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The effect of dust size distribution for vortex-like ion distribution dusty plasma
Physics Letters A 317 (2003) 275–279 www.elsevier.com/locate/pla
The effect of dust size distribution for vortex-like ion distribution dusty plasma Wen-Shan Duan Department of Physics, Northwest Normal University, Lanzhou 730070, China Received 15 December 2002; received in revised form 26 March 2003; accepted 21 August 2003 Communicated by A.R. Bishop
Abstract A reasonable normalization for vortex-like ion distribution dusty plasma with many different dust grains is adopted. For this system with N different dust grains, we have derived a modified Korteweg–de Vries equation for small, but finite amplitude dust acoustic waves. We have also studied the case that the dust grains satisfy the power law distribution and the ions satisfy the vortex-like ion distribution, and found that the soliton velocity is larger than that of mono-sized dusty plasma with average dust size of power law distribution. We have studied the amplitude and the width of a soliton existing in a dusty plasma as well, and concluded that if one soliton exists in a dusty plasma with power law distribution the smaller grains have larger velocities and propagate longer distances than that of larger grains. 2003 Published by Elsevier B.V. PACS: 52.35.Sb; 52.25.Vy; 05.45.Yv Keywords: Dust size distribution; Soliton; KdV equation
1. Introduction Dusty plasma has been the focus of intense research in the past decade. Dusty plasma is normal plasma consisting of ions and free electrons, which contain micro-sized particles. The dust grains become electrically charged due to interactions with the background plasma, causing them to act as a third charged plasma species. The dust grains can then significantly alter the behavior of the background plasma. Dusty plasma research has been driven by concerns in space, and here on earth. Dusty plasma is thought to play
E-mail address: [email protected] (W.-S. Duan). 0375-9601/$ – see front matter 2003 Published by Elsevier B.V. doi:10.1016/j.physleta.2003.08.041
an important role in star and planet formation, such as planetary rings, cometary surroundings, interstellar clouds and lower parts of Earth’s ionosphere [1,2]. The industrial community has also encouraged the study of dusty plasma. Plasma is used to produce microchip and hardened metals, etc. Understanding of how dust is trapped, and how it interacts with the background plasma, could improve manufacturing field. In the laboratory, the dust particles appear as impurities and can significantly influence the behavior of the surrounding plasma [3], with diverse applications in laboratory as well as in space plasma [4]. The dust acoustic wave (DAW) was firstly and theoretically reported in an unmagnetized dusty plasma by Rao [5]. Recent laboratory experiments on dusty
276
W.-S. Duan / Physics Letters A 317 (2003) 275–279
plasma have confirmed the existence of DAW [6–8]. Furthermore, low-frequency electrostatic ion-acoustic and ion-cyclotron waves have been studied in a magnetized dusty plasma [9]. It is noted that laboratory observations of low phase velocity dust acoustic waves are associated with significant depletion of electron number density [6], suggesting that the wave dynamics is governed by the inertia dust fluid and the pressure of the ions only. Recent numerical investigations on linear and nonlinear dust acoustic waves show a significant amount of ions trapping in the wave potential, indicating that there is a departure from Boltzmann ion distribution and one encounter vortex-like ion distribution in phase space. However, most of these works have focused attention on wave propagations in monosized dust grains because they are easier to study [5,9]. In a real case, the dust grains have many different sizes both in the space plasma and in the laboratory experiments [10–12]. As is widely accepted in space plasma the dust size distribution is given by a power law distribution, for charged dust grains with radii a in a given range [amin, amax ], the differential distribution is of the form [13,14] n(a) da = Ka −β da. It is found that the value of β = 4, 5 for the F-ring of Saturn, while for the G-ring the value of β = 7 or 6. For cometary environments, we recall a value of β = 3.4 [15]. Outside the dust size range a < amin or a > amax , we use n(a) = 0. It is noted that β = 1. The constant K can be determined by amax
Ntot =
n(a) da. amin
So K is given as follows K=
(1 − β)Ntot 1−β
1−β
amax − amin
.
In this Letter we study the dust acoustic waves in a dusty plasma by considering the different dust size distribution and the vortex-like ion distribution. We have investigated how the dust size distribution affect the propagation velocity, the amplitude and the width of a soliton for vortex-like ion distribution dusty plasma.
2. Discontinuous dust size distribution As is well known, there are many research works on dust acoustic waves for mono-sized dust grains. The governing equation is the Korteweg–de Vries equation (KdV equation) or MKdV equation [16–19]. By considering dust size distribution, for example, the power law distribution, we studied the dust acoustic wave (DAW). For simplicity, the plasma is assumed to be unmagnetized and wave is assumed to propagate in the x direction. In a dusty plasma, the dust grains are much heavier than ions. The dusty plasma consists of dust grains and ions. The dust grains have different sizes. First we assume that there are N different dust grains whose masses are mj (j = 1, 2, 3, . . . , N). For one-dimensional low-frequency dust acoustic motion, we have the following equations for the cold dust fluid ∂ndj ∂ + (ndj udj ) = 0, ∂t ∂x Qdj ∂φ ∂udj ∂udj + udj = ∂t ∂x mdj ∂x
(1) (j = 1, 2, . . . , N),
(2) where ndj , udj and mdj refer to the number density, the velocity and the mass of the j th dust grain, respectively. Qdj = −eZdj is the dust charge, Zdj the charge number of j th dust grain. The Poisson equation is given as follows N ∂ 2φ (3) = 4πe Zdj ndj − ni , ∂x 2 j =1
where Ntot is the total number density of dust grains ¯ which can be written by Ntot = N j =1 nd0j , Zd0 is the average value which can be determined by n Z , a ¯ is the the equation Z¯ d0 Ntot = N d0j d0j j =1 average radii of dust size given by equation of a¯ = N N j =1 aj nd0j / j =1 nd0j , where aj is the radii of the j th dust grain. The dust density is normalized by Ntot , Zd0j is normalized by Z¯ d0 . The space coordinate x, time t, velocity and electrostatic potential φ are normalized by the effective Debye length λDd = (Teff /4π Z¯ d0 Ntot e2 )1/2 , the inverse of effective dust −1 2 e 2 )1/2 , the = (m ¯ d /4πNtot Z¯ d0 plasma frequency ωpd effective dust acoustic speed Cd = (Z¯ d0 Teff /m ¯ d )1/2 , and Teff /e, respectively, where m ¯ d is the average mass of dustgrains which is determined by the equation of ¯ d. m ¯d = N j =1 nd0j mdj /Ntot , mdj is normalized by m Therefore, we obtained the following set of basic
W.-S. Duan / Physics Letters A 317 (2003) 275–279
In the small amplitude limit, namely, ψ 1, the ion number density takes the forms [22]
equations ∂ndj ∂ + (ndj udj ) = 0, ∂t ∂x ∂udj Zdj ∂φ ∂udj + udj = ∂t ∂x mdj ∂x
(4) (j = 1, 2, . . . , N), (5)
N
(6)
j =1
In order to study the effects of nonthermal ions on the nonlinear dust acoustic waves, we study the vortexlike ion distribution function of Schamel [20,21], which solve the ion Vlasov equation. We have 1 2 fif = √ e−(1/2)(v +2φ) , |v| > −2φ, 2π 1 −σ (1/2)(v 2+2φ) fit = √ e , |v| −2φ, 2π where fif and fit represents the free and trapped ion contribution. We note that ion distribution function, as presented above, is continuous in velocity space and satisfies the regularity requirements for an admissible BGK solution [20]. The electric potential is assumed to be negative, restricted by −ψ φ 0, where ψ plays the role of the amplitude. Furthermore, the velocity v is normalized by the ion thermal velocity vti , and σ , which is the ratio of free ion temperature (Tif ) to trapped ion temperature (Tit ), is a parameter determining the number of trapped ions. A plateau in the resonant ion region is given by σ = 0, and σ < 0 corresponds to a vortex-like excavated trapped ion distribution, which is of interest to us. The velocity of the nonlinear dust acoustic waves is assumed to be much smaller than the ion thermal velocity. Integrating the ion distribution functions over the velocity space, we readily obtain the ion number density as [22] e−σ φ ni = I (−φ) + √ erf( −σ φ ), σ where
I (−φ) = 1 − erf( −φ ) e−φ , 2 erf( −σ φ ) = √ π
e−y dy.
0
4a (−φ)3/2 + · · · , (7) 3 where a = (1 − σ )/π > 0. It may be noted here that this ion density ni is due to free and trapped ion distribution, the combined contribution of which cancels the terms containing (−φ)1/2 . In order to study the dust acoustic solitary waves, we must find an appropriate coordinate frame where the wave can be described smoothly. To find this frame we need to use the following stretched coordinates: [20–22] ξ = ' 1/4 (x − v0 t), and τ = ' 3/4 t, where ' is a small parameter characterizing the strength of nonlinearity, v0 is the velocity of dust acoustic waves. The dependent variables are expanded as follows ni = 1 − φ −
∂ 2φ = Zdj ndj − ni . ∂x 2
√ −σ φ
277
2
σ < 0,
ndj = nd0j + 'nd1j + ' 3/2 nd2j + · · · , udj = 'ud1j + ' φ = 'φ1 + '
3/2
3/2
ud2j + · · ·
(8)
(j = 1, 2, . . . , N),
φ2 + · · · .
(9) (10)
Substituting the expansions (8), (9) and (10) into Eqs. (4)–(6) and collecting the terms in different powers of ', we obtain the following equations at the lowest order nd1j = − ud1j v02 =
nd0j Zdj
φ1 , v02 mdj Zdj =− φ1 (j = 1, 2, . . . , N), v0 mdj N n 2 d0j Zdj j =1
mdj
,
(11) (12) (13)
at the higher order, we obtain the modified KdV equation ∂φ1 ∂ 3 φ1 ∂φ1 + A(−φ)1/2 + B 3 = 0, ∂τ ∂ξ ∂ξ
(14)
where A = av0 , (15) v0 B= . (16) 2 The second term of Eq. (14) is the nonlinear term, the third term is the dispersive term for the MKdV
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W.-S. Duan / Physics Letters A 317 (2003) 275–279
equation. The coefficients of both terms are depend on the v0 which is determined by the dust size distribution as shown by Eq. (13). In order to study the soliton solution of the MKdV equation of (14), we should study how both A and B depend on the v0 , namely, how they depend on the dust size distributions. When we assume that all dust grain sizes aj
λDd , we can express the mass and charge of dust particles as follows [15] mdj = km aj3 ,
(17)
Qdj = kq aj ,
(18)
or Zdj = kz aj ,
(19)
where km ≈ 4/3πρd , kq ≈ 4π'0 V0 , kz ≈ 4π'0 V0 /e, are approximately constants. ρd is the mass density of the dust grains (assumed to be constant and equal for all grains), V0 the electric surface potential at equilibrium, '0 the vacuum permittivity.
3. Continuous power law dust size distribution For the continuous case, we can easily change Eq. (13) into the following equation v02
k2 = z km
amax
n(a) da. a
(20)
amin
For the power law distribution, we find that kz2 K −β −β
a − amax , km β min K 1−β 1−β
amax − amin , Ntot = 1−β 2−β 2−β 1 − β amax − amin , a¯ = 1−β 1−β 2 − β amax −a
v02 =
(21) (22) (23)
min
kz2 Ntot (24) , km a¯ where v¯0 is the average velocity of the dust grains
v¯02 =
v02 v¯02
=
(β − 1)2 (c−β − 1)(c2−β − 1) , β(β − 2)(c1−β − 1)2
(25)
where c = amax /amin . The variations of v02 /v¯02 with respect to c for different β = 1.8, 2.2, 2.6, 3.0 are
Fig. 1. The variation of v02 /v¯02 with respect to c for different β = 1.8, 2.2, 2.6, 3.0, where β is a parameter for power law dust size distribution given by Eq. (1).
plotted in Fig. 1. It seems that the results are exactly same as that with standard Boltzmann ion distribution dusty plasma [19]. It seems that v02 /v¯02 increases as c increases, but it decreases as β increases. It can be concluded that as c increases, the soliton velocity increases. It can also be concluded from Fig. 1 that the soliton velocity for power law distribution is larger than that of mono-sized dusty plasma, for which dust grain size is assumed to be average dust size of the power law distribution determined by Eq. (23). It is well known that the stationary solitary wave solution of the MKdV equation (14) can be written as ξ − u0 τ , φ1 = −φm sech4 (26) w √ where φm = 15u0/(8A) and w = 8/u0 are the amplitude and the width of one soliton solution. We also find that φm = 15u0 /(8av0 ). It indicates that for power law dust size distribution of vortex-like ion distribution dusty plasma, the soliton amplitude is less than that of mono-sized dust grains with average dust size. It decreases as c increases, it increases as β increases. However, the soliton width does not depend on the soliton velocity. This results are completely different from those of Boltzmann ion distribution dusty plasma [23].
4. Discussion It is shown from Eqs. (11) and (12) that if there is a soliton or other nonlinear waves in this system the
W.-S. Duan / Physics Letters A 317 (2003) 275–279
perturbation velocities among different dust grains are different. They are inversely proportional to the square of the dust size a. This result also suggest that the propagation distances for different dust grains are also inversely proportional to the dust size. For a typical dusty plasma, it is reported that the size of the largest dust grain is, at least, about 10 times than that of the smallest dust grain, and therefore the velocity or propagation distance of the largest dust grain is 100 times less than that of the smallest dust grain. This is a interesting result since it indicates that if there is a soliton in dusty plasma the different dust grains have different velocities or propagation distances. It can be concluded that as c increases, the soliton velocity will increases, but as β increases it decreases. However, the soliton amplitude decreases as c increases, and increases as β increases. However, the soliton width has nothing to do with either c or β.
Acknowledgements This work is supported by the National Natural Science Foundation of China (NSFC) under Grant No. 10247008, Natural Science Foundation of Gansu Province of China under Grant No. YS021-A22-018, the Scientific Research Foundation (SRF) for the Returned Overseas Chinese Scholars (ROCS), State Education Ministry (SEM), and the Natural Science Foundation of Northwest Normal University (NWNUKJCXGC-215).
References [1] C.K. Goertz, Rev. Geophys. 27 (1989) 271. [2] O. Havnes, F. Melandso, C. La Hoz, T.K. Aslaksen, T. Hartquist, Phys. Scr. 45 (1992) 433.
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[3] S. Benkadda, V.N. Tsytovich, A. Verga, Comments Plasma Phys. Controlled Fusion 16 (1995) 321. [4] F. Verheest, Space Sci. Rev. 77 (1996) 267. [5] N.N. Rao, P.K. Shukla, M.Y. Yu, Planet. Space Sci. 38 (1990) 543. [6] A. Barkan, R.L. Merlino, N. D’Angelo, Phys. Plasmas 2 (1995) 2563. [7] G.E. Morfill, H. Thomos, J. Vac. Sci. Technol. A 14 (1996) 490. [8] R.L. Merlino, A. Barkan, C. Thompson, N. D’Angelo, Planet. Space Sci. 38 (1990) 1143. [9] N. D’Angelo, Planet. Space Sci. 38 (1990) 1143. [10] F. Verheest, Waves in Dusty Space Plasmas, Kluwer, Dordrecht, 2000. [11] V.W. Chow, D.A. Mendis, M. Rosenberg, J. Geophys. Res. 98 (1993) 19056. [12] M. Horanti, C.K. Goertz, Astrophys. J. 361 (1990) 155. [13] P. Meuris, Planet. Space Sci. 45 (1997) 449. [14] A. Brattll, Q. Havens, F. Melandso, J. Plasma Phys. 58 (1997) 691. [15] P. Meuris, Planet. Space Sci. 45 (1997) 1171. [16] Y.N. Nejoh, Phys. Plasmas 4 (1997) 2013. [17] J.X. Ma, J. Liu, Phys. Plasmas 4 (1997) 253. [18] B.S. Xie, K.F. He, Z.Q. Huang, Phys. Lett. A 247 (1998) 403. [19] W.S. Duan, Chaos Soliton Fractals 14 (2002) 503; W.S. Duan, Chaos Soliton Fractals 14 (2002) 1315; W.S. Duan, Chin. Phys. Lett. 19 (2002) 452; W.S. Duan, Chin. Phys. Lett. 18 (2001) 1088; W.S. Duan, Phys. Plasmas 8 (2001) 3583. [20] H. Schamel, J. Plasma Phys. 14 (1972) 905. [21] H. Schamel, J. Plasma Phys. 13 (1975) 139. [22] H. Schamel, S. Bujarbarua, Phys. Fluids 23 (1980) 2498; S. Bujarbarua, H. Schamel, J. Plasma Phys. 25 (1981) 515; H. Schamel, Phys. Rep. 140 (1986) 161. [23] W.S. Duan, Phys. Rev. E, submitted for publication.
The effect of dust size distribution for vortex-like ion distribution dusty plasma Wen-Shan Duan Department of Physics, Northwest Normal University, Lanzhou 730070, China Received 15 December 2002; received in revised form 26 March 2003; accepted 21 August 2003 Communicated by A.R. Bishop
Abstract A reasonable normalization for vortex-like ion distribution dusty plasma with many different dust grains is adopted. For this system with N different dust grains, we have derived a modified Korteweg–de Vries equation for small, but finite amplitude dust acoustic waves. We have also studied the case that the dust grains satisfy the power law distribution and the ions satisfy the vortex-like ion distribution, and found that the soliton velocity is larger than that of mono-sized dusty plasma with average dust size of power law distribution. We have studied the amplitude and the width of a soliton existing in a dusty plasma as well, and concluded that if one soliton exists in a dusty plasma with power law distribution the smaller grains have larger velocities and propagate longer distances than that of larger grains. 2003 Published by Elsevier B.V. PACS: 52.35.Sb; 52.25.Vy; 05.45.Yv Keywords: Dust size distribution; Soliton; KdV equation
1. Introduction Dusty plasma has been the focus of intense research in the past decade. Dusty plasma is normal plasma consisting of ions and free electrons, which contain micro-sized particles. The dust grains become electrically charged due to interactions with the background plasma, causing them to act as a third charged plasma species. The dust grains can then significantly alter the behavior of the background plasma. Dusty plasma research has been driven by concerns in space, and here on earth. Dusty plasma is thought to play
E-mail address: [email protected] (W.-S. Duan). 0375-9601/$ – see front matter 2003 Published by Elsevier B.V. doi:10.1016/j.physleta.2003.08.041
an important role in star and planet formation, such as planetary rings, cometary surroundings, interstellar clouds and lower parts of Earth’s ionosphere [1,2]. The industrial community has also encouraged the study of dusty plasma. Plasma is used to produce microchip and hardened metals, etc. Understanding of how dust is trapped, and how it interacts with the background plasma, could improve manufacturing field. In the laboratory, the dust particles appear as impurities and can significantly influence the behavior of the surrounding plasma [3], with diverse applications in laboratory as well as in space plasma [4]. The dust acoustic wave (DAW) was firstly and theoretically reported in an unmagnetized dusty plasma by Rao [5]. Recent laboratory experiments on dusty
276
W.-S. Duan / Physics Letters A 317 (2003) 275–279
plasma have confirmed the existence of DAW [6–8]. Furthermore, low-frequency electrostatic ion-acoustic and ion-cyclotron waves have been studied in a magnetized dusty plasma [9]. It is noted that laboratory observations of low phase velocity dust acoustic waves are associated with significant depletion of electron number density [6], suggesting that the wave dynamics is governed by the inertia dust fluid and the pressure of the ions only. Recent numerical investigations on linear and nonlinear dust acoustic waves show a significant amount of ions trapping in the wave potential, indicating that there is a departure from Boltzmann ion distribution and one encounter vortex-like ion distribution in phase space. However, most of these works have focused attention on wave propagations in monosized dust grains because they are easier to study [5,9]. In a real case, the dust grains have many different sizes both in the space plasma and in the laboratory experiments [10–12]. As is widely accepted in space plasma the dust size distribution is given by a power law distribution, for charged dust grains with radii a in a given range [amin, amax ], the differential distribution is of the form [13,14] n(a) da = Ka −β da. It is found that the value of β = 4, 5 for the F-ring of Saturn, while for the G-ring the value of β = 7 or 6. For cometary environments, we recall a value of β = 3.4 [15]. Outside the dust size range a < amin or a > amax , we use n(a) = 0. It is noted that β = 1. The constant K can be determined by amax
Ntot =
n(a) da. amin
So K is given as follows K=
(1 − β)Ntot 1−β
1−β
amax − amin
.
In this Letter we study the dust acoustic waves in a dusty plasma by considering the different dust size distribution and the vortex-like ion distribution. We have investigated how the dust size distribution affect the propagation velocity, the amplitude and the width of a soliton for vortex-like ion distribution dusty plasma.
2. Discontinuous dust size distribution As is well known, there are many research works on dust acoustic waves for mono-sized dust grains. The governing equation is the Korteweg–de Vries equation (KdV equation) or MKdV equation [16–19]. By considering dust size distribution, for example, the power law distribution, we studied the dust acoustic wave (DAW). For simplicity, the plasma is assumed to be unmagnetized and wave is assumed to propagate in the x direction. In a dusty plasma, the dust grains are much heavier than ions. The dusty plasma consists of dust grains and ions. The dust grains have different sizes. First we assume that there are N different dust grains whose masses are mj (j = 1, 2, 3, . . . , N). For one-dimensional low-frequency dust acoustic motion, we have the following equations for the cold dust fluid ∂ndj ∂ + (ndj udj ) = 0, ∂t ∂x Qdj ∂φ ∂udj ∂udj + udj = ∂t ∂x mdj ∂x
(1) (j = 1, 2, . . . , N),
(2) where ndj , udj and mdj refer to the number density, the velocity and the mass of the j th dust grain, respectively. Qdj = −eZdj is the dust charge, Zdj the charge number of j th dust grain. The Poisson equation is given as follows N ∂ 2φ (3) = 4πe Zdj ndj − ni , ∂x 2 j =1
where Ntot is the total number density of dust grains ¯ which can be written by Ntot = N j =1 nd0j , Zd0 is the average value which can be determined by n Z , a ¯ is the the equation Z¯ d0 Ntot = N d0j d0j j =1 average radii of dust size given by equation of a¯ = N N j =1 aj nd0j / j =1 nd0j , where aj is the radii of the j th dust grain. The dust density is normalized by Ntot , Zd0j is normalized by Z¯ d0 . The space coordinate x, time t, velocity and electrostatic potential φ are normalized by the effective Debye length λDd = (Teff /4π Z¯ d0 Ntot e2 )1/2 , the inverse of effective dust −1 2 e 2 )1/2 , the = (m ¯ d /4πNtot Z¯ d0 plasma frequency ωpd effective dust acoustic speed Cd = (Z¯ d0 Teff /m ¯ d )1/2 , and Teff /e, respectively, where m ¯ d is the average mass of dustgrains which is determined by the equation of ¯ d. m ¯d = N j =1 nd0j mdj /Ntot , mdj is normalized by m Therefore, we obtained the following set of basic
W.-S. Duan / Physics Letters A 317 (2003) 275–279
In the small amplitude limit, namely, ψ 1, the ion number density takes the forms [22]
equations ∂ndj ∂ + (ndj udj ) = 0, ∂t ∂x ∂udj Zdj ∂φ ∂udj + udj = ∂t ∂x mdj ∂x
(4) (j = 1, 2, . . . , N), (5)
N
(6)
j =1
In order to study the effects of nonthermal ions on the nonlinear dust acoustic waves, we study the vortexlike ion distribution function of Schamel [20,21], which solve the ion Vlasov equation. We have 1 2 fif = √ e−(1/2)(v +2φ) , |v| > −2φ, 2π 1 −σ (1/2)(v 2+2φ) fit = √ e , |v| −2φ, 2π where fif and fit represents the free and trapped ion contribution. We note that ion distribution function, as presented above, is continuous in velocity space and satisfies the regularity requirements for an admissible BGK solution [20]. The electric potential is assumed to be negative, restricted by −ψ φ 0, where ψ plays the role of the amplitude. Furthermore, the velocity v is normalized by the ion thermal velocity vti , and σ , which is the ratio of free ion temperature (Tif ) to trapped ion temperature (Tit ), is a parameter determining the number of trapped ions. A plateau in the resonant ion region is given by σ = 0, and σ < 0 corresponds to a vortex-like excavated trapped ion distribution, which is of interest to us. The velocity of the nonlinear dust acoustic waves is assumed to be much smaller than the ion thermal velocity. Integrating the ion distribution functions over the velocity space, we readily obtain the ion number density as [22] e−σ φ ni = I (−φ) + √ erf( −σ φ ), σ where
I (−φ) = 1 − erf( −φ ) e−φ , 2 erf( −σ φ ) = √ π
e−y dy.
0
4a (−φ)3/2 + · · · , (7) 3 where a = (1 − σ )/π > 0. It may be noted here that this ion density ni is due to free and trapped ion distribution, the combined contribution of which cancels the terms containing (−φ)1/2 . In order to study the dust acoustic solitary waves, we must find an appropriate coordinate frame where the wave can be described smoothly. To find this frame we need to use the following stretched coordinates: [20–22] ξ = ' 1/4 (x − v0 t), and τ = ' 3/4 t, where ' is a small parameter characterizing the strength of nonlinearity, v0 is the velocity of dust acoustic waves. The dependent variables are expanded as follows ni = 1 − φ −
∂ 2φ = Zdj ndj − ni . ∂x 2
√ −σ φ
277
2
σ < 0,
ndj = nd0j + 'nd1j + ' 3/2 nd2j + · · · , udj = 'ud1j + ' φ = 'φ1 + '
3/2
3/2
ud2j + · · ·
(8)
(j = 1, 2, . . . , N),
φ2 + · · · .
(9) (10)
Substituting the expansions (8), (9) and (10) into Eqs. (4)–(6) and collecting the terms in different powers of ', we obtain the following equations at the lowest order nd1j = − ud1j v02 =
nd0j Zdj
φ1 , v02 mdj Zdj =− φ1 (j = 1, 2, . . . , N), v0 mdj N n 2 d0j Zdj j =1
mdj
,
(11) (12) (13)
at the higher order, we obtain the modified KdV equation ∂φ1 ∂ 3 φ1 ∂φ1 + A(−φ)1/2 + B 3 = 0, ∂τ ∂ξ ∂ξ
(14)
where A = av0 , (15) v0 B= . (16) 2 The second term of Eq. (14) is the nonlinear term, the third term is the dispersive term for the MKdV
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W.-S. Duan / Physics Letters A 317 (2003) 275–279
equation. The coefficients of both terms are depend on the v0 which is determined by the dust size distribution as shown by Eq. (13). In order to study the soliton solution of the MKdV equation of (14), we should study how both A and B depend on the v0 , namely, how they depend on the dust size distributions. When we assume that all dust grain sizes aj
λDd , we can express the mass and charge of dust particles as follows [15] mdj = km aj3 ,
(17)
Qdj = kq aj ,
(18)
or Zdj = kz aj ,
(19)
where km ≈ 4/3πρd , kq ≈ 4π'0 V0 , kz ≈ 4π'0 V0 /e, are approximately constants. ρd is the mass density of the dust grains (assumed to be constant and equal for all grains), V0 the electric surface potential at equilibrium, '0 the vacuum permittivity.
3. Continuous power law dust size distribution For the continuous case, we can easily change Eq. (13) into the following equation v02
k2 = z km
amax
n(a) da. a
(20)
amin
For the power law distribution, we find that kz2 K −β −β
a − amax , km β min K 1−β 1−β
amax − amin , Ntot = 1−β 2−β 2−β 1 − β amax − amin , a¯ = 1−β 1−β 2 − β amax −a
v02 =
(21) (22) (23)
min
kz2 Ntot (24) , km a¯ where v¯0 is the average velocity of the dust grains
v¯02 =
v02 v¯02
=
(β − 1)2 (c−β − 1)(c2−β − 1) , β(β − 2)(c1−β − 1)2
(25)
where c = amax /amin . The variations of v02 /v¯02 with respect to c for different β = 1.8, 2.2, 2.6, 3.0 are
Fig. 1. The variation of v02 /v¯02 with respect to c for different β = 1.8, 2.2, 2.6, 3.0, where β is a parameter for power law dust size distribution given by Eq. (1).
plotted in Fig. 1. It seems that the results are exactly same as that with standard Boltzmann ion distribution dusty plasma [19]. It seems that v02 /v¯02 increases as c increases, but it decreases as β increases. It can be concluded that as c increases, the soliton velocity increases. It can also be concluded from Fig. 1 that the soliton velocity for power law distribution is larger than that of mono-sized dusty plasma, for which dust grain size is assumed to be average dust size of the power law distribution determined by Eq. (23). It is well known that the stationary solitary wave solution of the MKdV equation (14) can be written as ξ − u0 τ , φ1 = −φm sech4 (26) w √ where φm = 15u0/(8A) and w = 8/u0 are the amplitude and the width of one soliton solution. We also find that φm = 15u0 /(8av0 ). It indicates that for power law dust size distribution of vortex-like ion distribution dusty plasma, the soliton amplitude is less than that of mono-sized dust grains with average dust size. It decreases as c increases, it increases as β increases. However, the soliton width does not depend on the soliton velocity. This results are completely different from those of Boltzmann ion distribution dusty plasma [23].
4. Discussion It is shown from Eqs. (11) and (12) that if there is a soliton or other nonlinear waves in this system the
W.-S. Duan / Physics Letters A 317 (2003) 275–279
perturbation velocities among different dust grains are different. They are inversely proportional to the square of the dust size a. This result also suggest that the propagation distances for different dust grains are also inversely proportional to the dust size. For a typical dusty plasma, it is reported that the size of the largest dust grain is, at least, about 10 times than that of the smallest dust grain, and therefore the velocity or propagation distance of the largest dust grain is 100 times less than that of the smallest dust grain. This is a interesting result since it indicates that if there is a soliton in dusty plasma the different dust grains have different velocities or propagation distances. It can be concluded that as c increases, the soliton velocity will increases, but as β increases it decreases. However, the soliton amplitude decreases as c increases, and increases as β increases. However, the soliton width has nothing to do with either c or β.
Acknowledgements This work is supported by the National Natural Science Foundation of China (NSFC) under Grant No. 10247008, Natural Science Foundation of Gansu Province of China under Grant No. YS021-A22-018, the Scientific Research Foundation (SRF) for the Returned Overseas Chinese Scholars (ROCS), State Education Ministry (SEM), and the Natural Science Foundation of Northwest Normal University (NWNUKJCXGC-215).
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