Complex Korteweg-de Vries equation and Nonlinear dust-acoustic waves in a magnetoplasma with a pair of trapped ions

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by  A. P. Misra

Complex Korteweg-de Vries equation and Nonlinear dust-acoustic waves in a
magnetoplasma with a pair of trapped ions
A. P. Misra1,∗
1Department of Mathematics, Siksha Bhavana, Visva-Bharati University, Santiniketan-731 235, West Bengal, India
(Dated: 5 January 2015)
Thenonlinearpropagationofdust-acoustic(DA)wavesinamagnetizeddustyplasmawithapair
oftrappedionsisinvestigated. Startingfromasetofhydrodynamicequationsformassivedustfluids
as well as kinetic Vlasov equations for ions, and applying the reductive perturbation technique, a
Korteweg-de Vries (KdV)-like equation with a complex coefficient of nonlinearity is derived, which
governs the evolution of small-amplitude DA waves in plasmas. The complex coefficient arises due
5 to vortex-like distributions of both positive and negative ions. An analytical as well as numerical
1 solutionoftheKdVequationareobtainedandanalyzedwiththeeffectsofexternalmagneticfield,
0 the dust pressure as well as different mass and temperatures of positive and negative ions.
2

n
a I. INTRODUCTION nonlinearity. A stationary as well as numerical solutions
J oftheKdVequationareobtainedandanalyzedwiththe

5 Recently, there has been a renewed interest in inves- effects of external magnetic field, the dust pressure as
well as different mass and temperatures of ions.
tigating electrostatic disturbances in pair-plasmas and,
]
h in particular, plasmas with a pair of ions [1–7]. How-
p ever, nonthermal pair plasmas may frequently occur not
- only in semiconductors in the form of electron and ion II. BASIC EQUATIONS
m
holes, but also in many astrophysical environments, e.g.,
s pulsars, magnetars, as well as in the early universe, ac- Weconsiderthenonlinearpropagationofdust-acoustic
a
l tive galactic nuclei and supernova remnants in the form (DA)solitarywavesinamagnetizeddustyplasmawhich
p of electrons and positrons [8–11]. On the other hand, consists of positively or negatively charged mobile dusts
s. a number of experiments have been conducted to cre- and a pair of trapped ions with vortex-like distributions.
c ate pair-ion plasmas using fullerene as ion source [12]. The dust particles are assumed to have equal mass and
i
s Furthermore, it has been observed that the dust parti- constant charge. The collisions of all particles are also
y cles injected into a pair-ion plasma (e.g., K+/SF− plas- neglected compared to the dust plasma period. Fur-
h 6
mas) can become positively charged when the number thermore, in dusty pair-ion plasmas the ratio of electric
p
density of negative ions greatly exceeds that of electrons charge to mass of dust particles remains much smaller
[
(n (cid:38)500n ) [13, 14]. These pose some possibilities to thanthoseofpositiveandnegativeions. Wealsoassume
n0 e0
1 investigatecollectivebehaviorsaswellastheformationof that the size of the dust grains is small compared to the
v nonlinear coherent structures in pair-ion plasmas under average interparticle distance. The static magnetic field
6
6 controlledconditions. Theformationofphasespaceholes is considered along the z-axis, i.e., B = B0zˆ. While the
8 in pure pair-ion plasmas [5] as well as ion holes in dusty dynamics of massive charged dusts in the(cid:112)propagation of
0 pair-ionplasmas[6]inthepropagationoflargeamplitude DAwaves(vtd (cid:28)vp (cid:28)vp,n,wherevtj (= kBTj/mj)is
0 electrostatic waves have been investigated in the recent thethermalvelocityofj-thspeciesparticlesandvp isthe
. past in which ions have been treated as trapped in self- phase velocity of the wave) is described by a set of fluid
1
0 createdlocalizedelectrostaticpotentialsasprescribedby equations (1) and (2), the dynamics of singly charged
5 Schamel [15]. positive and negative ions are described by the Vlasov
1 In this paper we present a theoretical study on the equations (3).
:
v formation and the dynamics of small-amplitude solitary
∂n
i structures in a dusty plasma composed of charged dust d +∇·(n v )=0, (1)
X ∂t d d
particlesandapairofionswithoutelectrons. Inourthe-
r oretical model the massive charged dusts are described
a
by a set of fluid equations, while the dynamics of both ∂v q ∇P
d +(v ·∇)v = d (E+v ×B )− , (2)
positiveandnegativeionsaregovernedbykineticVlasov ∂t d d m d 0 m n
d d d
equations. Using the reductive perturbation technique
we show that the evolution of small-amplitude electro-
∂f q ∂f
static waves can be described by a Korteweg-de Vries j +v·∇f − j ∇φ· j. (3)
(KdV)-like equation with a complex coefficient of the ∂t j mj ∂v
The system of equations is then closed by the Poisson
equation
∗ [email protected];[email protected] ∇·E=4πe(n −n +αZ n ). (4)
p n d d
2
In Eqs. (1)-(4), q , n , v , f and m respectively, de- are constant of motion of the Vlasov Eq. (8), are cho-
j j j j j
note the charge, number density (with its equilibrium sen [15] for the excitation of localized solitary waves so
value n ), velocity, velocity distribution function, and that (i) they are continuous, and both the free particle
j0
mass of j-species particles. Also, q = αz e with α = ± distributions are Maxwellian distribution where φ → 0
d d
denoting for positively/negatively charged dusts and z at |ξ| → ±∞ and trapped particles are absent, (ii) both
d
thechargestate. Also,E=−∇φistheelectricfieldwith trapped particle distributions are Maxwellian (with also
φ denoting the electrostatic potential and P is the dust negative temperatures). Thus, f (for free and trapped
j
thermalpressuregivenbytheadiabaticequationofstate particles)are(withasuitablechoiceofthenormalization
P/P =(n /n )γ. Here, γ =5/3 is the adiabatic index constants) [15–18] for positive ions
0 d d0
for three-dimensional configuration and P =n k T is
tmhaenenquciolinbsrtiaunmtdaunsdtTpretshsuertehweritmhokdByndaemno0itcintgemtdhp0eeBrBaotldutrze- fpf(v)= √12π exp(cid:20)−12(cid:0)v2+2φ(cid:1)(cid:21), |v|>(cid:112)−2φ, (12)
j
of j-species particles. Furthermore, the ion densities are
given by f (v)= √1 exp(cid:104)−σp (cid:0)v2+2φ(cid:1)(cid:105), |v|≤(cid:112)−2φ, (13)
pt 2π 2
(cid:90) ∞
n = f dv. (5)
j j and for negative ions
−∞
(cid:114) (cid:20) (cid:18) (cid:19)(cid:21)
mσ mσ 2φ (cid:112)
In what follows, we recast Eqs. (1)-(4) in terms f (v)= exp − v2− , |v|> 2φ/m,
nf 2π 2 m
of dimensionless variables. To this end the physical
(14)
quantities are normalized according to n → n /n ,
j j j0
(v, vd) → (v, vd)/cd, φ → eφ/kBTp with e denot- (cid:114)mσ (cid:20) 1 (cid:18) 2φ(cid:19)(cid:21) (cid:112)
ing the elementary charge, f → f v /n , where c = f (v)= exp − mσσ v2− , |v|≤ 2φ/m,
(cid:112) j j tp j0 d nt 2π 2 n m
z k T /m = ω λ is the dust-acoustic speed with
d B p d pd D (15)
(cid:112) (cid:112)
ωpd = 4πnd0zd2e2/mdandλD = kBTp/4πnd0zde2de- wherem(=mn/mp (cid:38)1)isthemassratio,σ(=Tp/Tn (cid:38)
noting, respectively, the dust plasma frequency and the 1) is the temperature ratio and σ , for j =p,n, measure
j
plasma Debye length. The space and time variables are theinverseofthetrappedpositiveandnegativeiontem-
normalizedbyλD andωp−d1 respectively. Thus,fromEqs. peratures which may be negative (σj < 0) correspond-
(1)-(5) we have following set of normalized equations. ing to a depression in the trapped particle distribution.
The case of σ → 0 represents the plateau (constant or
∂n j
∂td +∇·(ndvd)=0, (6) dfliastt-rtiobpuptieodn) oafndionσsj.→Ne1xtc,orirnetsepgorantdisngtoththeepBarotlitczlme adnisn-
tributionfunctions(12)-(15)overthevelocityspace, i.e.,
dv 5 using Eq. (10) we obtain the number densities n for
d +α∇φ=αω v ×zˆ− σ n−1/3∇n , (7) j
dt c d 3 d d d positive and negative ions as
1 (cid:40)e−σpφ erf(cid:0)(cid:112)−σpφ(cid:1), σp ≥0
δ∂∂ftj +v·∇fj −ζjmmp∇φ· ∂∂fvj =0, (8) np(φ)=I(−φ)+(cid:112)|σp| √2πW (cid:0)(cid:112)σpφ(cid:1), σp <0,
j (16)
√
∇2φ=µnnn−µpnp−αnd, (9) nn(φ)=I(σφ)+ (cid:112)|1σn|(cid:26)e√σ2πσWnφ(cid:0)e√rf−(cid:0)σσσnσφn(cid:1)φ,(cid:1), σσnn <≥00,
(17)
(cid:90) ∞ √
n = f dv, (10) where I(x) = exp(x)[1−erf( x)]. The error and Daw-
j j
−∞ sonfunctionserf(x)andW(x)are, respectively, givenby
where d/dt ≡ ∂/∂t + vd · ∇, α = ±1 for posi- erf(x)= √2 (cid:90) xe−t2dt, W(x)=e−x2(cid:90) xet2dt. (18)
tively/negatively charged dusts, ωc = |qd|B0/mdωpd is π 0 0
the dust-cyclotron frequency normalized by the dust
(cid:112) In the small amplitude limit φ (cid:28) 1, so that σφ (cid:28) 1, we
plasma frequency, σ ≡ T /T z , δ = z m /m ,
d d p d d p d
obtain from Eqs. (16) and (17) the following expressions
ζ = ±1 for positive/negative ions and µ = n /Z n
j j j0 d 0
for the number densities [15–18]
are the density ratios (j =p,n) which satisfy the follow-
ing charge neutrality condition at equilibrium: 4(1−σ ) 1
n ≈1−φ− √ p (−φ)3/2+ φ2, (19)
p 3 π 2
µ +α=µ . (11)
p n
We neglect the ion inertial effects compared to the 4(1−σ ) 1
n ≈1+(σφ)− √ n (σφ)3/2+ (σφ)2. (20)
charged dusts, i.e., δ → 0 in Eq. (8). The distribu- n 3 π 2
tion functions f for positive and negative ions, which
j
3
FIG. 1. Profiles of |Φ| given by Eq. (33) are shown with respect to ξ (and with σ ∼ 1) for different values of the plasma
parameters as in the figure. The fixed parameter values for the subplots (a) to (d), respectively, are (a) α=1, µ =0.2, l =
p z
0.1, ω =0.5, σ =0.06 and u =0.1, (b) α=1, µ =0.2, l =0.1, σ =σ =0.3, σ =0.06 and u =0.1, (c) α=1, µ =
c d 0 p z p n d 0 p
0.2, l =0.1, σ =σ =0.3, ω =0.5, σ =0.06 and u =0.1, and (d) α=1, µ =0.2, σ =σ =0.3, ω =0.5, σ =0.06
z p n c d 0 p p n c d
and u =0.1.
0
FIG. 2. The space-time evolution of the soliton profile |Φ| [numerical solution of Eq. (30)] is shown at τ =100 [Subplots (a)
and (b)] and τ =160 [Subplots (c) and (d)]. While the plots (a) and (c) show the solton profiles with space, plots (b) and are
the corresponding contour plots. The parameter values are the same as for the dashed line in Fig. 1(d) and σ∼1.
III. EVOLUTION EQUATION where (cid:15) is a small parameter measuring the strength of
nonlinearity. The dependent variables are expanded as
In order to derive the evolution equation for the DA
waves,wetransformthespaceandtimevariablesaccord-
ing to [16]
ξ =(cid:15)1/4(l x+l y+l z−Mt), τ =(cid:15)3/4t, (21)
x y z
4
[16] (26), (27) and (28), and the coefficient of φ(2) vanishes
by Eq. (25). Thus, arranging the terms and using Eq.
n=1+(cid:15)n(1)+(cid:15)3/2n(2)+··· , (23) one obtains the following KdV-like equation
φ=(cid:15)φ(1)+(cid:15)3/2φ(2)+··· ,
(22) ∂Φ (cid:16) √ √ (cid:17)∂Φ ∂3Φ
vz =1+(cid:15)vz(1)+(cid:15)3/2vz(2)+··· , ∂τ + Ap −Φ+An Φ ∂ξ +B∂ξ3 =0, (29)
v =(cid:15)5/4v(1) +(cid:15)3/2v(2) +··· .
x,y x,y x,y where Φ ≡ φ(1). It follows that Eq. (29) has a complex
The anisotropy in Eq. (22) for the transverse velocity solution for Φ. Typically, for Φ ∼ rexp(iθ), where r is
components of dust fluids is introduced on the assump- a real√function√of ξ and τ, and θ is a constant, one can
tion that the dust gyromotion is a higher-order effect have −Φ=i Φ. Thus, Eq. (29) can be written as
than the motion parallel to the magnetic field. Next,
∂Φ √ ∂Φ ∂3Φ
we substitute Eqs. (21) and (22) into Eqs. (6)-(9) and +A Φ +B =0, (30)
equate different powers of (cid:15) successively. In the lowest ∂τ ∂ξ ∂ξ3
order ((cid:15)5/4), we obtain the following first-order quanti-
wherethecoefficientsofnonlinearity(A≡A +iA )and
ties n p
dispersion (B) are given by
l
n(1) =αMz vz(1) =(µp+σµn)φ(1), (23) α (1−σ )µ (cid:18)T (cid:19)3/2
A = √ j j p , (31)
j πM(µ +σµ ) T
p n j
l (cid:18)∂φ(1) 5 ∂n(1)(cid:19)
v(1) =∓ y,x + ασ , (24)
x,y ωc ∂ξ 3 d ∂ξ B = lz2 (cid:20)1+ M4(1−lz2)(µ +σµ )2(cid:21). (32)
2M ω2l4 p n
and the dispersion relation for the nonlinear wave speed c z
given by
The nonlinear coefficient A becomes complex due to
(cid:18)5 1 (cid:19)1/2 vortex-like distributions of two oppositely charged par-
M =l σ + . (25) ticles. In absence of one of them, A becomes real, and
z 3 d µ +σµ
p n one can then obtain solitary waves with positive or neg-
ative potential. A stationary soliton solution of Eq. (30)
ReplacingM byω/k,onecanobtainthesamedispersion
can easily be obtained with its absolute value as (For
relationafterFourieranalyzingthelinearizedbasicequa-
details see Appendix A)
tions (6)-(9), i.e., assuming the perturbations as oscilla-
tions with the wave frequency ω and the wave number |Φ|=Φ sech4[(ξ−u τ)/W], (33)
k. We find that the phase speed M (normalized by the 0 0
DIAspeedcd)canbelargerorsmallerthantheunityde- where u is a constant, and Φ =(15u /8|A|)2 and W =
0 0 0
pendingonthechoiceoftheparametervalues. Thevalue (cid:112)
16B/u are the amplitude and width of the soliton
of M increases with increasing values of both l and σ . 0
z d respectively.
However, its values can slowly decrease with increasing
valuesofthedensityratiosµ aswellasthetemperature
j
ratio σ. From Eq. (6), collecting the coefficients of (cid:15)7/4
IV. RESULTS AND DISCUSSION
we obtain
M∂n(2) = ∂n(1) + (cid:88) l ∂vj(2). (26) ferWenet npulamsmeraicaplalyramaneatleyrzseatsheshosowluntiionnF(i3g3.) 1w.ithSidnicfe-
∂ξ ∂τ j ∂ξ
j=x,y,z σj (j =p,n)representsthereciprocaltemperatureofthe
trapped positive and negative ions, and can be allowed
Similarly, equating the coefficients of (cid:15)3/2 from the x- from their negative to positive values corresponding to
and y-components of Eq. (7), and the coefficients of (cid:15)7/4 different trapped particle distributions, we consider neg-
from the z-component of Eq. (7) we successively obtain ative, zero as well as positive values of σ .
j
From Fig. 1(a), it is seen that as σ increases from
αv(2) =±M ∂vy(1,x) =(cid:20)Mlx,y + 5σ (µ +σµ )∂2φ(1)(cid:21), σj = 0 (corresponding to a constant orjflat-topped dis-
x,y ω ∂ξ ω2 3 d p n ∂ξ2 tribution of ions) to σ ∼1 (corresponding to the Boltz-
c c j
(27) mann distributions of ions), both the amplitude and
width of the soliton increase (See the solid and dashed
∂v(2) ∂v(1) (cid:18) ∂φ(2) 5 ∂n(2)(cid:19) lines). Note here that the values of σ >1, for which the
M z = z +l α + σ . (28) j
∂ξ ∂τ z ∂ξ 3 d ∂ξ influence of the trapped ions are inverted, may be physi-
callyunrealisticasthosecorrespondtoamoresteepened
From the coefficients of (cid:15)3/2 of Eq. (9), we obtain an wave which can become unstable due to more peaked
equation in which n(2) is eliminated by the use of Eqs. bump of the ion distributions. However, as the absolute
5
valueofAstartsincreasingforσ <0,whichcorresponds nonlinearity. Such complex coefficient appears due to
j
toadepressioninthetrappedparticledistribution,both vortex-like distributions of both the ion species. The
the amplitude and width of the soliton are reduced (See KdV equation is solved both analytically and numeri-
the dotted line). The same can further be enhanced for cally. The properties of the absolute value of Φ are only
values of σ satisfying σ σ < 0 (See the dash-dotted exhibitedgraphically. Itisshownthatwhiletheexternal
j p n
line). magnetic field only influences the width of the soliton,
Figure 1(b) shows the soliton profile with the influ- the trapped ion temperatures, the thermal pressures of
ence of the external magnetic field. Since ω contributes ionsanddusts,therelativeconcentrationofpositiveions
c
only to the dispersive coefficient B of Eq. (30), the ef- aswellastheobliquenessofpropagationhavesignificant
fect of the magnetic field with increasing its intensity is effects on both the amplitude and width of the solitons.
to reduce the width (without changing the amplitude) We stress that other solutions [19–22] than those pre-
of the soliton. Thus, the external magnetic field makes sentedhereofthecomplexKdVequationcouldofinterest
the solitary structure more spiky. However, for stronger but beyond the scope of the present work. To conclude,
magnetic fields with ω (cid:29) 1, the width remains almost thepresentresultsshouldbeusefulinunderstandingthe
c
unaltered as in this case B ∼l2/2M. nonlinear features of electrostatic localized disturbances
z
The thermal effects of charged dusts are shown in Fig. in laboratory and space plasmas.
1(c). It is found that the effect of the dust thermal pres-
sure σ is to enhance both the amplitude and width of
d
the soliton. The enhancement is due to the fact that as ACKNOWLEDGMENTS
σ increases, the values of |A| (B) decrease (increase),
d
andhencetheincreaseinboththeamplitudeandwidth. This work was partially supported by the SAP-DRS
However, an opposite trend occurs by the effects of the (Phase-II),UGC,NewDelhi,throughsanctionletterNo.
positive to negative ion temperature ratio σ (not shown F.510/4/DRS/2009 (SAP-I) dated 13 Oct., 2009, and
in the figure). Typically, it reduces both the soliton am- by the Visva-Bharati University, Santiniketan-731 235,
plitudeandwidthsignificantlywithasmallincrementof through Memo No. REG/Notice/156 dated 07 January,
its value. 2014. APM thanks Dr. M. M. Panja of Department
Figure 1(d) exhibits the effects of the obliqueness of of Mathematics, Visva-Bharati, Santiniketan, India for
propagation l and the relative (to dusts) concentration some useful discussions.
z
of positive ions µ . We find that both the amplitude
p
and width of the soliton are greatly enhanced by a small
increment of l [Since A (B) is inversely (directly) pro- Appendix A: Stationary solution of the KdV-like
z j
portional to l ]. However, as the positive ion concentra- equation
z
tion increases, the amplitude gets reduced but the width
increases. Equation (30) is recast as
Next, we numerically solve Eq. (30) by the Runge-
∂Φ √ ∂Φ ∂3Φ
KuttaschemewithaninitialconditionoftheformΦ(ξ)= +A Φ +B =0. (A1)
0.05 sech4(ξ/10)exp(−iξ/15) and time step dτ = 0.001. ∂τ ∂ξ ∂ξ3
The development of the wave form |Φ| after a finite in- Next, weapplythetransformationη =ξ−u τ toobtain
0
terval of time is shown in Fig. 2. The parameter values from Eq. (A1)
are considered as the same as for the dashed line in Fig.
(cid:18) (cid:19)
1(d). It is seen that the leading part of the initial wave d BΦ¨ −u Φ+ 2AΦ3/2 =0, (A2)
steepens due to positive nonlinearity. As the time goes dη 0 3
on the pulse separates into solitons and a residue due to
where the dot denotes differentiation with respect to η.
the wave dispersion [See the subplots (a) and (b)]. It is
Integrating Eq. (A2) with respect to η and using the
found that once the solitons are formed and separated,
boundary conditions Φ, Φ¨ →0 as ξ →±∞ we get
they propagate in the forward direction without chang-
ingtheirshapeduetothenicebalanceofthenonlinearity 2
BΦ¨ −u Φ+ AΦ3/2 =0. (A3)
and dispersion [See the subplots (c) and (d)]. 0 3
Multiplying Eq. (A3) by 2Φ˙ and integrating once with
respect to η, we obtain
V. CONCLUSION
8
BΦ˙2−u Φ2+ AΦ5/2 =0, (A4)
We have investigated the nonlinear propagation of 0 15
dust-acoustic waves in a magnetized plasma which con- where we have used the boundary conditions Φ, Φ˙ →0.
sists of warm positively charged dusts and a pair of free,
From Eq. (A4) we have
as well as, trapped ions. We have shown that the evo-
lution of small-amplitude DA waves can be described by (cid:114)u 8A √
Φ˙ =±Φ 0 − Φ, (A5)
a KdV-type equation with a complex coefficient of the
B 15B
6
(cid:90) dΦ (cid:90)
Thus, we obtain a soliton solution of Eq. (30) as
or, =± dη, (A6)
(cid:113) √
Φ u /B−(8A/15B) Φ
0 √
(cid:18) a (cid:19) b√
1−tanh2 η = Φ, (A9)
which gives (a=u /B and b=8A/15B)
0 4 a
(cid:115) √
4 a−b Φ
− √ tanh−1 =±η, (A7)
a a
(cid:18)15u (cid:19)2 (cid:18)(cid:114) u (cid:19)
or, Φ= 0 sech4 0 η . (A10)
8A 16B
(cid:115) √ √
(cid:18) (cid:19)
a−b Φ a
or, =∓tanh η . (A8)
a 4
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