wave has a monotonic profile (dissipation
dominant).
References
[1] M. Tribeche, H. Houili, and T. H. Zerguini, Phys. Plasmas 9, 419 (2002).
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34th EPS 2007; A.BERBRI et al. : Dust acoustic solitons and shocks in a charge varying dusty plasma 3 of 4
Figure1 Figure2
Constant and variable charge soliton Collisional shock wave profile
X
Monotonic and oscillatory shock
profile
Figure 3 Figure 4
Effect of dust radius on shock profile
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34th EPS 2007; A.BERBRI et al. : Dust acoustic solitons and shocks in a charge varying dusty plasma 4 of 4
