Commun. Theor. Phys. 63 (2015) 243–248
Vol. 63, No. 2, February 1, 2015
Cylindrical and Spherical Electron-Acoustic Shock Waves in Electron-Positron-Ion
Plasmas with Nonextensive Electrons and Positrons
A. Rafat,∗ M.M. Rahman, M.S. Alam, and A.A. Mamun
Department of Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh
(Received September 30, 2014; revised manuscript received November 24, 2014)
Abstract Electron-acoustic shock waves (EASWs) in an unmagnetized four-component plasma (containing hot electrons and positrons following the q-nonextensive distribution, cold mobile viscous electron fluid, and immobile positive
ions) are studied in nonplanar (cylindrical and spherical) geometry. With the help of the reductive perturbation method,
the modified Burgers equation is derived. Analytically, the effects of nonplanar geometry, nonextensivity, relative number
density and temperature ratios, and cold electron kinematic viscosity on the basic properties (viz. amplitude, width,
speed, etc.) of EASWs are discussed. It is examined that the EASWs in nonplanar geometry significantly differ from
those in planar geometry. The results of this investigation can be helpful in understanding the nonlinear features of
EASWs in various astrophysical plasmas.
PACS numbers: 52.27.Ep, 52.27.-h, 52.35.Mw, 52.35.Tc
Key words: electron-acoustic shock waves, electron-positron-ion plasmas, modified Burgers equation, nonplanar geometry, nonextensivity
1 Introduction
Electron-acoustic waves (EAWs) are one of the basic
wave processes in plasmas, which are typically high frequency plasma waves in comparison with the ion plasma
frequency. The ion dynamics does not influence the EAWs
and thus the positively charged ions may be assumed to
be at rest, and their charges to be uniformly distributed
over the entire plasma system. The propagation characteristics of nonlinear EAWs have been extensively studied both theoretically and experimentally. Attempts have
been made to investigate the linear and nonlinear properties of EAWs[1−5] in plasma literature.
In contrast to the usual two component plasma
(electron-ion plasma), it has been observed that the nonlinear waves in plasmas having an additional positron
component behave differently.[6] Thus it is important to
investigate the linear and nonlinear behaviour of plasma
waves in electron-positron-ion (e-p-i) plasmas because
of their significant dominance in the early universe,[7]
supernovas, active galactic nuclei,[8] and in the pulsar
magnetospheres.[9] A great deal of research work has been
carried out to study different types of nonlinear waves in
e-p-i plasmas during the last three decades.[10−22] Rahman et al.[23] studied the positron-acoustic shock waves
in a four-component e-p-i plasmas consisting of immobile
positive ions, inertial cold positrons, and nonthermal distributed electrons and hot positrons. Biswajit Sahu[24]
investigated the planar as well as nonplanar positronacoustic shock waves in e-p-i plasmas containing Boltzmann distributed electrons and hot positrons, inertial cold
positrons, and immobile positive ions.
Over the last two decades, it has been proven that
nonextensive plasma[25] is an interesting research topic
because of it’s wide relevance in astrophysical and cosmological scenarios viz. planetary rings, cometary tails,
solar winds,[26] stellar polytropes,[27] hadronic matter
and quark-gluon plasma,[28] proto-neutron stars,[29] etc.
Tsallis[25] proposed a new entropy for nonextensive velocity distribution. The nonextensive parameter is generally
denoted by q and in the q-nonextensive framework, the
one-dimensional equilibrium distribution function, fs (vs ),
is given by[30]
h
ms vs2 i1/(q−1)
,
fs (vs ) = Aq 1 − (q − 1)
2kB Ts
where the normalization constant is
s
ns0 Γ(1/(1 − q))
ms (1 − q)
Aq =
, for −1 < q < 1 .
Γ(1/(1 − q) − 1/2)
2πkB Ts
Here, ns0 , ms , vs , and Ts are equilibrium number density,
mass, speed, and temperature of the energetic particles,
respectively (s stands for electron or ion or positron). kB is
the Boltzmann constant. Many authors have been successfully studied different kinds of nonlinear waves in nonextensive plasmas by considering nonextensive distribution (characterizing by a parameter q) of electrons,[31−32]
ions,[33] or both electrons and positrons.[34−35] Hussain et
al.[34] derived the Kadomtsev–Petviashvili–Burgers equation by using the reductive perturbation technique and
studied the two-dimensional (2D) ion-acoustic (IA) shocks
in nonextensive and dissipative e-p-i plasmas consisting
of warm ions, and q-nonextensive distributed electrons
and positrons. Han et al.[36] investigated the electron-
∗ E-mail: rafat.plasma@gmail.com
c 2015 Chinese Physical Society and IOP Publishing Ltd
°
http://www.iop.org/EJ/journal/ctp http://ctp.itp.ac.cn
244
Communications in Theoretical Physics
acoustic shock waves (EASWs) in a nonextensive electronion plasma.
It is quite well-known that shock waves can appear
in dissipative plasma systems, where kinematic viscosity
among the plasma constituents, Landau damping, collisions between charged and neutral particles are responsible for the dissipation resulting in the formation of shock
structures in that systems.[37] Shock waves have been
studied because of their significant importance in space
and laboratory plasmas, experimentally and as well as
theoretically.[37−42] Roy et al.[40] studied the IA shocks
in a quantum e-p-i plasma and found that the formation of the monotonic or oscillatory shock structures depend on the value of the quantum parameter. However, all of these works[23,34,36,40−42] are limited to onedimensional (1D) planar geometry associated with shocks
in different plasma models which may not be a realistic situation in space and laboratory devices. Since in
many situations, the wave structures observed in space or
laboratory devices are certainly not infinite (unbounded)
in one dimension.[43] The examples of nonplanar (cylindrical and spherical) geometries of practical interest are
supernova explosions, capsule implosion (spherical geometry), shock tube (cylindrical geometry), etc.[44] Han
et al.[45] studied the nonplanar electron-acoustic solitary
waves and EASWs in a dissipative, superthermal space
plasmas.[46−47] By recalling, the authors of Ref. [34] investigated the 2D IA shocks in e-p-i plasmas with warm ions,
and q-nonextensive distributed electrons and positrons.
But up to now, it is our first attempt to present a theoretical investigation on the nonplanar EASWs in e-p-i
plasmas with immobile positive ions, inertial cold electrons, and hot electrons and positrons following the qnonextensive distribution. The aim of this paper is to
study the effects of nonplanar geometry, nonextensivity
of hot electrons and positrons, and kinematic viscosity of
cold electrons on the basic features (viz. amplitude, width,
speed, etc.) of such EASWs in e-p-i plasmas under consideration. The arrangement of this paper is as follows: The
basic governing equations are presented in Sec. 2. The
modified Burgers (MB) equation is derived and the basic
features of the shock waves are numerically analyzed in
Sec. 3. Finally, a brief discussion is provided in Sec. 4.
2 Governing Equations
We consider an unmagnetized four-component plasma
system consisting of q-distributed hot electrons and
positrons, inertial cold electrons, and immobile positive
ions to study the propagation of EASWs in nonplanar geometry. Hence, at equilibrium, nc0 + nh0 = np0 + ni0 ,
where nc0 and nh0 are the unperturbed cold electron
number density and hot electron number density, respectively. np0 (ni0 ) is the unperturbed number density of
positrons (ions). To model the nonextensivity of hot electron and positron components, we employ the following qnonextensive velocity distribution function[48−49] for hot
Vol. 63
electrons and positrons:
h
eφ i(q+1)/2(q−1)
nh = nh0 1 + (q − 1)
,
kB Th
h
eφ i(q+1)/2(q−1)
np = np0 1 − (q − 1)
.
kB Tp
Here, the parameter q indicates the strength of nonextensivity. nh (np ) is the perturbed number density of hot
electrons (positrons). Th (Tp ) is the temperature of hot
electrons (positrons). It is important to note that in case
of q < −1, the q-distribution is not normalizable[50] The
strength of nonextensivity, q varies as −1 < q < 1.[50] For
q ≥ 1,[49−51] the distribution function exhibits Maxwell–
Boltzmann velocity distribution.
The nonlinear dynamics of the electron-acoustic (EA)
waves propagating in such a plasma system, in a nonplanar geometry, is governed by
1 ∂ ν
∂nc
+ ν
(r nc uc ) = 0 ,
(1)
∂t
r ∂r
∂φ
1 ∂ ³ ν ∂uc ´
∂uc
∂uc
+ uc
=
+η ν
r
,
(2)
∂t
∂r
∂r
r ∂r
∂r
1 ∂ ³ ν ∂φ ´
r
= nc + µ1 [1 + (q − 1)φ](q+1)/2(q−1)
rν ∂r
∂r
− µ2 [(1 − (q − 1)σφ](q+1)/2(q−1) − µ3 .
(3)
It is to be noted that ν = 0 for 1D planar geometry, and
ν = 1 (2) for a nonplanar cylindrical (spherical) geometry. Here, nc is the cold electron number density normalized by its equilibrium value nc0 , uc is the cold electron
fluid speed normalized by Cc = (kB Th /mc )1/2 , φ is the
electrostatic wave potential normalized by kB Th /e, kB is
the Boltzmann constant, mc is the cold electron mass,
σ = Th /Tp , µ1 = nh0 /nc0 , µ2 = np0 /nc0 , µ3 = ni0 /nc0 ,
and η is the cold electron kinematic viscosity normalized
by mc nc0 ωpc λ2D . The time variable t is normalized by
−1
ωpc
= (mc /4πnc0 e2 )1/2 , and the space variable x is normalized by the Debye length λD = (kB Th /4πnc0 e2 )1/2 .
3 Derivation of MB Equation
To study the finite amplitude EASWs, we derive the modified Burgers (MB) equation by introducing the following
stretched coordinates:
ξ = ²(r − Vp t) ,
(4)
τ = ²2 t ,
(5)
where Vp is the phase speed of the EASWs and ² is a smallness parameter measuring the weakness of the dispersion
(0 < ² < 1). To obtain a dynamical equation, we also
expand the perturbed quantities nc , uc , and φ, in power
series of ². Let S be any of the system variables nc , uc ,
and φ, describing the systems’s state at a given position
and instant. We consider small deviations from the equi(0)
(0)
librium state S (0) which explicitly is nc = 1, uc = 0,
(0)
and φ = 0 by taking
∞
X
S = S (0) +
²n S (n) .
(6)
n=1
Communications in Theoretical Physics
No. 2
To the lowest order in ², Eqs. (1)–(6) give
1 (1)
u(1)
φ ,
c =−
Vp
n(1)
c =−
s
Vp =
(7)
1 (1)
φ ,
Vp2
(8)
2
.
(1 + q)(µ1 + µ2 σ)
(9)
Equation (9) represents the phase speed of the EASWs.
To the next higher order in ², we obtain a set of equations,
which, after using Eqs. (7)–(9), can be simplified as
(1)
(2)
(1)
(2)
(2)
(1)
∂nc
∂nc
∂uc
∂ (1) (1)
νuc
− Vp
+
+
[nc uc ] +
= 0 , (10)
∂τ
∂ξ
∂ξ
∂ξ
Vp τ
(1)
(1)
∂uc
∂uc
∂uc
∂φ(2)
∂ 2 uc
− Vp
+ u(1)
−
−
η
c
∂τ
∂ξ
∂ξ
∂ξ
∂ξ 2
= 0 , (11)
(2)
∂nc
1
∂φ(2)
+ (1 + q)(µ1 + µ2 σ)
∂ξ
2
∂ξ
∂φ(1)
1
− (1 + q)(−3 + q)(µ1 − µ2 σ)φ(1)
= 0.
(12)
4
∂ξ
Now, combining Eqs. (10)–(12), we obtain a new equation
of the form:
∂φ(1)
∂φ(1)
ν (1)
∂ 2 φ(1)
+ Aφ(1)
+
φ −C
= 0,
(13)
∂τ
∂ξ
2τ
∂ξ 2
where
Vp3 h 1
A=
µ1 (1 + q)(−3 + q)
2 4
1
3 i
− µ2 σ(1 + q)(−3 + q) − 4 ,
(14)
4
Vp
η
C= .
(15)
2
Equation (13) is known as the MB equation modified by
the extra term (ν/2τ )φ(1) which arises due to the effect
of the nonplanar cylindrical (ν = 1) or spherical (ν = 2)
geometry. The third and fourth term on the left hand side
of Eq. (13) represents the geometry effects and dissipation
respectively. An exact analytical solution of Eq. (13) is not
possible. Therefore, we have numerically solved Eq. (13),
and have studied the effects of cylindrical (ν = 1) and
spherical (ν = 2) geometry on EASWs in presence of qdistributed hot electrons and positrons. It is obvious from
Eq. (13) that the nonplanar geometrical effect is important when τ → 0 and weaker for larger value of |τ |.
At first, we consider 1D planar geometry (ν = 0) and
examine the basic features of the shock wave solution
of MB equation. The stationary shock wave solution of
Eq. (13) in a planar geometry (ν = 0) is
h
³ ξ ´i
,
(16)
φ(1) (ν = 0) = φ(1)
m 1 − tanh
∆
(1)
where φm = U0 /A is the shock amplitude and ∆ = 2C/U0
is the shock width. It is found that for A < (>)0, the
plasma system supports compressive (rarefactive) EASWs
245
which are associated with a positive (negative) potential,
and no shock waves exist at A = 0 and A ∼ 0. It is obvious that A is a function of µ1 , µ2 , q and σ. Therefore,
A(µ1 = µc ) = 0 and µ1 can be expressed as
p
P + P (P − 24µ2 σL)
− µ2 σ ,
(17)
µ1 = µc =
6L
where, P = (−3+q), L = (1+q) and µc is the critical value
of µ1 below (above) which shock waves with a positive
(negative) potential exist. We can find A = 0 for a certain (critical) value of µ1 ; i.e., A = 0 for µ1 = µc ' 0.114
(obtained from A(µ1 = µc ) = 0 for a set of plasma param(1)
eters µ2 = 0.5, σ = 1, and q = 0.02). Clearly, φm = ∞ at
µ1 = µc . This means that the existence of the small amplitude shock waves with a positive potential for µ1 < µc ,
and with a negative potential for µ1 > µc . However, this
modified Burgers equation is not valid for µ1 ∼ µc . This
is because the µ1 ∼ µc give rise to infinitely large amplitude structures which break down the validity of the
reductive perturbation method applied for deriving the
modified Burgers equation. This implies that EASWs are
formed for the values far above or below the critical value.
Fig. 1 (Color online) The effect of cylindrical (ν = 1)
geometry on EA shock structure for below the critical
value µ1 = 0.08 for µ2 = 0.5, σ = 1, η = 0.3, q = 0.02,
and U0 = 0.02.
Fig. 2 (Color online) The effect of cylindrical (ν = 1)
geometry on EA shock structure for above the critical
value µ1 = 0.15 for µ2 = 0.5, σ = 1, η = 0.3, q = 0.02,
and U0 = 0.02.
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Vol. 63
start with a large (absolute) value of τ (τ = −20), as for
a large value of τ , the term (ν/2τ )φ(1) is negligible. This
large value of τ we choose in Eq. (16) (which is the stationary solution of Eq. (13) without the term (ν/2τ )φ(1) )
as our initial aim.
Fig. 3 (Color online) The effect of spherical (ν = 2)
geometry on EA shock structure for below the critical
value µ1 = 0.08 for µ2 = 0.5, σ = 1, η = 0.3, q = 0.02,
and U0 = 0.02.
Fig. 6 (Color online) Variation of the amplitude of
positive shock structures with µ2 and µ1 . Here σ = 1,
q = 0.02, and U0 = 0.02.
Fig. 4 (Color online) The effect of spherical (ν = 2)
geometry on EA shock structure for above the critical
value µ1 = 0.15 for µ2 = 0.5, σ = 1, η = 0.3, q = 0.02,
and U0 = 0.02.
Fig. 7 (Color online) Variation of the width of shock
waves with U0 and η. Here µ1 = 0.08, µ2 = 0.5, σ = 1,
and q = 0.02.
Fig. 5 (Color online) Variation of the amplitude of positive shock structures with σ and q. Here µ1 = 0.08,
µ2 = 0.5, and U0 = 0.02.
Since an exact analytical solution of Eq. (13) is not
possible, we have numerically solved Eq. (13) and studied
the effects of cylindrical (ν = 1) and spherical (ν = 2) geometries on the time-dependent EA shock structures. We
The shock structures are depicted in Figs. 1–4, which
show how the effects of cylindrical (ν = 1) and spherical
(ν = 2) geometries modify the time-dependent EA shock
structures. It is found that as the magnitude of τ increases the shock height decreases, and the effect is more
pronounced in spherical geometry by comparison with the
cylindrical geometry as shown in Figs. 1–4. The amplitude
of positive shock structures decreases with the increase of
ratio of the hot electron temperature to the positron temperature σ and nonextensive parameter q as displayed in
Fig. 5. From Fig. 6, it is found that the amplitude of positive shock structures increases in increasing the number
density ratio of positrons (hot electrons) to the cold electrons µ2 (µ1 ). Figure 7 shows that the width of the shock
waves decreases (increases) with the increase of cold electron fluid speed U0 (cold electron kinematic viscosity η).
No. 2
Communications in Theoretical Physics
The effect of hot electron and positron nonextensivity on
the phase speed (Vp ) of EASWs is shown in Fig. 8.
Fig. 8 (Color online) Variation of the phase speed Vp
with σ and q for EASWs. Here µ1 = 0.08 and µ2 = 0.50.
4 Discussion
We have considered an unmagnetized four-component
plasma system consisting of hot electrons and positrons
following the q-nonextensive distribution, mobile cold
electrons, and immobile positive ions and investigated the
cylindrical and spherical EASWs. By using the reductive perturbation method we have derived the modified
Burgers equation and numerically analyzed that modified
Burgers equation. The results that have been found from
our present investigation can be pinpointed as follows:
(i) The shock waves are formed for above and below
the critical value (i.e., when µ1 > µc and µ1 < µc ) but do
not form for µ1 ' µc .
(ii) It is observed that the time evolution of the cylindrical and spherical EASWs significantly differs from the
1D planar EASWs. It is found that as time (τ ) decreases,
the amplitude of the cylindrical and spherical EASWs increases (shown in Figs. 1–4.)
(iii) The amplitude of positive shock structures decreases abruptly with the increase of relative temperature ratio σ and nonextensive parameter q as displayed
in Fig. 5. On the other hand, the amplitude of positive
shock structures increases gradually with the increase of
relative number density ratios µ2 and µ1 as depicted in
Fig. 6.
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