Open Journal of Acoustics, 2011, 1, 70-75
doi:10.4236/oja.2011.13009 Published Online December 2011 (http://www.SciRP.org/journal/oja)
Time-Dependent Nonplanar Dust-Ion-Acoustic Gardner
Double Layers
Farah Deeba*, A. A. Mamun
Department of Physi c s , Jahangirnagar University, Dhaka, Bangladesh
Received September 24, 2011; revised October 30, 2011; accepted Novembe r 10, 2011
Abstract
A theoretical investigation has been made on the nonplanar (cylindrical and spherical) dust-ion-acoustic (DIA)
double layers (DLs) in a dusty plasma system, containing inertial ions, Boltzmann electrons, and negatively
charged stationary dust. In this investigation, in order to analyze the time dependent nonplanar DIA DLs, we
have used the modified Gardner equation, which has been obtained by employing the reductive perturbation
method. It has been found that the behaviors of DIA DLs have been significantly modified by the time period
and the nonplanar geometry. The nonplanar DIA DLs has been found to be similar with planar DIA DLs
only at large time scale and the cylindrical DIA DLs have been found to be smaller than the spherical DLs,
but larger than the planar DLs.
Keywords: Dust-Ion-Acoustic, Double Layers, Nonplanar
1. Introduction
The existence of novel dust-ion-acoustic (DIA) waves
was first predicted by Shukla and Silin [1] about twenty
years ago. Nearly four years later, the prediction of Shu-
kla and Silin [1] was conclusively verified by a labo-
ratory experiment of Barkan et al. [2]. Then the linear
features of the DIA waves have been rigorously investi-
gated by a number of authors [1,3,4]. The linear proper-
ties of the DIA waves in dusty plasmas are now well
understood from both theoretical and experimental points
of view [1-5].
The nonlinear structures (viz., solitary waves, shock
structure, and double layers) associated with the DIA
waves have also received a great deal of interest because
they have a great impact in understanding the basic pro-
perties of the localized electrostatic perturbations not
only in space [6-9], but also in laboratory dusty plasmas
[10-14]. A number of investigations have been made on
these nonlinear structures [6-12], particularly DIA soli-
tary waves (SWs) [11,15-19], shock waves [12,13,20-24],
and double layers (DLs) [25-28]. All of these work [11,
12,15-19] are confined in planar geometry. Since the
waves observed in laboratory devices are certainly not
bounded in one-dimension, the investigations made on
1D (planar) nonlinear DIA waves, may not be appropri-
ate for realistic space or laboratory dusty plasma situa-
tions. Recently, a few investigations have been made on
finite amplitude nonplanar DIA solitary and shock struc-
tures [29-31]. But in all of these investigations [29-31],
authors have used the K-dV or Burgers equations, which
are not valid for a parametric regime corresponding to A
= 0 or A ~ 0 (where A is the coefficient of the nonlinear
term of the K-dV or Burger equation [29-31], and A ~ 0
means here that A is not equal 0, but A is around 0). This
is because, the latter gives rise to infinitely large ampli-
tude structures which break down the validity of the re-
ductive perturbation method [32].
On the other hand, to the best of our knowledge, no
attempt has been made on nonplanar DIA DLs. There-
fore, in our present work we are going to analyze the
nonplanar DIA DLs in dusty plasma system by deriving
e modified Gardner (mG) equation. th
2. Basic Equations
We consider the nonlinear propagation of the DIA waves
in an unmagnetized nonplanar (cylindrical and spherical)
dusty plasma, consisting of inertial ions, Boltzmann
electrons, and negatively charged stationary dust. Thus,
at equilibrium we have 00 0
, where d =
iedd
nnZn
Z
is
the number of electrons residing onto the dust grain
surface, and 0d(0i
n) is the dust (ion) number density at
equilibrium. The nonlinear dynamics of the DIA waves,
n

F. DEEBA ET AL.71
whose phase speed is much smaller (larger) than the
electron (ion) thermal speed, in a nonplanar geometry is
governed by

1=0,
v iii
nrnu
tr r



 (1)
=
ii
i
uu
u
tr , r




(2)
1=r
rr r



,






 (3)

=1exp()
i
n,

    (4)
where =0

for 1D planar geometry, and =1(2)

for
a nonplanar cylindrical (spherical) geometry; i is the
ion number density normalized by its equilibrium value
; is the ion fluid speed normalized by
n
0i
ni u
=
iBei
; CkTm

is the electrostatic wave potential
normalized by Be
kT e,

is the surface charge den-
sity normalized by Be
kT e; 00 =dd i
Z
nn

, is the
ratio of dust number density to the ion number density
(the quantity

is often used to measure the quantity of
dust particles present in the plasma system); i is
the mass of the ion (electron); is the ion (ele-
ctron) temperature, d
m()
e
m
()
ie
TT
Z
is the number of electrons resid-
ing on a dust grain surface;
B
k is the Boltzmann con-
stant, and e is the magnitude of the electron charge. The
time and space variables are in units of the ion plasma
period

12
1
0
=4e
pii i
mn 
2
and the Debye radius

12
2
0
=4πe
DmB ei
kT n 
, respectively.
3. Derivation of mG Equation
To study finite amplitude DIA DLs by employing the
reductive perturbation method [27,33,34], we first in-
troduce the stretched coordinates:


=
p
rVt  ,
,
(5)
3
=t


(6)
where

is a smallness parameter (0<<1

) measur-
ing the weakness of the dispersion, and
p
V
i
n
(normalized
by i) is the phase speed of the perturbation mode. We
then expand all the dependent variables (viz. , ,
C
i
u

,
and

) in power series of

:
(1)2 (2)3 (3)
=1 ,
iiii
nnnn    



(7)
(1)2 (2)3 (3)
=0 ,
iiii
uuuu     (8)
(1)2 (2)3 (3)
=0 ,     (9)
(1)2(2)3 (3)
=0 ,     (10)
Now, Expressings (1)-(4) in terms of

and

, and
substituting (7)-(10) into the resulting [Equations (1)-(4)
expressed in terms of

and

], one can easily deve-
lop different sets of equations in various powers of

.
To the lowest order in

, one obtains
(1) (1)
2
=,= ,
ii
p
p
un


(11)
VV
(1)2 =
p
1
=0, ,
e
V  (12)

where (1) =

 and =1
e

 . The expression for 2
p
V
in (12) represents the linear dispersion relation for the
DIA waves propagating in a dusty plasma under
consideration. This equation clearly indicates that the
DIA wave phase speed (
p
V) increases with the increase
of the dust charge density (0dd
Z
n). To the next higher
order in

, we obtain a set of equations, which, after
using (11) and (12), can be simplified as
2 (
V

2)2 (2)
(2) (
342
3
=,=,
22
ii
p ppp
un
VVV


2)
 (13)

(2) 2
1

0,123.
2AA===

  (14)
It is obvious from (14) that since
=0A0


. Now
by solving for =0A

, we found
A
can be zero for
both =1

and =32

. But when =1

, 0dd
Z
n
n
(dust number density at equilibrium) will be equal to 0i
(ion number density at equilibrium), which means that
there is no electron present in the system, i.e., all
electrons will be captured by dust particles. As our
model contains all of electron, ion, and dust, this is an
invalid condition for our present model. For our system
<1

. Therefore the only valid solution of for =0A

is given by
==23
c.

(15) 
For

around its critical value (c

), i.e. for
=
c

  corresponding to 0 =
A
A, we can express
0
A
as
0
=
=
c
c
.
A
A
ss


 

 


 (16)
where c

  is a small and dimensionless parameter,
and can be taken as the expansion parameter

, i.e.
c

   , and for =1s<c

 and =1s

for
>c

 . So, (2)

can be expressed as
(2) 2
1,
2s

(17)  
This means that for c



, (2)

must be included
in the third order Poisson’s equation. To the next higher
order in

, we obtain a set of equations:

F. DEEBA ET AL. 72
(3) (3)3
32
3
(2)
2
2
= 0,
ii
pp p
p
nu
VVV
V


 



 

 




(18)
(3) 3
2(
3
(3)
11
2
=0,
i
pp
p
p
u
VVV
V


 


 

 

 


2)
(19)
2
(3) (3)(2)3
2
2
1
6
=0.
2
ie ee
n
s


 


 


(20)
Now, using (11)-(14) and (18)-(20), we finally obtain
a nonlinear dynamical equation of the form:
3
2
3=0,
2s
 
 
 

 

 (21)
where

2
15 1
=11
21
 ,
5






(22)

3
2
1
=1
2


. (23)
Equation (21) is known as modified Gardner (mG)
equation. The modification is due to the extra term,
2



, which arises due to the effects of the nonplanar
geometry. Because of the existence of both 2

and 3

term, this equation supports both Sws and DLs solutions.
We have already mentioned that =0

corresponds to a
1D planar geometry which reduces (21) to a standard
Gardner (sG) equation. We are now going to numerically
analyze the mG equation. However, for a better under-
standing, before going to numerical solutions of mG
equation, we first briefly discuss the stationary double
layer (DL) solution of this standard Gardner equation [i.e.
(21) with =0

]. The stationary DL solution of the sG
equation [i.e. (21) with =0

] is obtained by con-
sidering a moving frame (moving with speed 0 U)
0
=U

 , and imposing all the appropriate boundary
conditions for DL solution, including 0

,
dd 0

 , 22 dd 0  at

. These boun-
dary conditions for the stationary DL solution allow us to
express the sG equation as
2
1d ()=0,
2d V





 (24)
where the pseudo-potential ()V

is
23 0
()= .
2612
Us
V

4

 

 (25)
We note here that 0 U and

are always positive
since 1


is always valid. It is obvious from (25) that
=0
=0
d()
() ==0,
d
V
V



 (26)
2
2
=0
d() <0.
d
V



(27)
The conditions (26) and (27) imply that DL solution of
(24) exist if and only if
=
=
d()
()| ==0.
d
m
m
V
V




(28)
where m

is the amplitude of the DLs. The latter can
be expressed as
2
0=
6
s
U,


 (29)
0
6
=.
m
U
s

(30) 
Now, using (25) and (30) in (24) we have

2
2
2
d=0,
dm
 




 (31)
where =6

 . Now, integrating (31) the stationary DL
solution of sG equation [i.e. (21) with =0

] can be
directly given by
=1tanh
2
m


,









 (32)
where

is the DL width of the DLs, and is given
2
=.
m


 (33)
From (32) and (33), it is clear that DLs can be formed
in the dusty plasma system if and only if <0

, i.e.
<<
L
U

 , where =0.74
L

(=1
U

), obtained
from =0

, is the lower (upper) limit of

above
(below) which DLs exist.
In Figures 1 and 2, the variations of m

with 0 U
(Figure 1) and e

(Figure 2) have been graphically
shown. On the other hand, since >0

and >0

, (32)
and (33) indicate that the DLs are associated with
positive potential if , i.e. =1s<c


=1
, and associated
with negative potential if , i.e. c s>

 . It is
obvious from Figures 1-3 that >
L
c

 which indicates
that our DLs are associated only with negative potential.
The parametric regimes for the existence of negative

F. DEEBA ET AL.73
Figure 1. Showing the variation of the amplitude of DIA
DLs with U0 at μ = 0.75.
Figure 2. Showing the variation of the amplitude of DIA
DLs with μ at U0 = 0.1.
Figure 3. Showing the parametric regime for the existence
of DLs. The upper (lower) surface plot represents the lower
(upper) limit of μ below (above) which DLs exist.
DLs are bounded by the lower and upper surface plot of
Figure 3, and DLs exist for parameters corresponding to
any point in between two (=0

) surface plots.
The point to be noted here that if we would keep only
the lower order nonlinear term of (21) (viz. the third term
of (21) or the term containing 2

) instead of the higher
order nonlinear term (viz. the fourth term of (21) or the
term containing 3

), we would obtain the solitary
structures. On the other hand, in our present work, we
have kept both the terms containing 2

and 3

, and
have obtained the DL structures (associated with 2

and 3

).
4. Numerical Analysis
Now in order to analyze the nonplanar DIA DLs, we turn
to (21) with the term


2

 , which is due to the
effects of the nonplanar (cylindrical or spherical) geo-
metry. An exact analytic solution of (21) is not possible.
Therefore, we have numerically solved (21), and have
studied the effects of cylindrical and spherical geome-
tries on time-dependent DIA DLs. The results are de-
picted in Figures 4 and 5. The initial condition, which
we have used in numerical analysis, is in the form of the
stationary solution of (21) without the term


2

 .
Figure 4 shows how the effects of a cylindrical geometry
modify the DIA DLs, and Figure 5 shows how the
effects of a spherical geometry modify the DIA DLs.
From the numerical solutions of (21) (displayed in
Figures 4 and 5) we may conclude that for a large value
of

(e.g. =40


), the cylindrical (=1

) and sphe-
rical (=2

) DLs are almost similar to 1D planar (=0

)
structures. This is because when the value of

is large,
the term


2

 , which is due to the effects of the
cylindrical or spherical geometry, is no longer dominant.
However, as the value of

decreases, the term


2

 becomes dominant, and spherical and cylin-
drical DL structures differ from 1D (planar) ones. It has
been observed that as the value of

decreases, the
amplitude of these localized pulses increases. It is also
found that the amplitude of cylindrical DIA DL stru-
ctures is larger than those of 1D planar ones, but smaller
than that of the spherical ones.
5. Discussion and Conclusions
In this paper we have investigated time-dependent non-
planar dust-ion-acoustic Gardner double layers in a dusty
plasma system (composed of inertial ions, Boltzmann
electrons, and negatively charged stationary dust), by
deriving modified Gardner (sG) equation. The outcomes,
which have been obtained from this investigation can be
pinpointed as follows:
1) The dusty plasma system under consideration su-
pports both finite amplitude planar and nonplanar DLs,
whose basic features (polarity, amplitude, width, etc.) de-
pend on the ion and dust number densities.
2) The DLs having large width exist for 0.74<<1

and only have negative potential, i.e., no positive DLs

F. DEEBA ET AL. 74
Figure 4. Showing the effects of cylindrical geometry on
DIA DLs at μ = 0.75.
Figure 5. Showing the effect of spherical geometry on DIA
DLs at μ = 0.75.
have been formed.
3) The magnitude of the amplitude of the DLs in-
creases with the increase of , but decreases with the
increase of
0
U

.
4) The magnitude of the amplitudes of both cylindrical
and spherical DLs increase with decrease of

.
5) The spherical DLs have larger amplitude and poten-
tial than the cylindrical and planar DLs.
We, finally hope that our results may be useful in un-
derstanding the localized electrostatic disturbances in
both space environments [6-9], and laboratory devices
[11-14].
6. Acknowledgements
The Third World Academy of Science (TWAS) Research
Grant for research equipment is gratefully acknowled-
ged.
7. References
[1] P. K. Shukla and V. P. Silin, “Dust Ion-Acoustic Wave,”
Physica Scripta, Vol. 45, No. 5, 1992, p. 508.
doi:10.1088/0031-8949/45/5/015
[2] A. Barkan, N. D'Angelo and R. L. Merlino, “Experiments
on Ion-Acustic Waves in Dusty Plasmas,” Planetary and
Space Science, Vol. 44, No. 1, 1996, pp. 239-242.
doi:10.1016/0032-0633(95)00109-3
[3] P. K. Shukla and M. Rosenberg, “Boundary Effects on
Dust-Ion-Acoustic and Dust-Acoustic Waves in Collisio-
nal Dusty Plasmas,” Physics of Plasmas, Vol. 6, No. 3,
1999, pp. 1038-1040. doi:10.1063/1.873345
[4] P. K. Shukla and A. A. Mamun, “Introduction to Dusty
Plasma Physics,” Institute of Physics Publishing Ltd., Bris-
tol, 2002. doi:10.1887/075030653X
[5] R. L. Merlino and J. Goree, “Dusty plasmas in the labo-
ratory, Industry, and Space,” Physics Today, Vol. 57, No.
7, 2004, pp. 32-39. doi:10.1063/1.1784300
[6] V. E. Fortov, et al., “Complex (Dusty) Plasmas: Current
Status, Open Issues, Perspectives,” Physics Report, Vol.
421, No. 1-2, 2005, pp. 1-103.
doi:10.1016/j.physrep.2005.08.007
[7] O. Ishihara, “Complex Plasma: Dusts in plasma,” Journal
of Physics D, Vol. 40, No. 8, 2007, p. R121.
doi:10.1088/0022-3727/40/8/R01
[8] P. K. Shukla and B. Eliasson, “Colloquium: Fundamentals
of Dust-Plasma Interactions,” Reviews of Modern Physics,
Vol. 81, No. 1, 2009, pp. 25-44.
doi:10.1103/RevModPhys.81.25
[9] G. E. Morfill and A. V. Ivlev, “Complex plasmas: An In-
terdisciplinary Research Field,” Reviews of Modern Phy-
sics, Vol. 81, No. 4, 2009, pp. 1353-1404.
doi:10.1103/RevModPhys.81.1353
[10] K. E. Lonngren, “Soliton Experiments in Plasmas,” Plas-
ma Physics, Vol. 25, No. 9, 1983, p. 943.
doi:10.1088/0032-1028/25/9/001
[11] Y. Nakamura and A. Sharma, “Observation of Ion-Acoustic
Solitary Waves in a Dusty Plasma,” Physics of Plasmas,
Vol. 8, No. 9, 2001, pp. 3921-3926.
doi:10.1063/1.1387472
[12] Y. Nakamura, H. Bailung and P. K. Shukla, “Observation
of Ion-Acoustic Shocks in a Dusty Plasma,” Physical Re-
views Letters, Vol. 83, No. 8, 1999, pp. 1602-1605.
doi:10.1103/PhysRevLett.83.1602
[13] Q. Z. Luo, N. D'Angelo and R. L. Merlino, “Experimen-
tal Study of Shock Formation in a Dusty Plasma,” Phys-
ics of Plasmas, Vol. 6, No. 9, 1999, pp. 3455-3458.
doi:10.1063/1.873605
[14] C. Charles, “A Review of Recent Laboratory Double La-
yer Experiments,” Plasma Sources Science and Technol-
ogy, Vol. 16, No. 4, 2007, pp. R1-R25.
doi:10.1088/0963-0252/16/4/R01
[15] R. Bharuthram and P. K. Shukla, “Large Amplitude Ion-
Acoustic Solitons in a Dusty Plasma,” Planetary and
Space Science, Vol. 40, No. 7, 1992, pp. 973-977.
doi:10.1016/0032-0633(92)90137-D
[16] S. I. Popel and M. Y. Yu, “Ion Acoustic Solitons in Impu-
rity-Containing Plasmas,” Contributions to Plasma Phys-
ics, Vol. 35, No. 2, 1995, pp. 103-108.

F. DEEBA ET AL. 75
doi:10.1002/ctpp.2150350202
[17] A. A. Mamun and P. K. Shukla, “Cylindrical and Spheri-
cal Dust Ionacoustic Solitary Waves,” Physics of Plasmas,
Vol. 9, No. 4, 1997, pp. 1468-1470.
[18] P. K. Shukla and A. A. Mamun, “Solitons, Shocks and Vor-
tices in Dusty Plasmas,” New Journal of Physics, Vol. 5,
2003, pp. 17.1-17.37.
[19] S. I. Popel, et al., “Weakly Dissipative Dust-Ion-Acoustic
Solitons,” Physical Review E, Vol. 67, No. 5, 2003, p.
056402(5). doi:10.1103/PhysRevE.67.056402
[20] P. K. Shukla, “Dust Ion-Acoustic Shocks and Holes,”
Physics of Plasmas, Vol. 7, No. 3, 2000, pp. 1044-1046.
doi:10.1063/1.873905
[21] S. I. Popel, A. P. Golub and T. V. Losseva, “Dust Ion-
Acoustic Shock-Wave Structures: Theory and Laboratory
Experiments,” JETP Lettrers, Vol. 74, No. 7, 2001, pp.
362-366. doi:10.1134/1.1427122
[22] A. A. Mamun and P. K. Shukla, “The Role of dust Charge
Fluctuations on Nonlinear Dust Ion-Acoustic Waves,”
IEEE Transactions on Plasma Science, Vol. 30, No. 2,
2002, pp. 720-724. doi:10.1109/TPS.2002.1024274
[23] A. A. Mamun, P. K. Shukla and B. Eliasson, “Arbitrary
Amplitude Dust Ion-Acoustic Shock Waves in a Dusty
Plasma with Positive and Negative Ions,” Physics of
Plasmas, Vol. 16, No. 11, 2009, pp. 114503-114506.
doi:10.1063/1.3261840
[24] S. S. Duha and A. A. Mamun, “Dust-Ion-Acoustic Shock
Waves Due to Dust Charge Fluctuation,” Physical Letters
A, Vol. 373, No. 14, 2009, pp. 1287-1289.
doi:10.1016/j.physleta.2009.01.059
[25] A. A. Mamun, P. K. Shukla and B. Eliasson, “Solitary
Waves and Double Layers in a Dusty Electronegative
Plasma,” Physical Reviews E, Vol. 80, No. 4, 2009, p.
046406(6). doi:10.1103/PhysRevE.80.046406
[26] H. Alinejad and A. A. Mamun, “Small Amplitude Dust Ion-
Acoustic Solitary Waves and Double Layers in a Dusty
Plasma with Flat-Topped Electron Distribution,” Physics
of Plasmas, Vol. 17, No. 12, 2010, p. 123704(5).
doi:10.1063/1.3526649
[27] S. Ghosh and R. Bharuthram, “Ion Acoustic Solitons and
Double Layers in Electronpositronion Plasmas with Dust
Particulates,” Astrophysics and Space Science, Vol. 314,
No. 1-3, 2008, pp. 121-127.
doi:10.1007/s10509-008-9748-0
[28] R. Roychoudhury and P. Chatterjee, “Arbitrary Amplitude
Double Layers in Dusty Plasma,” Physics of Plasmas, Vol.
6, No. 1, 1999, pp. 406-408. doi:10.1063/1.873294
[29] A. A. Mamun and P. K. Shukla, “Cylindrical and Spheri-
cal Dust Ionacoustic Solitary Waves,” Physics of Plasmas,
Vol. 9, No. 4, 2002, pp. 1468-1470.
doi:10.1063/1.1458030
[30] A. A. Mamun and P. K. Shukla, “Effects of Nonthermal
Distribution of Electrons and Polarity of Net Dust-Charge
Number Density on Nonplanar Dust-Ion-Acoustic Soli-
tary Waves,” Physical Reviews E, Vol. 80, No. 3, 2009,
pp. 037401(4). doi:10.1103/PhysRevE.80.037401
[31] A. A. Mamun and P. K. Shukla, “Cylindrical and Spheri-
cal Dust Ion-Acoustic Shock Waves in Dusty Plasmas,”
Europhysics Letters, Vol. 87, No. 2, 2009, p. 25001.
doi:10.1209/0295-5075/87/25001
[32] H. Washimi and T. Taniuti, “Propagation of Ion-Acoustic
Solitary Waves of Small Amplitude,” Physical Reviews
Letters, Vol. 17, No. 19, 1966, pp. 996-998.
doi:10.1103/PhysRevLett.17.996
[33] S. Maxon and J. Viecelli, “Spherical Solitons,” Physical
Reviews Letters, Vol. 32, No. 1, 1974, pp. 4-6.
doi:10.1103/PhysRevLett.32.4
[34] N. C. Lee, “Small Amplitude Electron-Acoustic Double
Layers and Solitons in Fully Relativistic Plasmas of Two-
Temperature Electrons,” Physics of Plasmas, Vol. 16, No.
4, 2009, p. 042316(10). doi:10.1063/1.3121242