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Journal of Modern Physics, 2012, 3, 604-609
http://dx.doi.org/10.4236/jmp.2012.37082 Published Online July 2012 (http://www.SciRP.org/journal/jmp)
Nonlinear Propagation of Dust-Ion-Acoustic Waves in a
Degenerate Dense Plasma
M. S. Zobaer, N. Roy, A. A. Mamun
Department of Physics, Jahangirnagar University, Dhaka, Bangladesh
Email: [email protected], niparo[email protected], [email protected]
Received April 28, 2012; revised May 25, 2012; accepted June 16, 2012
ABSTRACT
Nonlinear propagation of dust-ion-acoustic waves in a degenerate dense plasma (with the constituents being degenerate,
for both the limits non-relativistic or ultra-relativistic) have been investigated by the reductive perturbation method. The
Korteweg de-Vries (K-dV) equation and Burger’s equation have been derived, and the numerical solutions of those
equations have been analyzed to identify the basic features of electrostatic solitary and shock structures that may form
in such a degenerate dense plasma. The implications of our results in compact astrophysical objects, particularly, in
white dwarfs, have been briefly discussed.
Keywords: Degenerate Plasma; Dust-Ion-Acoustic Waves; K-dV Equation; Burzer’s Equation
In present days, most theoretical concerns are to under-
stand the environment of the compact objects having
their interiors supporting themselves via degenerate pre-
ssure. The degenerate pressure, which arises due to the
combine effect of Pauli’s exclusion principle (Wolfgang
Ernst Pauli, 1925) and Heisenberg’s uncertainty principle
(Werner Heisenberg, 1927), depends only on the fermion
number density, but not on it’s temperature. This dege-
nerate pressure has a vital role to study the electrostatic
perturbation in matters existing in extreme conditions
[1-7]. The extreme conditions of matter are caused by
significant compression of the interstellar medium. High
density of degenerate matter in these compact objects
(which are, in fact, “relics of stars”) is one of these
extreme conditions. These interstellar compact objects,
having ceased burning thermonuclear fuel and thereby no
longer generate thermal pressure, are contracted signi-
ficantly, and as a result, the density of their interiors
becomes extremely high to provide non-thermal pressure
through degenerate pressure of their constituent particles
and particle-particle interaction. The observational evi-
dence and theoretical analysis imply that these compact
objects support themselves against gravitational collapse
by degenerate pressure.
The degenerate electron number density in such a
compact object is so high (e.g. in white dwarfs it can be
of the order of 1030 cm−3, even more [8]) that the electron
Fermi energy is comparable to the electron mass energy
and the electron speed is comparable to the speed of light
in vacuum. The equation of state for degenerate electrons
in such interstellar compact objects are mathematically
explained by Chandrasekhar [4] for two limits, namely
non-relativistic and ultra-relativistic limits. The inter-
stellar compact objects provide us cosmic laboratories for
studying the properties of the medium (matter), as well
as waves and instabilities [9-22] in such a medium at
extremely high densities (degenerate state) for which
quantum as well as relativistic effects become important
[9,21]. The quantum effects on linear [16,18,22] and
nonlinear [17,20] propagation of electrostatic and elec-
tromagnetic waves have been investigated by using the
quantum hydrodynamic (QHD) model [9,21], which is an
extension of classical fluid model in a plasma, and by
using the quantum magneto-hydrodynamic (QMHD)
model [16-20], which involve spin 1
2
and one-fluid
MHD equations.
Recently, a number of theoretical investigations have
also been made of the nonlinear propagation of electro-
static waves in degenerate quantum plasma by a number
of authors, e.g. Hass [23], Misra and Samanta [24],
Mistra et al. [25] etc. However, these investigations are
based on the electron equation of state valid for the non-
relativistic limit. Some investigations have been made of
the nonlinear propagation of electrostatic waves in a
degenerate dense plasma based on the degenerate elec-
tron equation of state valid for ultra-relativistic limit [8].
To the best of our knowledge, no theoretical investiga-
tion has been developed to study the extreme condition
of matter for both non-relativistic and ultra-relativistic
limits. Therefore, in our present investigation, we con-
sider a degenerate dense plasma containing non-rela-
C
opyright © 2012 SciRes. JMP
M. S. ZOBAER ET AL. 605
tivistic degenerate cold ion fluid, both non-relativistic
and ultra-relativistic degenerate electrons, and negatively
charged static dust (as it is possible to be some heavy
element in the system) to study the basic features of the
solitary waves in such degenerate dense plasma. The
model is relevant to compact interstellar objects (e.g.,
white dwarf, neutron star, etc.).
We consider the propagation of electrostatic pertur-
bation in a degenerate dense plasma containing non-
relativistic degenerate cold ion and degenerate electron
fluids. Thus, at equilibrium we have 00ieddo
=nnZn
,
where 00ie ddo
is the ion (electron) dust number
density at equilibrium with

nnZn

ddo
Z
n be the charge per
dust grain (number of dust per unit volume). The non-
linear dynamics of the electrostatic waves propagating in
such a degenerate plasma is governed by

=0,
ii
nu
tx

i
n (1)
2
2=0,
i i
n u
xx

1
ii
i
i
uu K
u
txxn
 

 (2)
2=0,
e
n
nK
e
x
x

(3)

2
,
ei
nn
2=1
x


nn

00ie
nn i
u
(4)
where ie
is the ion (electron) number density nor-
malized by its equilibrium value , is the ion
fluid speed normalized by

1/2
2
iei
Cmcm e
m
c
= with
(i
m) being the electron (ion) rest mass mass and
being the speed of light in vacuum,
is the electro-
static wave potential normalized by 2ee
e with
being the magnitude of the charge of an electron, the
mc
time variable (t) is normalized by

1/2
2
0
πi
ne m=4
pi
,
and the space variable (x) is normalized by

1/2
22
0
=4πne
se
mc
. The coefficient of viscosity
is
a normalized quantity given by 0
2
p
is ss (with mn

=,
ie
) and
, the ratio the number density of charged
dust and ion. The constants 12
=
10iii
K
nKmc
and
1
=2
i20ee
K
nKm
=,
iii
PKn
c
. The equations of state used here are
given by
(5)
where
123,
5
ic
3
53ππ
=; =
353
K
c
m



 (6)
for the non-relativistic limit (where
10
=π=1.210 cmmc

2π
=,
eee
PKn
c, and is the Planck cons-
tant divided by ). While for the electron fluid,
(7)
where
=;= for non-relativistic limit, and
ei
KK (8)
1
23
43π3
=; =,
349 4
e
K
cc



 

1/2
=,
p
(9)
in the ultra-relativistic limit [1,2,4,8].
To examine electrostatic perturbations propagating in
the ultra-relativistic degenerate dense plasma due to the
effect of dispersion by analyzing the outgoing solutions
of (1)-(4), we first introduce the stretched coordinates
[26]
x
Vt


3/ 2
=,t

(10)
(11)
where
p
V is the wave phase speed (k with
being angular frequency and being the wave number
of the perturbation mode), and
k
is a smallness para-
meter measuring the weakness of the dispersion
(0< <1
). We then expand , , , and
i
ne
ni
u
, in
power series of
:
(1)2 (2)
=1 ,
iii
nnn


(1)2 (2)
=1 ,
eee
nnn

(12)

(1)2 (2)
=,
ii i
uu u

(13)

(1)2 (2)
=,

(14)
(15)

and develop equations in various powers of
. To the
lowest order in
, (1)-(15) give
(1)(1) 2
1
=
ip p
uV VK


(1) (1)2
1
=
ip
nVK
, ,

(1) (1)
2
=
e
nK
21
=1
p
VK K


, and where

11
=1KK

and
22
=1KK


. The relation
21
=1
p
VKK

nlinear propaga-
tio
represents the dispersion relation
for the dust ion-acoustic type electrostatic waves in the
degenerate plasma under consideration.
We are interested in studying the no
n of these dispersive dust ion-acoustic type electro-
static waves in a degenerate plasma. To the next higher
order in
, we obtain a set of equations
(1) (2)
nn(2) (1) (1)=0,
ii
piii
Vu
nu


 

 (16)


(1)(2)(1) (2)
(1)
2
(2) (1)
1
2=0,
2
ii i
pi
ii
uu u
Vu
Kn n
 
 
 
 




(17)
Copyright © 2012 SciRes. JMP
M. S. ZOBAER ET AL.
606

2
(1) =0
,
ee
n
(18)
(2)
(2)
2
2
2
Kn





(2) (2) .
ei
nn

(19)
we deduce a Kort
Vries equation
2(1)
2=1
Now, combining (16)-(19)eweg-de
(1) (1)
(1)
3 (1)
AB





w
3=0, (20)
here



 
2
2
VK
2
11
32
21
1
32
12
=,
2
pp
pp
VK
AVK
VK


 



(21)

2
2
1
=.
2
p
p
VK
V
(22)
Thlitary wave solution of (20) is
B
e stationary so
(1) 2
=s ,
mech

ere the special coordi



(23)
whnate, 0
=u

, the amplitude,
0
=3uA, and the width,
m

1/ 2
0
=4Bu. It is obvi-
ous from (21)e degenerate plasma under
consideration ressive e
waves which a positive potential. It is
amplitud
and (32) that t
pports comp
are associated
h
su
with
e
lectrostatic solitary
obvious from (21)-(32) that the [m
] of these
solitary structures depends on thrameter e density pa
,
i.e., the ratio of electron to ion number density. The
electrostatic solitary profiles are shown in Figures 1-2.
This is obvious that the profiles are quite different from
those obtained from the previous investigation [8]. And
the potential for non-relativistic degenerate ion fluid and
ultra-relativistic degenerate electron fluid is different
from that when both the particles follow the same limit.
We now turn to (20) with the term (1)
which
changes proportionally with the parameter
. We have
numerically solved (20), and have studied the effects of
on electrostatic solitary structures in both non-rela-
tivistic and ultra-relativistic degenerate electrons (ion
alwe reays being non-relativistic degenerate). Thsults of
the first case are depicted in Figures 1 and 2.
To examine electrostatic perturbations propagating in
the relativistic degenerate dense plasma due to the effect
of dissipation by analyzing the outgoing solutions of (1)-
(4), we now introduce the new set of stretched coor-
dinates [26]

=,
p
x
Vt

 (24)
2
=,t
(25)
(1)
Figure 1. The solitary profiles represented by (32) with u0 =
1 and both the constituent particles non-r e l ativistic .
(1)
Figure 2. (Colour online) The solitary profiles represented
by (32) with u0 = 0.1 and both the constituent particles
non-relativistic.
To the lowest order in
, (1)-(9), (24), (25), and (12)-
(15) give the same results as we have had for the solitary
waves.
To the next higher order in
, we obtain a set of
equations
(1) (2)
(2)(1) (1)=0,
ii
piii
nn
Vunu


 

 (26)


(1)(2)(1) (2) 2
2
(2) (1)
1
2=0,
2
(1) (1)
2
ii i
p
ii
uu u
Vu u
 
ii
Kn n
 
 
 




(27)


(2) 2
(2) (1)
2
2=0,
2
ee
Kn n







(28)
(2) (2)
0= 1.
ei
nn
 (29)
Now, combining (26)-(29) we deduce a Burger’s equa-
tion
Copyright © 2012 SciRes. JMP
M. S. ZOBAER ET AL. 607
(1) (
(1)




1)2 (1)
2
=,AC
whervalue of
(30)
e the
A
is the same
give
as before and C is
n by
0
2ii
mn
=.C
(31)
The stationary shock wave solution of (30) is
(1) =1tan,h





32)

 (
(1)
0
where (1)
00
=uA and
0
=2Cu
.
e proock wave c
between nonlinearity and dissipation are shown in
Fi 4 that
there is no effect of η (the value of η was chosen from the
experimental evidences [27] and it was
with our present investigation) on potential of the shock
ut a signifect of 0
uhown in Figures 5
Th files of shaused by the balance
gures 3-6. It is observed from Figures 3 and
made coincided
wave, bant eff (s
an
ic
d 6). The potential of wave profile for non-relativistic
degenerate ion fluid and ultra-relativistic degenerate
electron fluid is very small compared to that of for both
(1)
Figure 3. The effect of the variation of μ and η on the
potential of shock wave for both electron-ion being non-
relativistic degenerate with u0 = 10.
(1)
Figure 5. The effect of the variation of μ and u0 on the
potential of shock wave for both electron-ion being non-
relativistic degenerate.
F
pote
igure 6. The effect of the variation of μ and u0 on the
ntial of shock wave for electron being ultra-relativistic
and ion being non-relativistic degenera te .
non-relativistic electron-ion fluid (from Figures 3-6).
The existence of the nonlinear structures (both solitary
and shock waves) have been verified using the standard
values of different parameters related to our present sit-
uation [8]. The effect of some parameters on the plas- ma
system have been studied directly from this inves-
tigation.
To summarize, we have investigated electrostatic soli-
tary and shock waves in a degenerate dense plasma,
which is relevant to interstellar compact objects [6,28
33]. The degenerate dense plasma is found to suppo
width, and speed
ty,
-
rt
solitary structures whose basic features (amplitude, width,
speed, etc.) depend only on the plasma number density. It
as been shown here that the amplitude, h
increase with the increase of the plasma number densi
particularly, the maximum number of the light particles
(electrons). This work is very much effective and quite
different from others and is more general than the
relevant previous works [8]. We hope that our present
investigation will be helpful for understanding the basic
Figure 4. The effect of the variation of μ and η on the
potential of shock wave for electron being ultra-relativistic
and ion being non-relativistic degener ate wi th u0 = 10.
Copyright © 2012 SciRes. JMP
M. S. ZOBAER ET AL.
608
features of the localized electrostatic disturbances in
compact astrophysical objects (e.g. white dwarf stars).
Acknowledgements
The authors are very grateful to the TWAS research
institute for their research equipment through this work.
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Physics of Plasmas, Vol. 16, No. 12, 2009, Article ID:
122108. doi:10.1063/1.3272667