Cylindrical and spherical dust ion-acoustic Gardner solitons in a quantumplasmaM. M. Hossain, A. A. Mamun, and K. S. Ashrafi Citation: Phys. Plasmas 18, 103704 (2011); doi: 10.1063/1.3646738 View online: http://dx.doi.org/10.1063/1.3646738 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v18/i10 Published by the AIP Publishing LLC. Additional information on Phys. PlasmasJournal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors

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Cylindrical and spherical dust ion-acoustic Gardner solitonsin a quantum plasma

M. M. Hossain, A. A. Mamun, and K. S. Ashrafia)

Department of Physics, Jahangirnagar University, Savar, Dhaka 1342, Bangladesh

(Received 6 June 2011; accepted 12 September 2011; published online 14 October 2011)

The properties of nonplanar (cylindrical and spherical) quantum dust ion-acoustic (QDIA) solitary

waves in an unmagnetized quantum dusty plasma, whose constituents are inertial ions, Fermi elec-

trons with quantum effect, and negatively charged immobile dust particles, are investigated by

deriving the modified Gardner (MG) equation. The reductive perturbation method is employed to

derive the MG equation, and the basic features of nonplanar QDIA Gardner solitons (GSs) are ana-

lyzed. It has been found that the basic characteristics of GSs, which are shown to exist for the value

of Zdnd0=ni0 around 2=3 (where Zd is the number of electrons residing on the dust grain surface,

and nd0 and ni0 are, respectively, dust and ion number density at equilibrium), are different from

those of the Korteweg-de Vries solitons, which do not exist for the value of Zdnd0=ni0 around 2=3.

It is also seen that the properties of nonplanar QDIA GSs are significantly different from those of

planar ones. VC 2011 American Institute of Physics. [doi:10.1063/1.3646738]

I. INTRODUCTION

Recently, quantum plasmas have attracted baronial in-

terest due to their enormous applications in dense plasma,

particularly, in different astrophysical and cosmological

systems1–3 (e.g., interstellar or molecular clouds, planetary

rings, comets, interior of white dwarf stars, etc.), in nano-

structures,4 in microelectronic devices,5 as well as in the

next-generation intense lasers.6 Many authors include

the quantum corrections to the quantum plasma echoes,7 the

self-consistent dynamics of Fermi gases,8 quantum beam

instabilities,9 wave interactions in quantum magnetoplas-

mas,10 classical and quantum kinetics of the Zakharov sys-

tem,11 quantum corrections to the Zakharov equations,12

expansion of quantum electron gas into vacuum,13 quantum

ion acoustic waves,14 quantum Landau damping,15 magneto-

hydrodynamics of quantum plasmas,16 etc. Quantum plasmas

have extremely high plasma number densities and low tem-

peratures. At extremely low temperatures, the thermal de

Broglie wavelength becomes comparable to the interelectron

distance and the electron temperature becomes comparable

to the electron Fermi temperature (TFe), and the electrons

follow Fermi Dirac distribution law. In this condition, quan-

tum mechanical effects are expected to play a significant role

in the behavior of charged particles.17–25

As electrons are lighter than ions, the quantum behavior

of electrons is reached faster than ions. At room temperature

and standard metallic densities, the electron gas in an ordi-

nary metal is a good example of a quantum plasma system.

In such a plasma system, the quantum effects cannot be

ignored. The concept of quantum plasma is also applicable

in semiconductor physics. The electron density in semicon-

ductors is much lower than in metals, but the great degree of

miniaturization of today’s electronic components is such that

the de Broglie wavelength of the charge carriers can be com-

parable to the spatial variation of the doping profiles. In the

behavior of such electronic components, typical quantum

mechanical effects (e.g., quantum tunneling effects) are

expected to play a central role. Another possible application

of quantum plasmas arises from astrophysics. In astrophysi-

cal and cosmological compact objects, the density of charged

particles is extremely high (some ten orders of magnitude

larger than that of ordinary solids). The properties of matter

existing under such ultra-dense plasmas possess strong quan-

tum effects and exhibit fluid and crystal properties in a quan-

tum sea of electrons.2

The dust particles are quite common in various plasma

systems. The inclusion of immobile charged dust in electron-

ion plasmas leads to introduce a new mode. Shukla and

Silin26 have first theoretically shown the existence of low-

frequency dust ion-acoustic (DIA) waves in a dusty plasma,

which was later observed in laboratory experiments.27–29

The phase speed of the DIA waves is much smaller (larger)

than electron (ion) thermal speed. The inertia is provided by

the ion mass, while the restoring force comes from the elec-

tron thermal pressure. These waves differ from the usual ion-

acoustic waves30 due to the conservation of equilibrium

charge density ne0þ Zdnd0� ni0¼ 0 and the strong inequality

ne0 � ni0, where ns0 is the particle number density of the

species s with s¼ e(i) d for electrons (ions) dust, Zd is the

number of electrons residing onto the dust grain surface, and

e is the magnitude of an electronic charge. Therefore, a dusty

plasma cannot support the usual ion-acoustic waves but can

do the DIA waves.26,31–34 Dust impurities existing in the

quantum plasma form a quantum dusty plasma. Microelec-

tronic devices and metallic nanostructures are good exam-

ples of quantum dusty plasma which are usually doped or

contaminated by the presence of highly charged dust impur-

ities. Quantum dusty plasmas also appear in astrophysics

(e.g., supernova environments) and are likely to be found in

ultra-intense laser-solid material plasma (clusters) interaction

a)Permanent address: Department of Basic Science, Primeasia University,

Banani, Dhaka 1213, Bangladesh.

1070-664X/2011/18(10)/103704/6/$30.00 VC 2011 American Institute of Physics18, 103704-1

PHYSICS OF PLASMAS 18, 103704 (2011)

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experiments.35 The nonlinear waves associated with the

quantum ion-acoustic, quantum dust ion-acoustic (QDIA),

quantum dust-acoustic SWs,20,36–39 shocks,40 and double

layers23 have received a great deal of interest in understand-

ing the basic properties of localized electrostatic perturbation

in space and laboratory dusty plasmas in both planar20,36,37

and non-planar41–43 geometry. The finite amplitude QDIA

solitary (shock) structures in a quantum dusty plasma have

been intensively investigated by deriving the Korteweg-de

Vries (K-dV) and Burgers equations.

However, in most of the investigations on the QDIA

waves,36,39,40 the K-dV and Burgers equations, which have

been derived for the study of the QDIA solitary or shock

waves, are not valid for a parametric regime corresponding

to A� 0 (where A is the coefficient of the nonlinear term of

the K-dV or Burger equation, and A� 0 means here that A is

not equal to 0, but A is around 0). This is because, the latter

gives rise to infinitely large amplitude structures which break

down the validity of the reductive perturbation method.44

This means that to study finite QDIA SWs beyond this K-dV

limit, one must resort to the other type of nonlinear dynami-

cal equation which can be valid for A� 0. Therefore, in our

present work, we derive a higher order nonlinear equation,

known as modified Gardner (MG) equation, and study the

nonlinear features of the finite amplitude nonplanar QDIA

Gardner solitons (GSs) in a quantum dusty plasma contain-

ing inertial ions, Fermi electrons with quantum effect, and

negatively charged immobile dust.

The manuscript is organized as follows. The model equa-

tions are provided in Sec. II. The MG equation is derived by

using the reductive perturbation method in Sec. III. The

analytical (for some special limiting cases) and numerical sol-

utions are presented in Sec. IV. A brief discussion is finally

given in Sec. V.

II. MODEL EQUATIONS

We consider nonplanar (cylindrical and spherical) ge-

ometry40,45 and nonlinear propagation of the QDIA waves in

an unmagnetized quantum dusty plasma system composed of

inertial ions, massless Fermi electrons, and negatively

charged immobile dust. Thus, at equilibrium, we have

ni0¼ ne0þZdnd0. The nonlinear dynamics of the QDIA

waves, whose phase speed is much smaller (larger) than the

electron (ion) thermal speed, in a nonplanar geometry is gov-

erned by

@ni

@tþ 1

r�@

@rðr�niuiÞ ¼ 0; (1)

@ui

@tþ ui

@ui

@r¼ � @/

@r; (2)

@/@r� 2

5ne

@n5=3e

@rþ b

@

@r

@2

@r2

ffiffiffiffiffinep

ffiffiffiffiffinep

0BB@

1CCA ¼ 0; (3)

1

r�@

@rr�@/@r

� �¼ �q; (4)

q ¼ ni � ð1� lÞne � l; (5)

where �¼ 0 for planar geometry; �¼ 1(2) for a nonplanar cy-

lindrical (spherical) geometry; ni (ne) is the ion (electron)

number density normalized by its equilibrium value ni0 (ne0);

ui is the ion fluid speed normalized by the quantum ion-

acoustic speed Ci¼ (kBTFe=mi)1=2 (with mi being the ion rest

mass, kB being the Boltzmann constant, and TFe being the

Fermi temperature of the electron gas); / is the electrostatic

wave potential normalized by kBTFe/=e (with e being the

magnitude of the charge of an electron); q is the normalized

surface charge density, l¼Zdnd0=ni0; b¼ dH2=2 with

d¼ ni0=ne0; H¼ �hxpe=kBTFe is the ratio between the plasmon

energy and the electron Fermi energy. The time variable (t) is

normalized by x�1pi ¼ mi=4pni0e2ð Þ1=2

, and the space variable

(r) is normalized by kDi¼ (kBTFe=4pni0e2)1=2. We note that in

order to obtain the second term of Eq. (3), we have used the

following Fermi pressure law for the electron species:46,47

Pe ¼1

5

meV2Fen5=3

e

n2=3e0

; (6)

where VFe¼ (2kBTFe=me)1=2 is the electron Fermi speed at

temperature TFe. It may be noted here that Eqs. (1)–(5) can

represent either a bounded or unbounded (open) plasma sys-

tem depending on the value of r chosen (since no restriction

is imposed on r).

III. DERIVATION OF MG EQUATION

To study QDIA GSs by analyzing the ingoing solutions

of Eqs. (1)–(5), we first introduce the stretched coordinates48

f ¼ �ðr � VptÞ; (7)

s ¼ �3t; (8)

where Vp is the QDIA wave phase speed (x=k) and � is a

smallness parameter measuring the weakness of the disper-

sion ð0 < � < 1Þ. We then expand ni, ne, ui, /, and q in

power series of �

ni ¼ 1þ �nð1Þi þ �2nð2Þi þ �3n

ð3Þi þ � � � ; (9)

ne ¼ 1þ �nð1Þe þ �2nð2Þe þ �3nð3Þe þ � � � ; (10)

ui ¼ 0þ �uð1Þi þ �2uð2Þi þ �3u

ð3Þi þ � � � ; (11)

/ ¼ 0þ �/ð1Þ þ �2/ð2Þ þ �3/ð3Þ þ � � � ; (12)

q ¼ 0þ �qð1Þ þ �2qð2Þ þ �3qð3Þ þ � � � (13)

and develop equations in various powers of �. To the lowest

order in �, Eqs. (1)–(13) give

uð1Þi ¼

1

Vpw; (14)

nð1Þi ¼

1

V2p

w; (15)

103704-2 Hossain, Mamun, and Ashrafi Phys. Plasmas 18, 103704 (2011)

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nð1Þe ¼3

2w; (16)

Vp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

3ð1� lÞ

s; (17)

where w¼/(1). Equation (17) represents the linear disper-

sion relation for the QDIA waves. This clearly indicates that

the QDIA wave phase speed (Vp) increases with the increase

of the value of l. To the next higher order in �, we obtain a

set of equations, which, after using Eqs. (14)–(17), can be

simplified as

uð2Þi ¼

1

2V3p

w2 þ 1

Vp/ð2Þ; (18)

nð2Þi ¼

3

2V4p

w2 þ 1

V2p

/ð2Þ; (19)

nð2Þe ¼3

2w2 þ 9

8/ð2Þ; (20)

qð2Þ ¼ 1

2Aw2 ¼ 0; (21)

A ¼ 9

4ð1� lÞð2� 3lÞ: (22)

It is obvious from Eq. (21) that A¼ 0 since w= 0. One can

find that A¼ 0 at its critical value l¼lc¼ 2=3 (which is a

solution of A¼ 0). So, for l around its critical value (lc),

A¼A0 can be expressed as

A0 ’ s@A

@l

� �l ¼ lc

jl� lcj ¼ c1s�; (23)

where c1¼�9=4, jl�lcj is a small and dimensionless pa-

rameter and can be taken as the expansion parameter �, i.e.,

jl� lcj ’ �, and s¼ 1 for l>lc and s¼�1 for l< lc. So,

q(2) can be expressed as

�2qð2Þ ’ �3 1

2c1sw2; (24)

which, therefore, must be included in the third order Pois-

son’s equation. To the next higher order in �, and after some

mathematical calculations, we obtain a set of equations

@nð3Þi

@f¼ 15

2V6p

w2 @w@fþ 3

V4p

@

@f½w/ð2Þ� þ �

V2ps

w

þ 1

V2p

@/ð3Þ

@fþ 2

V2p

@w@s

; (25)

@nð3Þe

@f¼ 81

16w2 @w

@fþ 9

4

@

@f½w/ð2Þ� þ 3

2

@/ð3Þ

@fþ 9b

8

@3w

@f3; (26)

@2w

@f2¼ � 1

2c1sw2 � n

ð3Þi þ ð1� lÞnð3Þe : (27)

Now, combining Eqs. (25)–(27), we obtain an equation of

the form

@w@sþ �

2swþ c2sw

@w@fþ a1w

2 @w@fþ a2

@3w

@f3¼ 0; (28)

where

c2 ¼1

2c1V3

p ; (29)

a1 ¼1

2V3

p

15

2V6p

� 81ð1� lÞ16

( ); (30)

a2 ¼1

2V3

p 1� 9H2

16

� �: (31)

Equation (28) is known as MG equation. The modification is

due to the extra term (viz., �2sw), which arises due to the effects

of the nonplanar geometry. It is important to note that if we

neglect w3 term and put c2s¼V3pA=2¼9V3

p 1�lð Þ 2�3lð Þ=8,

the MG equation reduces to a modified K-dV equation which

can be derived by using a lower order stretching, viz.,

f¼ �1=2 r�Vpt� �

, s¼ �3=2t. However, in this modified K-dV

equation, the nonlinear term vanishes at l¼lc and is not

valid near l¼lc which makes soliton amplitude large enough

to break down the validity of the reductive perturbation

method. But the MG equation derived here is valid for l near

its critical value.

IV. NUMERICAL ANALYSIS

Before going to numerical solutions of MG equation, we

will first analyze the stationary GSs solution of Gardner

equation [Eq. (28) with �¼ 0]. To do so, we first introduce a

transformation n¼ f�U0s which allows us to write

Eq. (28), under the steady state condition, as

1

2

dwdn

� �2

þ VðwÞ ¼ 0; (32)

where the pseudo-potential V(w) is

VðwÞ ¼ � U0

2a2

w2 þ c2s

6a2

w3 þ a1

12a2

w4: (33)

It is obvious from Eq. (33) that

VðwÞjw ¼ 0 ¼dVðwÞ

dw

����w ¼ 0

¼ 0; (34)

d2VðwÞdw2

����w ¼ 0

< 0: (35)

The conditions of Eqs. (34) and (35) imply that SW solution

of Eq. (32) exists if

VðwÞjw ¼ wm¼ 0: (36)

The latter can be solved as

U0 ¼c2s

3wm1;2 þ

a1

6w2

m1;2; (37)

103704-3 Dust ion-acoustic Gardner solitons Phys. Plasmas 18, 103704 (2011)

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wm1;2 ¼ wm 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ U0

V0

r� ; (38)

where wm¼�c2s=a1 and V0 ¼ c22s2=6a1. Now, using

Eqs. (33) and (38) in Eq. (32), we have

dwdn

� �2

þ cw2ðw� wm1Þðw� wm2Þ ¼ 0; (39)

where c¼ a1=6a2. The SW solution of Eq. (32) or (39) is,

therefore, directly given by

w ¼ 1

wm2

� 1

wm2

� 1

wm1

� �cosh2 n

d

� �� �1

; (40)

where wm1,2 are given in Eq. (38), and SWs width d is

d ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�cwm1wm2

p : (41)

We now turn to Eq. (28) with the term (�=2s)w, which is

due to the effects of the nonplanar (cylindrical or spherical)

geometry. An exact analytic solution of Eq. (28) is not possi-

ble. Therefore, we have numerically solved Eq. (28) and have

studied the effects of cylindrical and spherical geometries on

time-dependent QDIA GSs. The results are depicted in

Figures 1–8. The initial condition, which we have used in our

numerical analysis, is in the form of the stationary solution of

Eq. (28) without the term (�=2s)w. Figures 1–6 show how the

effects of a cylindrical geometry modify the QDIA GSs. On

the other hand, Figures 7 and 8 show how the effects of a

spherical geometry modify the QDIA GSs.

The numerical solutions of Eq. (28) (displayed in

Figures 1–8) reveal that for a large value of s, the spherical

and cylindrical solitary waves are similar to planar struc-

tures. This is because for a large value of s (e.g., s¼ 30), the

term (�=2s)w, which is due to the effects of the cylindrical

or spherical geometry, is no longer dominant. However, as

the value of s decreases, the term (�=2s)w becomes domi-

nant, and both spherical and cylindrical SW structures differ

from planar ones. It is found that as the value of s decreases,

the amplitude (the magnitude of the amplitude) of these

localized pulses increases. It is also found that the amplitude

of cylindrical QDIA SWs is larger than those of planar ones

but smaller than that of the spherical ones. We have also

found that the amplitude of positive and negative GSs does

not vary with the quantum diffraction parameter (H), but the

widths of both positive and negative GSs vary with it. From

Figure 1 (for H¼ 0.3) and Figure 2 (for H¼ 0.9), it is clear

that the width of positive GSs decreases with the increase of

H. Again from Figure 5 (for H¼ 0.3) and Figure 6 (for

H¼ 0.9), it is found that the width of negative GSs decreases

FIG. 1. (Color online) Showing the effects of cylindrical geometry on

QDIA positive GSs for l¼ 0.66, U0¼ 0.1, and H¼ 0.3.

FIG. 2. (Color online) Showing the effects of cylindrical geometry on

QDIA positive GSs for l¼ 0.66, U0¼ 0.1, and H¼ 0.9.

FIG. 3. (Color online) Showing the effects of cylindrical geometry on

QDIA positive GSs for l¼ 0.5, U0¼ 0.1, and H¼ 0.3.

FIG. 4. (Color online) Showing the effects of cylindrical geometry on

QDIA negative GSs for l¼ 0.67, U0¼ 0.1, and H¼ 0.3.

103704-4 Hossain, Mamun, and Ashrafi Phys. Plasmas 18, 103704 (2011)

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with the increase of H, which means that the quantum effect

makes both positive and negative GSs more spiky. From

Figure 1 (for l¼ 0.66) and Figure 3 (for l¼ 0.5), we have

observed that the amplitude of the positive GSs decreases

with the increase of l, but the width of the positive GSs

increases with the increase of l. Similarly, from Figure 4

(for l¼ 0.67) and Figure 5 (for l¼ 0.8), it is found that the

amplitude of the negative GSs decreases with the increase of

l, but the width of the negative GSs increases with the

increase of l.

V. DISCUSSION

We have investigated the cylindrical (�¼ 1) and spheri-

cal (�¼ 2) QDIA GSs in quantum dusty plasma by deriving

the MG equation. The K-dV solitons are not valid for

l� 2=3, which vanishes the nonlinear coefficients of the

K-dV equation. However, the QDIA GSs investigated in our

present work are valid for l� 2=3. The results, which have

been obtained from this investigation, can be summarized as

follows:

1. The quantum dusty plasma system under consideration

supports finite amplitude GSs, whose basic features (po-

larity, amplitude, width, etc.) depend on the ion and dust

number densities, and quantum diffraction (tunneling) pa-

rameter, H.

2. GSs are shown to exist for l� 2=3 and are found to be

different from K-dV solitons, which do not exist for

l� 2=3.

3. It is found that at l< 2=3, positive GSs exist, whereas at

l> 2=3, negative GSs exist.

4. We have seen that the amplitude of positive and negative

GSs decreases with l, whereas it is independent of H.

5. It is observed that the width of the GSs increases with lbut decreases with the increase of H. This means that the

quantum effect makes the GSs more spiky.

6. For a large value of s, the nonplanar term (�=2s)w is not

dominant. As a result, the spherical and cylindrical GSs

are found to be similar to planar structures for a large

value of s.

7. As the value of s decreases, the term (�=2s)w becomes

dominant and the amplitude of GSs increases. The ampli-

tude of the cylindrical QDIA GSs is larger than those of

the planar ones but smaller than that of the spherical ones.

It should be mentioned here that in our present investi-

gation, we have neglected the quantum effect of ions since

ions are heavier than electrons. However, QDIA solitary

waves in quantum dusty plasma with or without the effects

of obliqueness and external magnetic field are also problems

of recent interest for many space and laboratory dusty

plasma situations but beyond the scope of our present

FIG. 6. (Color online) Showing the effects of cylindrical geometry on

QDIA negative GSs for l¼ 0.67, U0¼ 0.1, and H¼ 0.9.

FIG. 5. (Color online) Showing the effects of cylindrical geometry on

QDIA negative GSs for l¼ 0.8, U0¼ 0.1, and H¼ 0.3.FIG. 7. (Color online) Showing the effects of spherical geometry on QDIA

positive GSs for l¼ 0.66, U0¼ 0.1, and H¼ 0.3.

FIG. 8. (Color online) Showing the effects of spherical geometry on QDIA

negative GSs for l¼ 0.67, U0¼ 0.1, and H¼ 0.3.

103704-5 Dust ion-acoustic Gardner solitons Phys. Plasmas 18, 103704 (2011)

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investigation. In conclusion, we propose that a new experi-

ment may be designed based on our results to observe such

waves and the effects of nonplanar geometry on these waves

in both laboratory and space quantum dusty plasma systems.

ACKNOWLEDGMENTS

The Third World Academy of Science (TWAS) Research

Grant for research equipment is gratefully acknowledged. The

constructive suggestions of the anonymous reviewer are also

gratefully acknowledged.

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