Modulation of dust acoustic waves with non-adiabatic dust charge fluctuations

Physics Letters A 320 (2003) 226–233 www.elsevier.com/locate/pla

Modulation of dust acoustic waves with non-adiabatic dust charge fluctuations Ju-Kui Xue College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, People’s Republic of China Received 2 September 2003; received in revised form 10 November 2003; accepted 11 November 2003 Communicated by F. Porcelli

Abstract The modulational instability of dust acoustic waves in a dusty plasma with non-adiabatic dust charge fluctuation is studied. Using the perturbation method, a modified nonlinear Schrödinger equation containing a damping term that comes from the effect of dust charge variation is derived. It is found that the modulational instability of the wave packet and the propagation characters of the envelope solitary waves are modified significantly by the non-adiabatic dust charge fluctuation.  2003 Elsevier B.V. All rights reserved. PACS: 52.35.Sb; 52.35.Mw; 52.25.Wz Keywords: Modulational instability; Dust-acoustic waves; Non-adiabatic dust charge fluctuations

1. Introduction It is well known that the presence of the charged dust grains in a plasma would modify the collective behavior of a plasma, as well as excite new modes [1–10]. In reality, the charge on the dust grain varies both with space and time due to electron and ion currents flowing into or out of the dust grain, as well as other processes such as photo-emission of electrons. Those cause the dust charge fluctuations. So, the dust charge becomes a new dynamic variable and its dynamic nature would be important for studying the behavior of dusty plasma. Under the assumption of adiabatic (τch /τd = 0) and non-adiabatic (τch /τd is small but finite) dust charge variation, where τch is the charging time scale and τd is the hydrodynamical time scale, recent nonlinear studies indicate [3–10] that the dust charge fluctuation would affect the wave mode properties and lead to some new aspects of dusty plasma. But up to now, there is no theoretical consideration about the modulational instability of dust acoustic waves in a dusty plasma with non-adiabatic dust charge fluctuation. The slow modulation of a monochromatic plane wave in plasmas can lead to the formation of an envelope soliton, which is described by the nonlinear Schrödinger equation (NLSE). During the recent years, the modulational instability of dust acoustic waves and the propagation of dust acoustic envelope solitary waves in dusty plasma have received a considerable attention [11–15] because of their E-mail address: [email protected] (J.-K. Xue). 0375-9601/$ – see front matter  2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2003.11.018

J.-K. Xue / Physics Letters A 320 (2003) 226–233

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relevance to observations of electromagnetic fluctuations in space and laboratory plasma. Space observations show [16–18] a clear signature of envelope solitary wave like structures. Hence, it is worthful to study the effects of dust charge variation on wave modulation properties. Therefore, in present Letter, the modulational instability of dust acoustic waves and the propagation of dust acoustic envelope solitary waves in a plasma with non-adiabatic dust charge fluctuation are investigated. By using the perturbation method, a modified NLSE with a damping term which represents the effect of non-adiabatic dust charge variation is deduced. The effects of non-adiabatic dust charge variation on the modulational instability of dust acoustic waves are discussed in detail.

2. Governing equations We consider the dust acoustic waves propagating in a collisionless plasma whose constituents are Bolzmanndistributed electrons, ions, and massive highly negatively charged warm dust grains. The non-adiabatic dust charge fluctuations are also considered. Then the dust dynamics can be described by the following one-dimensional set of continuity, momentum, and Poisson’s equations: ∂(nd ud ) ∂nd + = 0, ∂t ∂x ∂φ ∂ud ∂nd ∂ud + ud = −(q − 1) − 3σd nd , ∂t ∂x ∂x ∂x ∂ 2φ 1 φ δ −φ/σi e − e = − nd (q − 1), 1−δ ∂x 2 1 − δ where σd = Td /zd Te , δ = ni0 /ne0 , σi = Ti /Te . The variables are normalized as follows: t = ωpd t  , Here


ωpd =

x=

zd2 e2 nd0 4πmd

x , λD

ud =

1/2

ud , cd


,

λD =

4πTe zd nd0 e2

φ=

eφ  , Te

(ne , ni ) =

1/2


,

cd =

zd Te md

(ne , ni ) , zd nd0

(1) (2) (3)

nd =

nd . nd0

1/2

are the dust plasma frequency, dust Debye length, and dust acoustic velocity, respectively. The dust charge Q = −zd e + q, where q is the fluctuating dust charge, becomes q − 1, normalized in units of the equilibrium dust charge zd e. The normalized charge variable q is determined by the following normalized orbital motion limited charge current balance equation [6,10,19,20]
 τch ∂q ∂q τch + ud (Ie + Ii ), (4) = τd ∂t ∂x zd e where Ie and Ii are the electron and ion current, respectively. The normalized expressions for the electron and ion currents for spherical dust grains with radius a are 
  8Te 1/2 ne0 exp(φ) exp z(q − 1) , Ie = −πa 2 e (5) πme  

 z 8Ti 1/2 z − q , Ii = πa 2 e (6) ni0 exp(−φ/σi ) 1 + πmi σi σi −1 where z = zd e2 /aTe . τd = ωpd is the dust oscillation time scale, and



2 ωpi a (1 + z + σi ) τch = (2π)1/2 Vt hi

−1

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is the charging time scale. Here Vt hi is the ion thermal velocity. Eqs. (1)–(6) govern the dynamics of fully nonlinear dust acoustic waves in a warm dust plasma.

3. Derivation of the modified NLSE In order to investigate the modulation of nonlinear dust acoustic wave in the plasma, we employ the standard reductive perturbation technique [21] to obtain the NLSE. The independent variables are stretched as ξ = "(x −vg t) and τ = " 2 t, where " is a small parameter and vg is the group velocity of the wave. The dependent variables are expanded as nd = 1 +

∞ 

φ= q=

∞ 

"n

n=1 ∞ 

"n

n=1

(7)

∞ 

  u(n) l (ξ, τ ) exp i(kx − ωt)l ,

(8)

l=−∞ ∞ 

"n

n=1 ∞ 

  (n) nl (ξ, τ ) exp i(kx − ωt)l ,

l=−∞

n=1

ud =

∞ 

"n

l=−∞ ∞ 

  (n) φl (ξ, τ ) exp i(kx − ωt)l ,

(9)

  ql(n) (ξ, τ ) exp i(kx − ωt)l ,

(10)

l=−∞ ∗

(n) (n) (n) (n) where n(n) satisfy the reality condition A(n) and the asterisk denotes complex l , ul , φl , and ql −l = Al conjugate. Because in many experiments [7,8], it is seen that τch /τd ≈ 10−6 ∼ 10−5 , i.e., τch τd , so τch /τd is small and we can assume that τch /τd is proportional to " 2 . Thus we take

τch = µ" 2 , τd

(11)

where µ is a finite quantity of the order of unity. Hence, according to Eq. (4) and Eqs. (8)–(11), the non-adiabatic dust charge fluctuation expressed by the left-hand side of Eq. (4) would be small (∼ " 3 ) but finite. Substituting the expression (7)–(11) into Eqs. (1)–(6) and collecting the terms in the different powers of ", we √ can obtain each nth-order √ reduced equations. For the zeroth order, the equilibrium currents balance equation Ti /mi ni0 (1 + z/σi ) = Te /me ne0 exp(−z) and the equilibrium charge neutrality condition ni0 = ne0 + zd nd0 (1) are deduced. For the first-order (n = 1) equation with l = 1, we can obtain the first-order quantities in terms of φ1 as k2 (1) φ , ω2 − 3σd k 2 1 ωk φ (1) , u(1) 1 =− 2 ω − 3σd k 2 1 (1)

n1 = −

(1)

(12) (13)

(1)

q1 = −β1 φ1 .

(14)

The dispersion relation is also obtained as ω2 = 3σd k 2 +

k2 k2 +

1 1−δ

+

δ σi (1−δ)

+ β1

,

(15)

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where δ = ni0 /ne0 and β1 =

(z + σi )(1 + σi ) . zσi (1 + z + σi )

(16)

The variable β1 stand for the dust charge fluctuation effect. For the second-order (n = 2) reduced equation with l = 1, we get (1)

n(2) 1 = 2ki

∂φ1 k2 − 2 φ (2) , ∂ξ ω − 3σd k 2 1

(17)

(1)

u(2) 1 =i

ωk ω2 + 3σd k 2 ∂φ1 − 2 φ (2) , ω ∂ξ ω − 3σd k 2 1

q1(2) = −β1 φ1(2)

(18) (19)

and the compatibility condition ∂ω ω2 − (ω2 − 3σd k 2 )2 (20) = . ∂k ωk Similarly, the zeroth and second harmonic mode (n = 2) of the carrier wave with l = 0 and l = 2 is also obtained (1) in terms of φ1  (1) 2 (2) φ2 = A φ 1 , (21)   (2) (1) 2 n2 = B φ1 , (22)   3  (1) 2 ωk ω (2) φ1 , B− 2 u2 = (23) 2 2 k (ω − 3σd k )  (1) 2 (2) q2 = (−β1 A + β2 ) φ1 , (24) vg =

and  (1) 2 (2) φ 0 = C φ 1  ,  (1) 2   , n(2) 0 = D φ1    (1) 2 2ωk 3 φ  , u(2) = v D + g 0 1 2 2 2 (ω − 3σd k )   2 q (2) = (−β1 C + 2β2 )φ (1) , 0

1

(25) (26) (27) (28)

where [−zβ1 + (z + σi )/2σi ] − σi (z + σi )[1/2 − zβ1 + z2 β12 /2] , zσi (1 + z + σi ) 
  1 δ 1 1 (2ω2 + 3σd k 2 )k 2 β1 k 2 A= − − 2β − , − − β 1 2 6(ω2 − 3σd k 2 ) (ω2 − 3σd k 2 )2 6k 2 1 − δ σi2 (1 − δ) ω2 − 3σd k 2   k2 (2ω2 + 3σd k 2 )k 2 B =− , − 2A + β 1 2(ω2 − 3σd k 2 ) (ω2 − 3σd k 2 )2 1−δ C= 2 1 − δ + (3σd − vg )(1 + δ/σi + β1 (1 − δ)) β2 =

(29) (30) (31)

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k 2 (2ωvg k − ω2 − 3σd k 2 ) δ 1 2β1 k 2 + 2 + 2β2 + 2 − β , (32) 3σd − vg2 − − 1 1 − δ σi (1 − δ) ω − 3σd k 2 (ω2 − 3σd k 2 )2   1 δ δ 2β1 k 2 1 D=− (33) + 2 + 2 + β1 C − + 2β2 + 2 . 1 − δ σi (1 − δ) 1 − δ σi (1 − δ) ω − 3σd k 2

×

Finally, we obtain the l = 1 component of the third-order (n = 3) part of the reduced equations

(2) (2) 

(1) (2) ∂ vg n1 − u1 ∂n(1) (3) (3) (1) (2) (2) (1) (2) (1)  1 − iωn1 + iku1 = −ik n1 u0 + n−1 u2 + n0 u1 + n2 u−1 + , ∂τ ∂ξ

(34)

(1)

∂u1 (3) (3) − iωu(3) 1 − ikφ1 + i3σd kn1 ∂τ

(1) (2)  (1) (2) (1) (2) (2) (1) (2) (1) (2) (1) (1) (2)  = −ik u1 u0 + u−1 u2 + q0 φ1 − q2 φ−1 + 2φ2 q−1 + 3σd n1 n0 + n−1 n2

(2) (2)  ∂ vg u(2) 1 + φ1 − 3σd n1 , + ∂ξ 
1 δ (3) (3) (3) + φ − n1 + q1 − k2 + 1 − δ σi (1 − δ) 1



(1) (2) δ δ 1 1 1 (1) (2)  (1)  (1) 2 − 2 − 3 φ1 φ0 + φ−1 φ2 + = φ 1 φ 1  1 − δ σi (1 − δ) 2 1 − δ σi (1 − δ)


(1) (2) ∂φ1(1) ∂ (1) (2) (2) (1) (2) (1)  (2) − n1 q0 + n−1 q2 + n0 q1 + n2 q−1 − 2ikφ1 + , ∂ξ ∂ξ (3) (3) (1) (1)  (1) 2 −β1 φ1 − q1 = iωµq1 + β3 φ1 φ1  ,

(35)

(36) (37)

where


1 z β3 = (A + C) 1 + − 2β1 z + 3β2 z zσi (1 + z + σi ) σi 
  z − σi (σi + z) (A + C) 2 + − 2zβ1 + 3β2 z . σi

(38)

Substituting the above derived expressions of Eqs. (12)–(14), (17)–(19), and (21)–(28) into Eqs. (34)–(37) and (3) (3) (3) (3) (2) eliminating the term n1 , u1 , φ1 , q1 , and φ1 then we deduce a modified NLSE for dust acoustic wave in the plasma with dust charge fluctuation effect i

∂ 2φ ∂φ + P 2 + Q|φ|2 φ = −iΓ φ, ∂τ ∂ξ

(39)

where φ ≡ φ1(1) , and

  ω + 2vg k 3 3σd k 2 β1 (ω2 − 3σd k 2 )2 D − ωB − − Q=− (B + D) 2 2 2ω 2ωk 2
  

2 δ 1 ω2 − 3σd k 2 ωk 4 2 2 − k − ω − 3σ k 2β − (A + C) − 2 1 d 2 2 2ωk 1 − δ σi (1 − δ) (ω − 3σd k 2 )2

  

2 ω2 − 3σd k 2 1 1 δ 2 2 β + β + ω − 3σ k + − 4k , 2 d 3 2 1 − δ σi3 (1 − δ) 2ωk 2

(40)

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 σd vg k 2 3 ω2 − 3σd k 2 ωvg − 4σd k + , (41) 2 ωk ω 2 µβ1

Γ = 2 ω2 − 3σd k 2 . (42) 2k It is clear from Eqs. (11), (39), and (42) that the additional term −iΓ φ occurred in the modified NLSE (39) represents the non-adiabatic dust charge fluctuation effects. We can see from Eqs. (40) and (41) that the nonlinear term Q and the dispersion term P are also effected by the dust charge fluctuations because Q and P are functions of β1 , β2 , and β3 . Hence, the modulational instability of dust acoustic waves and wave propagation in the plasma will be modified. P =−

4. Modulational instability Now we analyze the modulational instability of dust acoustic waves described by the non-adiabatic dust charge fluctuation modified NLSE. Investigating the development of the small modulation δφ according to

τ      φ = φ0 + δφ(τ, ξ ) exp −i ∆(τ ) dτ − Γ τ , (43) 0

where ∆ is a nonlinear frequency shift. φ0 (real constant) is the amplitude of the pump carrier wave. Substituting Eq. (43) into Eq. (39) and collecting terms in the zeroth and first order (linearization), we obtain ∆(τ ) = −Q|φ0 |2 exp(−2Γ τ ), ∂ 2 δφ ∂δφ +P + Q|φ0 |2 exp(−2Γ τ )(δφ + δφ ∗ ) = 0, i ∂τ ∂ξ 2

(44) (45)

where δφ ∗ is the complex conjugate of δφ. Letting δφ = U + iV in Eq. (45), and separating the real and imaginary parts, one obtains the following two coupled equations: ∂ 2U ∂V = P 2 + 2Q|φ0 |2 exp(−2Γ τ )U, ∂τ ∂ξ ∂ 2V ∂U = −P 2 . ∂τ ∂ξ Assuming that the amplitude perturbation varies as 

 τ 

U = Re U0 exp i Kξ − τ

Ω(τ ) dτ 0



,

(46) (47)



τ

V = Re V0 exp i Kξ −

 

Ω(τ ) dτ



.

0

Ω(τ  ) dτ 

Here Kξ − 0 is the modulation phase with K and Ω are, respectively, the wave number and the frequency of the modulation. We then obtain from Eqs. (46) and (47) the following nonlinear dispersion relation for the amplitude modulation of the dust ion-acoustic wave modes:
 

Kc2 (τ ) 2 2 2 1− , Ω = PK (48) K2 where Kc2 (τ ) =

2Q|φ0 |2 P

exp(−2Γ τ ). From Eq. (48), the local-instability growth rate is given by 1/2
2 2 Kc (τ ) −1 . Im Ω(τ ) = P K K2

(49)

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J.-K. Xue / Physics Letters A 320 (2003) 226–233

In Eq. (49), the instability conditions of P Q > 0 and K 2
Kc2 (τ ) is satisfied. That is, the instability growth will cease when
 |φ0 |2 Q 1 ln 2 2 . τ  τmax = (50) 2Γ K P That is, there is a modulational instability period in the dust plasma with non-adiabatic dust charge variations. This modulational instability period not exist in plasma without dust variation [11]. Eq. (50) indicates that the instability period is proportional to the pump carrier wave amplitude φ0 and nonlinearity Q, but inversely proportional to modulation wave number K, the dispersion P , and dust charge fluctuation Γ . The total growth of the modulation during the unstable period is determined by exp(G), where


τmax 1/2
1/2   2Q|φ0 |2 2Q|φ0 |2 P K2   Im Ω(τ ) dτ = −1 − arctan −1 . G= (51) Γ P K2 P K2 0

The maximum value of G, i.e.,
 π Q|φ0 |2 max(G) = 1 − 4 Γ

(52)

occurs at K = Q|φ0 |2 /P . The maximum instability growth rate is proportional to the pump carrier wave amplitude φ0 and nonlinearity Q, but inversely proportional to the dust charge fluctuation Γ . The above results show that the modulational instability becomes a possible limiting factor for coherent transmission in plasma with non-adiabatic dust charge fluctuation. We must bear in mind that the modulational instability results given above are just suitable for the case of Γ = 0. That is, the modulation properties for the case of Γ = 0 must be re-discussed according to Eq. (39) by setting Γ = 0 [11]. Now let us consider the wave propagation in the plasma with the presence of the non-adiabatic dust charge fluctuation effect. An exact analytic solution of Eq. (39) is not possible. If the non-adiabatic dust charge fluctuation effect is weaker, we can take the term Γ in Eq. (39) as a perturbation. The perturbation solution of Eq. (39) can be obtained by the inverse scattering transforming method [22]. For P Q > 0, we have the perturbation solution of Eq. (39) (please refer to Eqs. (2.41), (3.21) and (1.8a) of Ref. [22]):

    Q

ξ + 2P Q cτ φ = 2iφ0 exp(−2Γ τ ) sech 2φ0 exp(−2Γ τ ) 2P 

     φ02 P Q

Q 2 PQ τ− exp(−4Γ τ ) − 1 , cξ + c × exp −i (53) 2P 2 Γ 2 where c is a constant. Eq. (53) indicates that the amplitude of the envelope solitary wave described by the modified NLSE (39) decreases exponentially with time. The non-adiabatic dust charge fluctuation gives rise to a damping effect. A soliton wave excited in plasma will be damped due to the dissipation caused by the non-adiabatic charge variation of the dust particles.

5. Conclusions A modified NLSE describing the slow modulation of the dust acoustic waves in dust plasma with non-adiabatic dust charge fluctuation is derived by the standard reductive perturbation method. The properties of the modulational instability of dust acoustic waves in the system is studied. The non-adiabatic charge variation of the dust particles modifies the instability character of dust acoustic waves. A critical threshold modulation wave number to establish

J.-K. Xue / Physics Letters A 320 (2003) 226–233

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the modulational instability is related to time τ . On the other hand, there is a modulational instability period determined by (50) for each modulation wave number K. When τ > τmax , the modulational of dust acoustic waves will becomes to stable. Meanwhile, the non-adiabatic dust charge fluctuation produces a damping effect for the envelope soliton waves excited in the system. With reference to the present study, we can understand the properties of dust acoustic solitary waves in space and laboratory.

Acknowledgement This work is supported by the National Natural Science Foundation of China (Grant No. 10347006).

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