CHINESE JOURNAL OF PHYSICS
VOL. 47, NO. 5
OCTOBER 2009
Roles of Adiabaticity and Dynamics of Electrons and Ions
on Dust-Acoustic K-dV Solitons
A. A. Mamun and M. S. Rahman
Department of Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh
(Received January 10, 2009)
A theoretical investigation has been made of the roles of the adiabaticity and dynamics
of the electrons and ions on the dust-acoustic Korteweg-de Vries (K-dV) solitons that are
found to exist in an adiabatic hot dusty plasma (containing non-inertial adiabatic electron
and ion fluids, and a negatively charged inertial adiabatic dust fluid). The basic properties
of dust-acoustic (DA) K-dV solitons, which exist in such an adiabatic hot dusty plasma, are
explicitly examined by the reductive perturbation method. Comparing the basic properties
(speed, amplitude, and width) of the DA K-dV solitons observed in such a dusty plasma
with those observed in a dusty plasma containing isothermal electron and ion fluids and an
adiabatic dust fluid, it is found that the adiabatic effect of the inertia-less electron and ion
fluids may significantly modify the basic properties of the DA K-dV solitons, and that on
the basic properties of the DA solitary waves, the adiabatic effect of electron and ion fluids
is much more significant than that of the dust fluid.
PACS numbers: 52.35.Fp, 52.35.Mw, 52.35.Sb
The dynamics of charged dust, which is ubiquitous in space, viz. mesosphere,
cometary tails, planetary rings, planetary megnetospheres, interplanetary space, interstellar
media, etc. [1–4], and laboratory [1, 5, 6] plasmas, has attracted a great deal of attention
aimed at understanding the electrostatic density perturbations and potential structures
that are observed in space environments and laboratory devices.
It has been shown both theoretically [7] and experimentally [8] that in an unmagnetized dusty plasma the dynamics of charged dust introduces a new eigenmode, namely
dust-acoustic (DA) waves [7, 8]. Mamun et al. [9] considered a two-component unmagnetized dusty plasma system consisting of a negatively charged cold dust fluid and an
inertia-less isothermal ion fluid, and investigated the DA solitary waves in such a dusty
plasma. The work of Mamun et al. [9] is valid only when a complete depletion of electrons
onto the dust grain surface is possible. A number of theoretical investigations [10–13] have
been made of the DA solitary waves in order to generalize the work of Mamun et al. [9] by
assuming a three-component unmagnetized dusty plasma consisting of a negatively charged
cold dust fluid and inertia-less isothermal electron and ion fluids. These works are valid
only for a cold dust and isothermal electrons and ions. Recently, the effects of the dust fluid
temperature on the DA solitary waves have been investigated by a number of authors [14–
16]. Mendonza-Briceño et al. [14] assumed a two-component dusty plasma containing the
adiabatic dust fluid and non-adiabatic ions following the non-thermal distribution of Cairns
et al. [17], and studied the effect of the dust fluid temperature on the DA solitary waves
http://PSROC.phys.ntu.edu.tw/cjp
654
c 2009 THE PHYSICAL SOCIETY
OF THE REPUBLIC OF CHINA
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A. A. MAMUN AND M. S. RAHMAN
655
by the pseudo-potential approach [18]. Gill et al. [15] assumed a dusty plasma containing
the adiabatic dust fluid and non-adiabatic ions following the bi-Maxwellian distribution of
Nishihara and Tajiri [19], and studied the effect of the dust fluid temperature on the DA
solitary waves by the pseudo-potential approach [18]. Sayed and Mamun [16] assumed a
dusty plasma containing the adiabatic dust fluid and non-adiabatic (isothermal) inertialess electron and ion fluid, and studied the effect of the dust fluid temperature on the DA
solitary waves by the reductive perturbation method [20]. It is obvious that all these investigations [14–16] are concerned with different dusty plasma models which are not consistent
(appropriate). The inconsistency of all these dusty plasma models arises from the consideration of one component (dust) being adiabatic, and other components (electrons or ions
or both) being non-adiabatic.
Therefore, in the present work a consistent dusty plasma model, which assumes a
dusty plasma containing non-inertial adiabatic electron and ion fluids, and a negatively
charged inertial adiabatic dust fluid, has been considered in order to perform a proper
investigation of the basic properties of small amplitude DA K-dV solitons by the reductive
perturbation method [20].
The dynamics of the DA waves in one dimensional form in such an adiabatic hot
dusty plasma is governed by
∂
∂ns
+
(ns us ) = 0,
∂t
∂x
∂ps
∂ps
∂us
+ us
+ γps
= 0,
∂t
∂x
∂x
∂pe
∂Ψ
= ne α
,
∂x
∂x
∂pi
∂Ψ
= −ni
,
∂x
∂x
∂ud
∂ud
∂Ψ
σ ∂pd
+ ud
=
−
,
∂t
∂x
∂x
nd ∂x
∂2Ψ
= µe n e − µi n i + n d ,
∂x2
(1)
(2)
(3)
(4)
(5)
(6)
where ns is the number density of species s (with s = e for the electron fluid, s = i for the
ion fluid, and s = d for the dust fluid) normalized by its equilibrium value ns0 , us is the
fluid speed normalized by cd = (Zd kB Ti0 /md )1/2 , Ψ is the wave potential normalized by
kB Ti0 /e, ps is the fluid thermal pressure normalized by ns0 kB Ts0 , γ is the adiabatic index,
α = Ti0 /Te0 , σ = Td0 /Zd Ti0 , µ = ne0 /ni0 , ns0 is the equilibrium fluid number density, Ts0
is the equilibrium fluid temperature, Zd is the number of electrons residing on a dust grain
surface, md is the dust particle mass, kB is the Boltzmann constant, −e is the electronic
charge, µe = µ/(1 − µ), and µi = 1/(1 − µ). The time and space variables t and x are
−1
normalized by ωpd
= (md /4πe2 Zd2 nd0 )1/2 and λD = (kB Ti /4πe2 Zd nd0 )1/2 , respectively.
It is important to mention here that for an isothermal process γ = 1 and ps = ns with
constant Ts (i.e., Ts = Ts0 ), and hence (1) and (2) are identical. It is also important to note
that for isothermal processes, (3) and (4) reduce to ne = exp(αΨ) and ni = exp(−Ψ), which
ROLES OF ADIABATICITY . . .
656
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were used by Mendonza-Briceño et al. [14], Gill et al. [15], and Sayed and Mamun [16]. To
consider an adiabatic hot dusty plasma, one cannot use γ = 1 and ps = ns with constant
Ts . Therefore, in the present work (1D problem) γe = γi = γd = γ and Ts 6= constant are
used to study small amplitude DA solitary waves in an adiabatic hot dusty plasma by the
reductive perturbation method [20].
It is also important to note here that Mamun [21] has studied the adiabatic effects
of inertialess electrons and inertial ions on dust-ion-acoustic (DIA) solitary waves by the
pseudo-potential approach, and Mamun and Jahan [22] have investigated the effects of
electron dynamics and adiabaticity of inertialess electrons and inertial ions on DIA solitary
waves by the reductive perturbation method. These earlier works [21, 22] on nonlinear DIA
waves [23], where the inertia is provided by the ion mass and the restoring force comes
from the pressure of inertialess electrons while the equilibrium charge neutrality condition
is maintained by the stationary charged dust, are very different from our present work on
the DA waves, where the dust particle mass provides the inertia and the pressures of the
inertia-less electrons and ions give rise to the restoring force.
We investigate the basic features of small amplitude DA solitary waves by the reductive perturbation technique and the stretched coordinates [20] ζ = 1/2 (x − Vp t) and
τ = 3/2 t, where is a smallness parameter measuring the weakness of the dispersion, and
Vp is the phase speed (ω/k) of the DA waves normalized by cd , i.e., Vp = ω/kcd ). We can
expand the variables ns , us , ps , and Ψ in power series of as
2 (2)
ns = 1 + n(1)
s + ns + · · ·,
(7)
2 (2)
us = 0 + u(1)
s + us + · · ·,
2 (2)
ps = 1 + p(1)
s + ps + · · ·,
(1)
2 (2)
(8)
Ψ = 0 + Ψ
+ Ψ
+ · · ·.
(9)
(10)
Now, expressing (1)–(6) in terms of ζ and τ and substituting (7)–(10) into them, one can
easily develop different sets of equations in various powers of . To the lowest order in one obtains
(1)
n(1)
s =
(1)
ps
αs Ψ(1)
us
=
=
,
Vp
γ
γ
(1)
(1)
p
ud
Ψ(1)
= d =
,
Vp
γ
σγ − Vp 2
1−µ
2
Vp = γ σ +
.
1 + αµ
(1)
nd =
(11)
(12)
(13)
Equation (13) is the linear dispersion relation for the DA waves propagating in a dusty
plasma under consideration. It implies that, for inertialess isothermal
electrons and ions
p
(γ = 1) and cold dust fluid (σ = 0), the phase speed (ω/k = cd (1 − µ)/(1 + αµ) ) is
exactly the same as obtained by Rao [7] and Mamun [11]. We note that 1 ≤ γ ≤ 3,
0 < α ≤ 1, and 0 ≤ µ < 1. Therefore, due to the adiabaticity of the electrons and ions, the
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A. A. MAMUN AND M. S. RAHMAN
657
1.2
1
Vp 1
0.8
0.6
1
0.8
0.6
Α
1.5
0.4
2
0.2
2.5
Γ
3
FIG. 1: The variation of the phase speed Vp of the DA waves with γ and α for σ = 0.0001 and
µ = 0.5.
1.5
Vp
0.1
1
0.5
0.075
0.05 Σ
0.2
0.4
0.025
0.6
Μ
0.8
0
FIG. 2: The variation of the phase speed Vp of the DA waves with µ and σ for α = 0.5 and γ = 3.
phase speed of the DA waves can be increased significantly. Similarly, to the next order
in one gets another set of equations which, after using (11)–(13), can be reduced to a
well-known Korteweg-de Vries (K-dV) equation:
∂Ψ(1)
∂Ψ(1)
∂ 3 Ψ(1)
= 0,
+ Ad Ψ(1)
+ Bd
∂τ
∂ζ
∂ζ 3
(14)
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-0.12
-0.14
Ym -0.16
-0.18
-0.2
3
2.5
2 Γ
0.2
0.4
1.5
0.6
Α
0.8
11
FIG. 3: The variation of the amplitude Ψm of the DA solitary waves with α and γ for σ = 0.0001
and µ = 0.5.
0
Ym -0.1
0.1
-0.2
0.075
0.05 Σ
0.2
0.4
0.025
0.6
Μ
0.8
10
FIG. 4: The variation of the amplitude Ψm of the DA solitary waves with µ and σ for α = 0.5 and
γ = 3.
where the coefficients Ad and Bd are given by
1
2
2
Ad = −Bd Γ(µe α − µi ) + 6 (3Vµ + Γσ ) ,
Vµ
Bd =
Vµ4
,
2Vp
in which Vµ2 = γ(1 − µ)/(1 + αµ), Γ = (2 − γ)/γ 2 , and Γσ = σγ(1 + γ).
(15)
(16)
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A. A. MAMUN AND M. S. RAHMAN
D
5
4
3
2
1
659
1
0.8
0.6
Α
1.5
0.4
2
Γ
0.2
2.5
3
FIG. 5: The variation of the width ∆ of the DA solitary waves with γ and α for σ = 0.0001 and
µ = 0.5.
Now, transforming the independent variables ζ and τ to ξ = ζ − U0 τ 0 and τ = τ 0
(where U0 is a constant velocity normalized by cd ) and imposing the appropriate boundary
conditions (viz. Ψ(1) → 0, ∂Ψ(1) /∂ξ → 0, ∂ 2 Ψ(1) /∂ξ 2 → 0 at ξ → ±∞), one can express
the stationary solution of the K-dV equation (14) as
Ψ(1) = Ψm sech2 (ξ/∆),
(17)
where the amplitude Ψm (normalized by kB Ti0 /e) and the width ∆ (normalized by λD ) are
given by
3U0
,
Ψm =
Ad
p
∆ = 4Bd /U0 .
(18)
(19)
It is obvious from (17) and (18) that the DA solitary waves will be associated with positive
(negative) potential when Ad > 0 (Ad < 0). We note that for isothermal electrons and ions
(γ = 1) and cold dust fluid (σ = 0) we can express Ad as Ad = −[Vp /(1 − µ)2 ][1 + (3 +
αµ)αµ + µ(1 + α2 )/2], which is always negative. This means that for isothermal electrons
and ions (γ = 1) and cold dust fluid (σ = 0) DA solitary waves exist only with negative
potential. This completely agrees with Mamun [11].
To include the effects of the adiabaticity of the electrons and ions on the polarity
of the DA solitary wave potential, we numerically analyze Ad and find that Ad is always
negative. This means that the DA solitary waves are associated only with the negative
potential, and that the effects of the adiabaticity of the electrons and ions do not have any
role in changing the polarity of the solitary potential. However, these can have a significant
role in modifying the other basic properties (viz. speed, amplitude, and width) of the DA
solitary waves. These are displayed in Figures 1–6.
660
ROLES OF ADIABATICITY . . .
8
D 6
4
2
0
0.2
VOL. 47
0.1
0.075
0.05 Σ
0.4
0.025
0.6
Μ
0.8
0
FIG. 6: The variation of the width ∆ of the DA solitary waves with µ and σ for α = 0.5 and γ = 3.
A consistent and realistic plasma system containing inertia-less adiabatic electrons
and ions and negatively charged mobile dust is considered, in order to perform a proper
investigation of the basic properties of small amplitude DA K-dV solitons by the RP
method [20]. It is found that the effects of the adiabaticity of the electrons and ions
significantly modify the basic properties (speed, amplitude, and width) of the DA K-dV
solitons. It is also found that, due to the effect of the adiabatic electrons and ions, one cannot have negative DA solitary waves for any possible set of plasma parameters [1 ≤ γ ≤ 3,
0 < α ≤ 1, and 0 ≤ µ < 1]. It may be noted that for γ = 1 and σ = 0 the basic features
of the DA solitary waves found in the present investigation completely agree with Rao [7]
and Mamun [11].
The ranges of different plasma parameters used in this investigation are very wide
(α = 0.1–0.9 and σ = 0.1–0.9) and are relevant to both space [1, 3, 4] and laboratory [1,
5, 6] plasmas. Thus, the results of the present investigation should help us to explain the
basic features of localized electro-acoustic perturbations propagating in space [1, 3, 4] and
laboratory [1, 5, 6] dusty plasmas.
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