Planar and non-planar ion acoustic shock waves in electron–positron–ion plasmas

Planar and nonplanar ion acoustic shock waves in relativistic degenerateastrophysical electron-positron-ion plasmasAta-ur-Rahman, S. Ali, Arshad M. Mirza, and A. Qamar Citation: Phys. Plasmas 20, 042305 (2013); doi: 10.1063/1.4802934 View online: http://dx.doi.org/10.1063/1.4802934 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v20/i4 Published by the American Institute of Physics. 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

Planar and nonplanar ion acoustic shock waves in relativistic degenerateastrophysical electron-positron-ion plasmas

Ata-ur-Rahman,1,2,a) S. Ali,2 Arshad M. Mirza,3 and A. Qamar1,2

1Institute of Physics & Electronics, University of Peshawar, Peshawar 25000, Pakistan2National Centre for Physics, QAU Campus, Shahdrah Valley Road, Islamabad 44000, Pakistan3Theoretical Plasma Physics Group, Physics Department, Quaid-i-Azam University, Islamabad 45320,Pakistan

(Received 5 March 2013; accepted 9 April 2013; published online 25 April 2013)

We have studied the propagation of ion acoustic shock waves involving planar and non-planar

geometries in an unmagnetized plasma, whose constituents are non-degenerate ultra-cold ions,

relativistically degenerate electrons, and positrons. By using the reductive perturbation technique,

Korteweg–deVries Burger and modified Korteweg–deVries Burger equations are derived. It is

shown that only compressive shock waves can propagate in such a plasma system. The effects of

geometry, the ion kinematic viscosity, and the positron concentration are examined on the ion

acoustic shock potential and electric field profiles. It is found that the properties of ion acoustic

shock waves in a non-planar geometry significantly differ from those in planar geometry. The

present study has relevance to the dense plasmas, produced in laboratory (e.g., super-intense

laser-dense matter experiments) and in dense astrophysical objects. VC 2013 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4802934]

I. INTRODUCTION

Collective interactions in electron-positron (EP) plasma

have received fundamental importance due to its potential

applications in a variety of astrophysical environments, such

as in the bipolar outflows (jets), in active galactic nuclei

(AGN),1 in pulsars magnetosphere,2 in the polar regions of

neutron stars,3 in the inner region of accretion disc in the

vicinity of black holes,4 and at the centre of our own galaxy.5

It is believed that after the Big Bang, EP plasmas occurred in

the MeV-epoch of the early universe.6 Although experimen-

tal production of EP plasma is quite rare. However, it is

appropriate to mention here that the interaction of intense

laser pulse with a solid target can be regarded as an active

source of EP plasmas.7 Quite recently, Ridgers et al.8 carried

out numerical simulations of a 10 PW laser pulse striking a

solid material and demonstrated the production of EP plasma

by the same mechanism as expected to occur in high-energy

astrophysical environments with a maximum positron den-

sity of 1026 m�3. In particular, positrons can be produced in

large tokamaks including JET and JT-60U due to collisions

of several MeV runaway electrons and thermal particles.9 It

is also believed10 that EP pairs exist in white dwarfs during

the collapse of white dwarfs to neutron stars. Apart from

prolific amount of EP plasmas in the aforementioned astro-

physical environments, a small number of ions are also likely

to be present there. In this context, the investigations11,12

exhibit the presence of ions in the interior of white dwarfs,

as the white dwarf loses its thermal energy to space and stel-

lar material cools down. As a result, the ions behave differ-

ently and can be assumed as non-degenerate and cold.13 The

presence of ions not only modifies the response of EP plasma

but also introduces new time scales in electron-positron-ion

(EPI) plasmas. Very recently, the latter has been recognized

as an active area of research (see, e.g., Refs. 12, 14, and 15).

One of the most interesting quantum phenomena is the

structure of white dwarf stars, which couple the degener-

acy of electrons due to the Pauli exclusion principle and

Heisenberg’s uncertainty principle to the stability of the

white dwarf star on macroscopic scales via the balance

between gravitational pull and the degeneracy pressure of

the electrons. When the electron Fermi energy,

eFeð¼ ð�h2=2meÞð3p2ne0Þ2=3Þ, becomes comparable to or

higher than the electron rest energy (mec2), the relativistic

effects become important and the equation of state changes

from P� n5=3 (for non-relativistic case) to P � n4=3 (for

ultra-relativistic case), making the white dwarf gravitation-

ally unstable for masses larger than about Mc ¼ 1:4M�,

where M� is the solar mass.16 This critical mass is called

Chandrasekhar Mass and most of the stars collapse above

this limit.16,17 In recent years, the study of relativistic

degenerate dense plasmas has attracted much interest (see,

for example, Refs. 12, 14, 15, 18, and 19).

It is well known that the small but finite amplitude ion-

acoustic solitary waves can be described by Korteweg–de

Vries (KdV) equation, where the balance between disper-

sion and nonlinearity results in the formation of stable local-

ized structures. In a dissipative system, a Korteweg–de

Vries Burger (KdVB) equation is obtained, which governs

the shock wave formation. The dissipation in plasmas may

be caused by several mechanisms, e.g., the inter-particle

interaction, the multi-ion streaming, the Landau damping,

the anomalous viscosity, etc. Ion acoustic shock waves

(IASWs) were first observed experimentally in a double

plasma device by Taylor et al.20 More recently, Zobaer

a)Author to whom correspondence should be addressed. Electronic mail:

[email protected]

1070-664X/2013/20(4)/042305/7/$30.00 VC 2013 AIP Publishing LLC20, 042305-1

PHYSICS OF PLASMAS 20, 042305 (2013)

et al.18 investigated IASWs in a degenerate dense plasma

containing non-relativistic degenerate cold ion fluid with

non-relativistic and ultra-relativistic degenerate electrons.

They examined the basic features of the shock structures

and pointed out that the shock wave potential increases

more rapidly for ultra-relativistic electrons and non-

relativistic degenerate ions in comparison to the case of

electron-ion both being non-relativistic degenerate.

Pakzad21 studied IASWs in EPI plasma and showed that the

effects of ion kinematic viscosity, positron concentration,

and suprathermal character of plasma modify the amplitude

and steepness of shock waves, significantly. Roy et al.22

studied the propagation and characteristics of IASWs in

EPI quantum plasma demonstrating the effects of different

plasma parameters on the propagation of shock waves.

However, all the above studies deal with a one-dimensional

planar geometry, which may not be a realistic situation in

space and laboratory plasmas. There are numerous cases of

practical importance where planar geometry does not work

and one would have to consider a non-planar geometry. In

this regard, capsule implosion (spherical geometry), shock

tube (cylindrical geometry), star formation, supernova

explosions are some well-known examples, where non-

planar geometry plays a vital role. Hussain et al.23 explained

cylindrical and spherical IASWs in non-Maxwellian EPI

plasma and derived a modified KdVB equation by using

reductive perturbation technique. It is found that the posi-

tron concentration, the spectral indices of electrons and

positrons, as well as the ion kinematic viscosity significantly

modify the characteristics of IASWs in both cylindrical and

spherical geometries. They also showed that spherical

shocks are strongest in comparison with the cylindrical and

planar shocks. Waqas et al.24 investigated non-planar

IASWs in EPI quantum plasmas by taking into account

the effects of geometry, quantum parameter involving

Bohm potential, and kinematic viscosity. Later, Mamun

and Shukla19 considered the propagation of non-planar ion

shock waves in a strongly coupled degenerate plasma to

obtain a modified Burger equation, where the strong correla-

tion among ions is a source of dissipation. Sahu and

Roychoudhury25 studied planar and non-planar quantum

IASWs in unmagnetized plasma. Through numerical com-

putation of modified KdVB equation, they explained the

effects of geometry on the dynamics of IASWs. To the best

of our knowledge, no attempt has been made to investigate

the nonlinear properties of IASWs in planar and non-planar

plasma geometries incorporating non-degenerate cold ions,

relativistic degenerate electrons, and positrons. In this paper,

our main objective is to study planar and non-planar IASWs

in the said plasma by using the reductive perturbation tech-

nique. The effects of ion kinematic viscosity, positron con-

centration, and geometry are examined on the shock wave

potential and electric field profiles.

The manuscript is organized in the following manner.

The basic model equations are presented in Sec. II. The

planar KdVB equation is derived in Sec. III and a modified

KdVB equation for non-planar geometry in Sec. IV. The

numerical results are discussed in Sec. V, while the summary

is given in Sec. VI.

II. BASIC MODEL EQUATIONS

We consider nonlinear propagation of finite amplitude

IASWs in an unmagnetized, homogeneous, dissipative

plasma, whose constituents are the relativistic degenerate

inertialess electrons and positrons, and non-degenerate iner-

tial cold ions. At equilibrium, the charge-neutrality condition

demands ne0 ¼ Zini0 þ np0, where ns0 is the number density

of sth species (s¼ e for electrons, p for positrons, and i for

ions) and Zi is the charge state of ions. We assume that the

wave phase speed is much smaller than the Fermi speeds of

electrons and positrons but much larger than the ion thermal

speed, viz. vti � vph � vFe, vFp.

The dynamics of ions is governed by the following ion

continuity and momentum equations, respectively, as:

@ni

@tþ @ðniviÞ

@x¼ 0 (1)

and

@vi

@tþ vi

@vi

@x¼ � e

mi

@/@xþ li

@2vi

@x2: (2)

Here, ni, vi, mi, e, /, and li are the ion number density, ion

fluid velocity, ion mass, electronic charge, electrostatic

potential, and ion dynamic viscosity, respectively. Note that

in case of singly ionic charging state, we have Zi ¼ 1. On the

other hand, the momentum equations for inertialess electrons

and positrons can be expressed, respectively, as

0 ¼ ene@/@x� @PeR

@x(3)

and

0 ¼ enp@/@xþ @PpR

@x; (4)

where ne (np) is the electron (positron) number density.

Equations (1)–(4) may be closed with the help of Poisson’s

equation, as

@2/@x2¼ �4peðni þ np � neÞ: (5)

The degeneracy pressure16,17 for fully degenerate relativistic

electrons and positrons is given by

PjR ¼pm4

j c5

3h3frjð2r2

j � 3Þðr2j þ 1Þ1=2 þ 3 sinh�1ðrjÞg; (6)

where j¼ (e, p) denotes the electrons and positrons, respec-

tively, rjð¼ pFj=mjcÞ is the normalized relativistic parameter,

and pFj is the momentum on the Fermi surface.

Expanding (6) around an equilibrium density value nj0

by Taylor series expansion, we obtain26

PjR ’ Pj0 þ@PjR

@nj

� �nj¼nj0

dnj þ1

2

@2PjR

@n2j

!nj¼nj0

dn2j (7)

042305-2 Rahman et al. Phys. Plasmas 20, 042305 (2013)

or

PjR ¼ Pj0 þ2

3cj0

eFjdnj þ3r2

j0 þ 2

9c3j0nj0

eFjdn2j ; (8)

where cj0 ¼ ð1þ r2j0Þ

1=2is the relativistic gamma factor with

rj0 ¼ ðpj0=mjcÞ; pj0 ¼ pFj ¼ ð3h3nj0=8pÞ1=3and dnj is the

perturbed density.

To normalize the above set of equations, the following

rescaling is employed:

x! �xkFi; ns ! �ns ns0; /! eFe

e�/;

vi ! �ui Ci; t! �t x�1pi ; (9)

where Ci ¼ ðeFe=miÞ1=2;x�1pi ¼ ðmi=4pni0e2Þ1=2

, and kFi

¼ ðeFe=4pni0e2Þ1=2are the quantum ion acoustic speed, the

ion plasma period, and the quantum analog of the Debye

length, respectively. The normalized set of fluid equations in

a planar geometry by taking into account the effects of rela-

tivistic degeneracy pressure can be written as

@ni

@tþ @

@xðniuiÞ ¼ 0; (10)

@ui

@tþ ui

@ui

@x¼ � @/

@xþ gi

@2ui

@x2; (11)

0 ¼ ne@/@x� ae

@dne

@x� be

@dn2e

@x; (12)

0 ¼ np@/@xþ ap

@dnp

@xþ bp

@dn2p

@x; (13)

and

@2/@x2¼ bne � anp � ni; (14)

where we have a ¼ np0=ni0, b ¼ ne0=ni0, ae ¼ 2=ð3ce0Þ,be¼ð3g2

e0þ2Þ=9c3e0, ap¼ðeFp=eFeÞf2=3cp0g, bp¼ðeFp=eFeÞ

fð3g2p0þ2Þ=9c3

p0g, and gi¼ li=xpik2Fi. Moreover, the ratios

a and b can be expressed in terms of positron concentration

(p ¼ np0=ne0) as a¼ p=ð1�pÞ and b¼ 1=ð1�pÞ and are

connected via the charge neutrality condition at equilibrium

as b¼ 1þa.

III. KORTEWEG-DE VRIES BURGERS EQUATION

In order to investigate IASWs in a relativistic degenerate

dense plasma, we employ the standard reductive perturbation

technique27 to derive a KdVB equation. For this, we describe

the independent variables into a moving frame f in which the

shock wave moves with a phase speed of k, as

f ¼ ��1=2ðxþ ktÞ and s ¼ �3=2t; (15)

where � is a small expansion parameter, i.e., �� 1. The de-

pendent variables can be expanded in the following form:

ns ¼ 1þ �ns1 þ �2ns2 þ � � � ;ui ¼ 0þ �ui1 þ �2ui2 þ � � � ;/ ¼ 0þ �/1 þ �2/2 þ � � � :

(16)

We assume a weak damping due to the ion kinematic viscos-

ity gi, so that we may let gi ¼ �1=2gi0.

Using Eqs. (15) and (16) into Eqs. (10)–(13) and collecting

the lowest powers in �, the following equations are obtained:

ni1 ¼ui1

k; ui1¼

/1

k; ne1¼

/1

ae

; and np1¼�/1

ap

: (17)

Solving for k, we get

k ¼ ð1� pÞaeap

ap þ pae

� �1=2

: (18)

It is clear from Eq. (18) that the phase speed has been signifi-

cantly affected by the relativistic factor cj0, due to the

electron-positron degeneracy pressure. Also, in the absence

of positron concentration ðviz: p ¼ 0Þ, the phase speed

k(¼ ffiffiffiffiffiaep

) of Ref. 26 is recovered for relativistic degenerate

dense EI plasma.

Now, the next higher order equations in � are given by

k@ni2

@f� @ui2

@f� @ni1

@s� @ðni1ui1Þ

@f¼ 0; (19)

k@ui2

@f� @/2

@f� @ui1

@s� ui1

@ui1

@fþ gi0

@2ui1

@f2¼ 0; (20)

@/2

@f� ae

@ne2

@f� be

@ðne1Þ2

@fþ ne1

@/1

@f¼ 0; (21)

@/2

@fþ ap

@np2

@fþ bp

@ðnp1Þ2

@fþ np1

@/1

@f¼ 0; (22)

and

@2/1

@f2� bne2 þ anp2 þ ni2 ¼ 0: (23)

Eliminating second order quantities (ne2; ni2; np2, and /2)

from Eqs. (19)–(23) by using Eq. (17), we obtain a KdVB

equation in a planar geometry describing the nonlinear

IASWs in a relativistic degenerate EPI plasmas

@/1

@sþ A/1

@/1

@fþ B

@3/1

@f3¼ C

@2/1

@f2: (24)

The second term represents the nonlinearity (responsible for

wave steepening), the third term corresponds to the disper-

sion (causing wave broadening), and the fourth term shows

the dissipation (resulting in wave decay) and their associated

coefficients are defined as

A ¼ B

1� p

2be

a3e

� 1

a2e

þ p

a2p

�2pbp

a3p

þ 3ð1� pÞk4

( );

B ¼ k3

2; and C ¼ gi0

2: (25)

042305-3 Rahman et al. Phys. Plasmas 20, 042305 (2013)

In the limit gi0 ¼ 0 (i.e., the dissipationless case), one can

retrieve the KdV equation corresponding to nonlinearity and

dispersion coefficients as reported in Ref. 28 for relativistic

degenerate EPI plasma. Further, ignoring the positron con-

centration ðp ¼ 0Þ, we obtain the earlier result26 for relativis-

tic degenerate dense EI plasma.

IV. MODIFIED KORTEWEG-DE VRIES BURGERSEQUATION

In a non-planar geometry (cylindrical or spherical geom-

etry), the nonlinear dynamics of IASWs in a relativistic

degenerate dense EPI plasma can be described by the follow-

ing set of equations:24

@ni

@tþ 1

rv

@

@rðrvniviÞ ¼ 0; (26)

@vi

@tþ vi

@vi

@r¼ � e

mi

@/@rþ li

1

rv

@

@rrv @vi

@r

� �� �; (27)

0 ¼ ene@/@r� @PeR

@r; (28)

0 ¼ enp@/@rþ @PpR

@r; (29)

and

1

rv

@

@rrv @/@r

� �¼ �4peðni þ np � neÞ; (30)

where v ¼ 0 for a planar geometry and v ¼ 1 (v ¼ 2) for a

non-planar cylindrical (spherical) geometry.

Using the rescaling as in Eq. (9), one obtains a normal-

ized set of equations in a non-planar geometry, as

@ni

@tþ 1

rv

@

@rðrvniviÞ ¼ 0; (31)

@ui

@tþ ui

@ui

@r¼ � @/

@rþ gi

1

rv

@

@rrv @ui

@r

� �� �; (32)

ne@/@r� ae

@dne

@r� be

@dn2e

@r¼ 0; (33)

np@/@rþ ap

@dnp

@rþ bp

@dn2p

@r¼ 0; (34)

and

1

rv

@

@rrv @/@r

� �¼ bne � anp � ni: (35)

Following the same procedure as described in Sec. III, a

modified KdVB equation can be derived in EPI plasma

accounting for the relativistic degenerate electrons and posi-

trons and geometrical effects, as

@/1

@sþ �

2s/1 þ A/1

@/1

@fþ B

@3/1

@f3¼ C

@2/1

@f2; (36)

where we have now used the stretched coordinates, as

f ¼ ��1=2ðr þ ktÞ, and s ¼ �3=2t. Equation (36) has been

modified by an extra term ð�=2sÞ/1, which arises due to the

effect of the cylindrical geometry (� ¼ 1) or spherical geom-

etry (� ¼ 2). Note that, if we set � ¼ 0 in the above equation,

a one dimensional planar KdVB equation (24) is obtained.

V. NUMERICAL RESULTS AND DISCUSSIONS

In order to solve the nonlinear partial differential equa-

tion (36), numerous methods have been used in the literature,

for instance, inverse scattering method,29 Hirota Bilinear for-

malism,30 Backlund transformation,31 tanh method,32 etc.

However, when a partial differential equation contains the

combined effect due to dispersion and dissipation, the most

efficient method for solving such an equation is tanh method.

In numerical analysis, we have used the following analytical

solution33 of KdVB equation as our initial profile

/1ðn; sÞ ¼ /0shock 1þ 1

2sec h2v� tanh v

� �; (37)

where /0shockð¼ 6C2=25ABÞ is the shock wave amplitude,

v ¼ jðn� u0sÞ, j ¼ C=10B, j�1 denotes the shock width,

and u0ð¼ 6C2=25BÞ is the shock speed. Employing Eq. (37),

we may derive an exact expression34 for the electric field

ðE ¼ �r/1Þ profile as

E ¼ /0shock

j�1sec h2vð1þ tanh vÞ: (38)

By taking into account the geometrical effect (v 6¼ 0), an

exact analytical solution of (36) is not possible; therefore, we

solve it numerically to understand the nature of the non-

planar shock waves. For this purpose, we choose Eq. (37) as

our initial profile at time s ¼ �21. We consider some typical

plasma density which is consistent with the relativistic

degenerate astrophysical plasmas, e.g., ne0 ’ 1030cm�3, and

compute the plasma parameters as ce0 � 1:55; ae � 0:429;be � 0:185, and eFe � 5:80� 10�7erg. These values are

found to be relevant to the interior of white dwarfs having

mass density qm � 107 g=cm3.12,15,19 For low positron

FIG. 1. Shock wave potential /1 versus f for variation of ion kinematic vis-

cosity gi0. We have used other parameters as ne0 ¼ 1030cm�3, p¼ 0.1, and

eFe ¼ 5:80� 10�7erg.

042305-4 Rahman et al. Phys. Plasmas 20, 042305 (2013)

concentration (p¼ 0.1), we find cp0 � 1:14; k � 0:54;/0shock � 0:04; j�1 � 3:86;A � 2:87;B � 0:08, and C¼ 0.2.

Moreover, for high value of positron concentration (p¼ 0.7),

we obtain cp0 � 1:46; k � 0:27;/0shock � 0:18; j�1 � 0:47;A � 5:66;B � 0:009, and C¼ 0.2 with fixed ion kinematic

viscosity (gi0). Thus, the relevant characteristic parameters

of IASWs are strongly influenced by the positron concentra-

tion. In Fig. 1, we have plotted the shock wave potential

/1 against f for varying the ion kinematic viscosity

(0 � gi0 � 0:6). It is observed that as the kinematic viscosity

increases (i.e., to increase dissipation effects in the system),

the strength of the shock wave potential also increases.

Physically, the strength of the shock, i.e., 6C2=25AB is

directly related with kinematic viscosity, consequently the

strength of the shock is enhanced with increasing values of

kinematic viscosity. The effect of positron concentration on

the propagation characteristics of shock waves in a planar

geometry is displayed in Fig. 2. Note that the strength of the

shock wave potential enhances by increasing positron

concentration (0:1 � p � 0:7) and the shock wave front

becomes more steepened at higher values of positron

concentration.

Figure 3 exhibits a comparison between planar and non-

planar shock wave profiles at time s ¼ �3. It is noticed that

the strength of the shock potential is maximum for spherical

geometry (� ¼ 2), minimum for planar geometry (� ¼ 0),

and is intermediate for cylindrical geometry (� ¼ 1). It is

also seen that the steepness of the shock front follows the

same trend as that of strength of the shock. The difference in

the steepness and strength of the shock waves in planar and

non-planar geometries is due to the presence of a term

ð�=2sÞ/1. For large values of jsj, the aforementioned geo-

metrical term becomes negligible and the profiles of cylindri-

cal and spherical shock waves become identical to one

dimensional planar shock waves.

Figure 4 displays the time evolution of non-planar cylin-

drical and spherical shock waves. It is found that for smaller

values of jsj, the strength and steepness of the shock wave

increase in both cylindrical and spherical geometries. It is

worth mentioning here that in non-planar geometries, the

shock wave propagates faster in comparison to the shocks in

planar geometry. However, ion acoustic shocks move more

FIG. 2. Shock wave potential /1 versus f for different values of positron

concentration p, with gi0 ¼ 0:2. Other parameters are the same as in Fig. 1.

FIG. 3. A comparison of the numerical solution of Eq. (36) for planar

(� ¼ 0) and non-planar geometries (v 6¼ 0) at time s ¼ �3. Other parameters

are the same as in Fig. 1.

FIG. 4. The profile of shock wave potential for varying time s in (a) cylindrical geometry (v ¼ 1) and (b) spherical geometry (v ¼ 2). Other parameters are

being taken as ne0 ¼ 1030cm�3, p¼ 0.1, eFe ¼ 5:80� 10�7erg, and gi0 ¼ 0:2.

042305-5 Rahman et al. Phys. Plasmas 20, 042305 (2013)

faster in spherical geometry as compared to cylindrical ge-

ometry, showing that compressions or large densities can be

achieved using spherical shocks, which can be important due

to its applications in astrophysical plasmas as well as in iner-

tial confinement fusion plasmas.

Figure 5 examines the effects of positron concentration

and ion kinematic viscosity on the electric field profiles of

IASWs. For smaller values of positron concentration, the

electric field is spread out with a small finite amplitude but

as the positron concentration increases, the electric field pro-

files become more localized with enhanced amplitude as can

be seen from Fig. 5(a). Similarly, the variation of ion kine-

matic viscosity comparatively leads to more wider electric

field profiles [Fig. 5(b)] associated with IASWs. Note that

the electric field is the negative gradient of the potential and

hence more localized shocks with steeper slopes result for

higher values of positron concentration and ion kinematic

viscosity.

VI. SUMMARY

To summarize, we have presented planar and non-

planar IASWs in an unmagnetized and dissipative plasma

system comprising of non-degenerate cold ions, relativistic

degenerate electrons, and positrons. By using the reductive

perturbation technique, KdVB and modified KdVB equa-

tions are derived and analyzed both analytically and

numerically. It has been found that the ion kinematic vis-

cosity, positron concentration, and geometrical effects

significantly modify the profiles of IASWs in relativistic

degenerate EPI plasmas. The strength and propagation

speed of shock wave potential are found to be maximum for

spherical geometry case, intermediate for cylindrical geom-

etry, and minimum for one dimensional planar geometry.

These results are helpful to understand the propagation of

localized shock structures in relativistic degenerate dense

plasmas, such as, those found in dense astrophysical objects

(e.g., white dwarfs), as well as, in extreme states of matter

on Earth and in space.35

ACKNOWLEDGMENTS

Ata-ur-Rahman is thankful to Higher Education

Commission for financial support through Indigenous 5000

Ph.D. Scholarship Scheme.

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FIG. 5. Plot of E versus f for different values of (a) positron concentration, p¼ 0.1 (dotted curve), p¼ 0.3 (dotted-dashed curve), p¼ 0.5 (dashed curve), and

p¼ 0.7 (solid curve), with gi0 ¼ 0:2 and (b) ion kinematic viscosity, gi0 ¼ 0:2 (dotted curve), gi0 ¼ 0:4 (dotted-dashed curve), gi0 ¼ 0:6 (dashed curve), and

gi0 ¼ 0:8 (solid curve), with p¼ 0.1.

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