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Appl. Appl. Math.
ISSN: 1932-9466
Applications and Applied
Mathematics:
An International Journal
(AAM)
Vol. 5, Issue 1 (June 2010) pp. 26 - 41
(Previously, Vol. 5, No. 1)
Improved dust acoustic solitary waves in two temperature dust fluids
E. K. El-Shewy and H. G. Abdelwahed
Theoretical Physics Group
Faculty of Science
Mansoura University
Mansoura, Egypt
emadshewy@yahoo.com
M. I. Abo el Maaty and M. A. Elmessary
Engineering Mathematics and Physics Department
Faculty of Engineering
Mansoura University
Mansoura, Egypt
e_k_el_shewy@mans.edu.eg
Received: April 5, 2010; Accepted: May 12, 2010
Abstract
A theoretical investigation is carried out for contribution of the higher-order nonlinearity to
nonlinear dust-acoustic solitary waves (DASWs) in an unmagnetized two types of dust fluids
(one cold and the other is hot) in the presence of Bolltzmannian ions and electrons. A KdV
equation that contains the lowest-order nonlinearity and dispersion is derived from the lowest
order of perturbation and a linear inhomogeneous (KdV-type) equation that accounts for the
higher-order nonlinearity and dispersion is obtained. A stationary solution for equations resulting
from higher-order perturbation theory has been found using the renormalization method. The
effects of hot and cold dust charge grains are found to significantly change the higher-order
properties (viz. the amplitude and width) of the DASWs.
Keywords: Dust-acoustic waves; Hot and cold dust charge; KdV-type equation; Higher order
nonlinearity
AMS (2010) #: 82D10; 34G20; 35G20
26
AAM: Intern. J., Vol. 5, Issue 1 (June 2010) [Previously, Vol. 5, No. 1]
27
1. Introduction
The study of the dynamics of dust contaminated plasmas (mixtures of ordinary plasma particles
and charged dust grains) has recently received considerable interest due to their occurrence in
real charged particle systems, e.g., in interstellar clouds, in interplanetary space, in cometary
tails, in ring systems of giant planets (like Saturn F- ring's), in mesospheric noctilucent clouds, as
well as in many Earth bound plasma, see for instance Verheest (2000), Shukla and Mamun
(2002). Dusty plasma applications range from astrophysics to strongly coupled dusty plasmas
and dusty plasma crystals to technology plasma etching and deposition Mendis and Rosenberg
(1994), Horonyi (1996), Verheest (1996), Shukla and Mamun (2003).
In dusty plasma, the dust grains may be charged negatively by plasma electron and ion currents
or positively by secondary electron emission, UV radiation, or thermionic emission, etc Hornayi
(1996), Whipple (1981). From theoretical perspective, Chow et al. (1993) have shown that due to
the size effect on secondary emission insulating dust grains with different sizes in space plasmas
can have the opposite polarity (smaller ones being positive and larger ones being negative).
Therefore, it is so important and interesting to deal with the dust plasma systems if in addition to
the electrons and ions, there are two types of dust particles with different masses, charges and
temperatures. Consquently, there will be two basic dust-acoustic modes propagating with two
different velocities El-Wakil et al. (2006a), Mowafy et al. (2008).
The dust-acoustic modes have also been investigated in the presence of hot dust in unmagnetized
plasmas Roychoudhury and Mukherjee (1997), Mahmood and Saleem (2003). In most of the
previous investigations of DAW, the dust has been assumed to be cold or hot. Recently, Akhtar
et al. (2007), derived the Sagdeevs pseudopotential for dust acoustic waves (DAWs) in an
unmagnetized two types of dust fluids (one cold and the other is hot) in the presence of
Bolltzmannian ions and electrons, and studied the existence of rarefactive solitons. Investigations
of small-amplitude (DAWs) in dusty plasma usually describe the evolution of the wave by the
Korteweg--de Vries (KdV) equation. In fact, this equation contains the lowest-order nonlinearity
and dispersion, and consequently can only describe a wave restricted to small amplitude. In other
words, (the first-order solution would underestimate the amplitude of the solitary wave by as
much as 20%.). As the wave amplitude increases, the width and velocity of a soliton deviate
from the prediction of the KdV equation, i.e., the KdV approximation does not apply anymore.
Therefore, in order to overcome this deviation, higher-order nonlinear and dispersive effects
have to be taken into account El-Labany (1995), El-Wakil et al. (2004a), El-Taibany and
Moslem (2005), El-Shewy (2005), El-Shewy et al. (2008), El-Wakil et al. (2006b), El-Wakil et
al. (2004b), Attia et al. (2010).
In the present work, our main concern is to examine the effect of both higher-order depressive
and nonlinear corrections on the amplitude and width of (DA) solitary waves propagated in an
unmagnetized two types of dust fluids (one cold and the other is hot) in the presence of
Bolltzmannian ions and electrons. Also, we aim to study the effect of hot and cold dust charge
grains on the formations of both the improved rarefactive and compressive solitons. This paper is
organized as follows; in Section 2, we present the basic set of fluid equations governing our
plasma model which reduced to the well-known KdV equation by employing the reductive
perturbation theory. In Section 3, a linear inhomogeneous (KdV-type) equation is obtained. In
28
El-Shewy et al.
Section 4, higher-order solution is obtained by using the renormalization method. Finally,
discussions and conclusions are given in Sections 5 and 6, respectively.
2. Basic equations and KdV equation
We consider homogeneous unmagnetized plasma having electrons, singly charged ions, hot and
cold dust species. The electrons and ions are assumed to follow the Boltzmann distribution with
T e ,T i  T d , where Te and Ti are the temperatures of the electrons and ions, respectively, and
T d is the temperature of hot dust. The hot dust is assumed to be heated adiabatically. The onedimensional continuity and momentum equations for cold dust, respectively, are,
nc

(ncu c )  0,

t x
(1.a)
uc
Z 

 (uc )uc  c
 0.
t
x
mc x
(1.b)
Corresponding equations for hot dust particles are:
n h

(n hu h )  0,

t x
(2.a)
Z 
uh
1 Ph


 (uh )uh  h
 0.
mh x
mh nh x
t
x
(2.b)
For adiabatic hot dust, we have
Ph  Ph 0 (
nh 
) ,
Nh0
(2.c)
where   ( N  2) / N , and N is the number of degrees of freedom. The present work,
N  1 ,   3 and Ph 0  N h 0T h . The Boltzmann distributed electrons and ions follow the
density equations, respectively, as
ne  ne 0 exp(e  / T e )
(3.a)
n i  n i 0 exp( e  / T i ) .
(3.b)
and
The Poisson equation can be written as
AAM: Intern. J., Vol. 5, Issue 1 (June 2010) [Previously, Vol. 5, No. 1]
 2
 4 e (ne  n i  Z c nc  Z h n h ).
x 2
29
(4)
In equilibrium, we have,
ni 0  ne 0  Z c N c 0  Z h N h 0 .
(5)
In the earlier equations,  is the electrostatic potential, Z h and Z c are the charge numbers for
negatively charged hot and cold dust, respectively. uc (uh ) is the cold (hot) dusty plasma
velocity. nh , nc , ne , ni , N h 0 , N c 0 , ne 0 , and ni 0 are the perturbed and equilibrium number densities of
the species, respectively.
According to the general method of reductive perturbation theory for small amplitude, we
introduce the slow stretched co-ordinates Kodama and Taniuti (1978):
   t ,    (x  t ) ,
3
2
1
2
(6)
where  is a small dimensionless expansion parameter measuring the strength of nonlinearity
and  is the wave speed. All physical quantities appearing in (1) and (2) are expanded as power
series in  about their equilibrium values as:
nc  N c 0   nc1   2 nc 2   3 nc 3  ...,
uc   uc1   2uc 2   3uc 3  ...,
nh  N h 0   nh1   2 nh 2   3 nh 3  ...,
(7)
uh   uh1   2uh 2   3uh 3  ...,
  1   2 2   33  ....
We impose the boundary conditions that as   , nc  N c 0 , n h  N h 0 , u c  u h  0,   0.
Substituting (6) and (7) into (1-4), and equating coefficients of like powers of  . Then, from the
lowest-order equations in  , the following results are obtained:
nc 1  
eN c 0 Z c
eZ
1 , u c 1   c 1 ,
2
 mc
 mc
n h1  
eN h20 Z h
,
 2m h N h 0  3p0 1
u h1  
e N h 0Z h
.
 m h N h 0  3p0 1
2
Poisson's equation gives the linear dispersion relation
(8)
30
El-Shewy et al.
N c 0 Z c2
N h20 Z h2
n
n

 e 0  i 0  0.
2
2
 m c  m h N h 0  3 p 0 Te Ti
(9)
Proceeding to order of  2 and by eliminating the second order perturbed quantities
nc2 , nh 2 , uc2 , uh 2 and 2 , we obtain the desired KdV equation for the first-order perturbed
potential:
1
1 B  31
 A 1

 0,

 2  3
where
2(
12 N c 0 Z c3e 3

A
4
mc2

(10)

 m N
2
h
(

12 N h3 0 m h N h 0 2  p0 Z h3e 3
h 0 3 p0
8e mh Z h2N h3 0
2
 m N
2
h
h 0 3 p0

2


3

8e 2 N c 0 Z c2
3mc


4 Ti 2N e 0 Te 2ni 0 e 3
2
Te Ti
2
)
,
)
and
2
B 
8e  m h Z
2

2
3
hN h0
 2m h N h 0 3 p0

2

8e 2 N c 0 Z c2
.
 3m c
3. Linear inhomogeneous KdV-type equation
In this section we start by writing down the second-order perturbed quantities nc2 , uc2 , nh 2
and uh 2 in terms of 1 and 2 as:
nc 2 

Nc0 Zce 
B  21
2
m
A
m




 3 Z c e12  2  2 mc 2  ,
2
4
1
c
c
4
2 
2
2  mc 
2 


Zce 
B  21
2
uc 2 
 mc A1  2  mc
 Z c e12  2  2 mc2  ,
3
2 
2
2  mc 
2 

AAM: Intern. J., Vol. 5, Issue 1 (June 2010) [Previously, Vol. 5, No. 1]
nh 2 
31
Z he Nh0
2 ( mh  9 Th mh 2  4  27 Th 2 mh  2  27 Th 3 )
6
3
B  21
(3  Z h emh1  2  mh A  4  mh
 6  mhTh A12
2
2 
2
2
3
2
2
1
3
2
B  21
12  mhTh
 2 mh 2  42  12 mh  2Th 2  18 Th 2 2  3Th Z h e12 ),
2
2 
eZ h
uh 2 
6
3
2 4
2( mh  9 Th mh   27 Th 2 mh  2  27 Th 3 )
(18 Th 2
B  21
 18 Th 2  2  9 Th 2 A12  9 Th Z h e 12  12 Th mh  32
2  2
2 mh 2  5 2  Z h e 3 mh12  mh 2  4 A12  2 mh 2  4
(11)
B  21
).
2  2
If we now go to the higher order in  , we obtain the following equations
Ze
Ze
N Ze





nc 2  c 1
nc 2  ( c nc 2  c 02 c uc 2 ) 1  N c 0
uc 3  
nc 3




 mc  
 mc
 mc
N Ze

uc 2  0,
 c20 c 1
 mc

Z e
Z e
Z e 3




  u c 3  0,
u c 2  c u c 2 1  c 1 u c 2  c

  m c 

m c 
 mc
 Zhe
 Z he
Z he Nh0



nh 2 
nh 2  (
n 
u ) 1

2 1
2 h2
2 h2



3 Th  mh 
3 Th  mh 
3 Th  mh 
Zhe Nh0



 nh 0
uh 3 
uh 2  
nh 3  0,

2 1



3Th  mh 
 Zhe
T
Z e 3




uh 2 

uh 2  
uh 3  3 h
nh 3  h
2 1
3 Th  mh 



N h 0 mh 
mh 
Th Z h e
 Z he


1
(nh 2 1  1
3
nh 2 ) 
u
 0,
2 h2
2
3Th  mh 



N h 0 mh  3Th  mh  
 e 2 ne 0
 e 2 ni 0
 22



3  4  eZ h n h 3  4  eZ c nc 3  0 .
4
4
3
 2
Te
Ti
(12.a)
(12.b)
(12.c)
(12.d)
(12.e)
Eliminating nc 3 ,u c 3 , n h 3 ,u h 3 and 3 from (12), and with the aid of both (8) and (10), a linear
32
El-Shewy et al.
inhomogeneous equation for the second-order perturbed potential 2 can be obtained

2
 (12 ) B  32
L (1 )2 
A

 S (1 )  0,


2  3
(13)
where
S (1 )  A1
13
 2
 3
 51

,
 A2 ( 1 21 )  A31 ( 31 )  A4

 

 5
(14)
where the coefficients Ai (where i  1, 2,..., 4 ) are listed as:


A1  (4 mc mh N h30 Z h2  4  N c 0   2 mh N h 0  3 p0  Z c2 ) 1
2
 mc   mh N h 0  3 p0  (
3
2
2
3e 2  5mh2 N h20  4  30mh N h 0 p0  2  9 p02  Z h4 N h40
 m N  3 p 
  3p Z N
6A m m N   p  Z

3p 
 m N  3 p 
Z  6 A  m  20 Ae Z m  15e Z 
,
5
2
h0
h
(
20 Ae mh  mh N h 0
 m N
2
h
2
3
h
0
4
h0
2
2
h
4
h0
e 2 ne 0 e 2 ni 0 N c 0
 3  3 
Te
Ti
h0
h
3
h0
h
2
0
2
0
2
c
0
2
2
c
2
c
c
 6 mc3


A2  (16 emc mh N h30 Z h2  4  eN c 0   2 mh N h 0  3 p0  Z c2 ) 1
2
 (4emc2 mh N h40  mh N h 0 2  3 p0  Z h3 6
2
 9 Amc2 mh N h30   2 mh N h 0  3 p0  mh N h 0  2  p0  Z h2  5
 N c 0   2 mh N h 0  3 p0  Z c2  9 A mc  4eZ c  ,
4

A3  (8 emc mh N h30 Z h2  4  eN c 0   2 mh N h 0  3 p0  Z c2
2
 (4emc2 mh N h40  mh N h 0  2  3 p0  Z h3 6
)
2
3 Amc2 mh N h30   2 mh N h 0  3 p0  mh N h 0  2  p0  Z h2  5
 N c 0   2 mh N h 0  3 p0  Z c2  3 A mc  4eZ c  ,
4
and
1
0
2
c
2
h
N h30
AAM: Intern. J., Vol. 5, Issue 1 (June 2010) [Previously, Vol. 5, No. 1]

33

A4  (128e 4 2 mc mh N h30 Z h2  4  N c 0   2 mh N h 0  3 p0  Z c2 ) 1
2

3

3 5 mc2   2 mh N h 0  3 p0  mc mh N h30  mh N h 0  2  p0  Z h2  4  N c 0   2 mh N h 0  3 p0  Z c2 .
3
3
In summary, we have reduced the basic set of fluid (1-4) to a nonlinear KdV equation (10) for 1
and a linear inhomogeneous differential equation (13) for 2 ; for which the source term is
described by a known function 1 .
4. Stationary Solution
To obtain a stationary solution from (10) and (13), we adopt the renormalization method
introduced by Kodama and Taniuti (1978), and El- Labany (1995) to eliminate the secular behavior
up to the second-order potential. According to this method, (10) can be added to (13) to yield

K (1 )   n L (1 )n   n S n ,
n 2
(15)
n 2
where S 2 represents the right-hand side of (13). Adding  n 1 n  n
to both sides of (15),
where   1    2   3  ... , with coefficients to be determined later,  n are determined
successively to cancel out the resonant term in S n . Then, (10) and (13) may be written as
2
3




  1 1  3  1
 1
 1
 A 1
 B
 
 0,

 2  3

(16)





 2
  
1  3 2
 2
 1
A
( 1 2 )  B
 
 S 2 ( 1 )  1
.


2  3


(17)
The parameter  in (15) and (16) can be determined from the conditions that the resonant terms


in S 2 ( 1 ) may be cancelled out by the terms ( 1 /  ) in (16) see for instance El-Labany
(1995).
Let us introduce the variable,
    (  ),
where the parameter  is related to the Mach number M   / C d by
    M  1  M ,
(18)
34
El-Shewy et al.
with M being the soliton velocity and C d is the dust-acoustic velocity.
Integrating equations (16) and (17) with respect to the new variable  and using the appropriate


vanishing boundary conditions for  1 ( ) and  2 ( ) and their derivatives up to second order
as    , one gets



d 21
1

B
(
A

2

)


1
1 0,
d 2
(19)






d 2 2
d 1
1
1
 2B (A  1  ) 2  2B  [S 2 ( 1 )  1
]d  .

d 2
d
(20)
The one-soliton solution of (19) admits

 1  1m sech 2 (D ),
(21)
where the soliton amplitude 1m and the soliton width D 1 are given
1m 
3
,
A
D 1 
2B
.

(22)
Substituting (21) in (19), then the source term of (19) becomes

d 1
2B  [S 2 ( 1 )  1
]d   2B 1 (1  16D 4 A 4 )1m sech 2 (D ) 

d
1


212m 
12m 
4
sech
(
)



 2 sech 6 (D ),
D
1
2
2
B
B
(23)
with
1  2 A 2  A 3 
2 
10A
A4 ,
B
2B
20A
A1  6A 2  2A 3 
A4.
A
B

In order to cancel out the resonant terms in S ( 1 ) , the value of   should be
1  16D 4 A 4 .
and for (23), we introduce the independent variable
(24)
AAM: Intern. J., Vol. 5, Issue 1 (June 2010) [Previously, Vol. 5, No. 1]
  tanh( D),
35
(25)
to recast (23) into

 
d 
d

22  
2
2
2 2
2
1   
  3(3  1) 
  2   3 (1   )   4 (1   ) ,
d 
d  
1 2 
(26)
where
412m
1 ,
B
2 2
 4  1m  2 .
B
3 
Note that the two independent solutions of homogeneous part of (26) can be represented in terms
associated Legendre function of first and second kind,
P32  15 (1   2 ),
Q 32 
(27)
15
 1  
2 1
2 2
 (1   2 ) ln 
  2(1   )  15(1   )  5,
2
 1  
(28)
the complementary solution of (26) admits the following form

15
 1  
2 1
2
 (1   2 ) ln 
  2(1   )  15(1   )  5).
2
 1  
 2c  C1 (15 (1   2 ))  C2 (
(29)
By applying the method of variation of parameters, the particular solution of (26) takes the form

 2 p  L1 ( )P32 ( )  L 2 ( )Q 32 ( ),
where L1 ( ) and L 2 ( ) given by
L1 ( )   
Q 32 ( )T ( )
d ,
(1   2 )W (P32 ( ),Q 32 ( ))
P32 ( )T ( )
L 2 ( )  
d ,
(1   2 )W (P32 ( ),Q 32 ( ))
with
(30)
36
El-Shewy et al.
T ( )   3 (1   2 )   4 (1   2 ) 2 ,
W ( P32 , Q32 )  P32
dQ32
dP 2
120
 Q32 3 
.
d
d  1 2
Using P32 and Q 32 , then L1 ( ) and L 2 ( ) reduced to
1
1
1
1 
1
(1   2 )3 (  3   4 (1   2 )) ln(
)  ( 5 )  ( 6 ) 3  ( 7 ) 5   7 ,
32
3
4
1 
64
1
1
1
L2 ( )   (1   2 )3 (  3  (1   2 ) 4 ).
16
3
4
L1 ( ) 
with
1 1
1 1
 ) 3  (  ) 4 ,
48 15
64 15
1
33
1
 ) 4 ,
 6  ( ) 3  (
18
360 64
1
3
1
 ) 4 .
 7  ( ) 3  (
48
320 15
 5  (
The particular solution is given by

 2p 
12m D 2

(1   2 )[ 8   2 (1   2 )],
(31)
where
8 
6BA1 10
4
20A
 A 2  A3 
A4.
A
3
3
3B
In (23) the first term is the secular one, which can be eliminated by renormalization the

amplitude. Also the boundary conditions  2  0 as    produce C 2  0. Thus, only the

particular solution contributes to  2 .
Expressing (31) in terms of the old variable  the solution of (26) is given by

2p 
12m D 2

sech 2 ( D ) 10   2 tanh 2 ( D )  .
The stationary soliton solution for DA waves is given by
(32)
AAM: Intern. J., Vol. 5, Issue 1 (June 2010) [Previously, Vol. 5, No. 1]


37

 ( )   1 ( )   2 ( )
 (1m 
12m D 2

11 ) sech (D )   2
2
12m D 2

(33)
sech 4 (D ).
where
3BA1 1
2
10A
 A 2  A3 
A4 ,
2
3
3B
A
11  10   2 ,
10 
  B 2 [(1 
16A 4 M 12
)  1] / (8A 4 ).
B2
and the soliton width is given by
D 1  (
2B 12
2A  M
) (1  4 2 ).
M
B
0
f
`
f
0.0004
-0.0001
0.0003
-0.0002
0.0002
-0.0003
0.0001
0
-0.04
f
`
f
-0.0004
-0.0005
-0.02
0
h
0.02
0.04
Figure 1. The comparison between the
lowest order potential  ( ) and higherorder correction ˆ ( ) of the compressive
soliton
with
respect
to

for,
14
m h  m c  10 m i , N c 0  6, N h 0  5, ne 0  10
, n i 0  10010 , T i  T e  1eV , Z c  1000 ,
Z h  1000 and   0.4 .
-0.02 -0.01
0
h
0.01
0.02
Figure 2. The comparison between the
lowest order potential  ( ) and higher-order
correction ˆ ( ) of the rarefactive soliton
with respect to  for N c 0  6, N h 0  11,
ne 0  10, ni 0  10010 ,
m h  m c  1014 m i ,
T i  T e  1eV , Z c  1000 , Z h  1000 , and
  0.4 .
5. Numerical Results and Discussion
Nonlinear DAWs in a homogeneous unmagnetized plasma having electrons, singly charged ions,
hot and cold dust grains have been investigated. We have assumed that the effect of gravity in
the system is neglected ( rd  1 ) as well as there are no neutrals. Saturn F-ring's are one of the
space plasma observations that satisfy our conditions: (i) there are no neutrals, (ii) the ratio
38
El-Shewy et al.
between the inter-grain distances between dust particles to plasma Debye radius is less than one,
(iii) coupling parameter  is less than one, and (iv) rd is smaller than 1 m . Hence, numerical
studies have been made using plasma parameters close to those values corresponding to Saturn
F-ring's i.e., the equilibrium electron and dust densities are ne 0  10cm 3 , N h 0  10cm 3 and
dust charge and mass are taken as Z h  102  103 , m h  m c  1012 m i , respectively, as given
in Refs. Akhtar et al. (2007), Shukla and Mendis (1997), Farid et al. (2001). Generally speaking,
the present system supports compressive and rarefactive soliton i.e. the amplitude of KdV
solitons can be positive ( A  0 ) or negative ( A  0 ). Since our objectives was to study the effect
of higher-order nonlinearity on the formation of solitary waves, we specifically elucidated to
what extent the higher-order solution modifies the soliton amplitude. The comparison between
the lowest order potential  ( ) and higher-order correction ˆ ( ) of the compressive
(rarefactive) solitons were depicted in Figures (1, 2).
0.0009
0.0007
0.0008
Nh0=4
Nh0=5
0.0007
0.0006
0.0005
f 0c
f
0c
0.0006
Nc0=5
Nc0=6
0.0004
0.0005
0.0004
0.0003
0.0003
0.0002
0.0002
900 925 950 975 1000 1025 1050 1075
Zh
900 925 950 975 1000 1025 1050 1075
Zc
Figure 3. The variation of the amplitude Figure 4. The variation of the amplitude 0c
0c of the compressive soliton for higher- of the compressive soliton for higher-order
order correction with Z h for different values correction with Z c for different values of
for
N h 0  4, ne 0  10, ni 0  10010,
N
for N  6, n  10, n  10010, N c 0
of
h0
c0
e0
m h  m c  10 m i , Ti  Te  1eV ,
Z c  1000 and   0.4 .
14
i0
mh  mc  1014 mi , Ti  Te  1eV , Z h  1000
and   0.4 .
It is found that, the higher-order correction increasing the soliton amplitude for compressive and
rarefactive solitons. On the other hand, it is important to study effects of the dusty plasma density
n h (nc ) and the charge numbers for negatively charged hot and cold dusty grains on the
existence of improved solitary waves. It is obvious from Figures (3, 4) that compressive soliton
amplitude increases with enhancing of N h 0 , Z h and decreases with N c 0 , Z c . Finally, the
rarefactive soliton amplitude increases with N c 0 , Z c and decreases with N h 0 , Z h as shown in
Figures 5 and 6.
These results show that the presence of cold (hot) dusty plasma density nc (n h ) and the charge
numbers for negatively charged cold (hot) dust Z c ( Z h ) as well as the higher-order correction
modify significantly the properties of dust-acoustic solitary waves. The results presented here
AAM: Intern. J., Vol. 5, Issue 1 (June 2010) [Previously, Vol. 5, No. 1]
39
should be useful in understanding salient features of localized electrostatic perturbations in space
and laboratory plasmas.
-0.0005
-0.000275
-0.001
-0.0003
0r
f
-0.0015
f
0r
-0.000325
Nh0=10
Nh0=11
-0.002
-0.0025
900
920
940
960
980
-0.00035
-0.000375
-0.0004
1000
Zh
Nc0=10
Nc0=11
-0.000425
700 720 740 760 780 800 820 840
Zc
Figure 5. The variation of the amplitude 0 r
of the rarefactive soliton for higher-order
correction with Z h for different values of
N h 0 for N c 0  6, ne 0  10, n i 0  10010 ,
Figure 6. The variation of the amplitude 0 r
of the rarefactive soliton for higher-order
correction with Z c for different values of
N c 0 for N h 0  10, ne 0  10, n i 0  10010 ,
m h  m c  1014 m i , T i  T e  1eV ,
Z c  1000 and   0.4 .
m h  m c  1014 m i , T i  T e  1eV ,
Z h  1000 and   0.4 .
6. Conclusions
In this work, we have investigated the properties of nonlinear DAWs in a unmagnetized plasma
having electrons, singly charged ions, hot and cold dust grains. The reductive perturbation
method has been used to reduce the basic set of fluid equations to the well-known KdV equation;
however, as the wave amplitude increases, the width and velocity of a soliton deviate from the
prediction of the KdV equation. In particular, we study the next-order of perturbation theory, a
linear inhomogeneous (KdV-type) equation that accounts for the higher-order nonlinearity and
dispersion is obtained. A stationary solution for equations resulting from higher-order
perturbation theory has been found using the renormalization method. The consideration of
higher-order approximation was found to increase the amplitude of DAWs solitons. In other
word, we have shown that the presence of cold (hot) dusty plasma density nc (n h ) and the
charge numbers for negatively charged cold (hot) dust Z c ( Z h ) do not only modify the basic
properties of the improved dust acoustic solitary potential structures, but also causes two
different potential profiles, namely compressive and rarefactive pulses.
We inject to that the analytical model demonstrated here can provide a useful basis for the
interpretation of recent observations of solitary wave in dusty plasma environments. For
example, the results presented may be applicable to dusty plasma existing in Saturn F-ring's
region.
40
El-Shewy et al.
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