- Email: [email protected]
Effects of dust temperature and trapped ions on the formation of dust-acoustic solitary waves
Physics Letters A 374 (2010) 1855–1859
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Effects of dust temperature and trapped ions on the formation of dust-acoustic solitary waves H. Alinejad a,b,∗ a b
Department of Basic Science, Babol University of Technology, Babol 47148-71167, Iran Research Institute for Astronomy and Astrophysics of Maragha, P.O. Box 55134-441, Maragha, Iran
a r t i c l e
i n f o
Article history: Received 14 December 2009 Received in revised form 7 February 2010 Accepted 18 February 2010 Available online 21 February 2010 Communicated by F. Porcelli Keywords: Dust temperature Trapped ions Higher-order nonlinearities Solitary waves
a b s t r a c t The effects of dust temperature and trapped ions are incorporated in the study of dust-acoustic solitary waves. An energy integral equation involving the Sagdeev potential is derived, and the basic properties of large amplitude solitary structures are investigated. It is shown that the effects of dust temperature, resonant ions and equilibrium free electron density significantly change the regions of the existence of large amplitude solitary waves. Expanding the Sagdeev potential to include higher-order nonlinearities of electric potential, an exact steady state solution is also obtained which confirms the possibility of dust-acoustic soliton in the small amplitude limit. Furthermore, two asymptotic cases of the stationary solution are found which are related to the contribution of trapped ions. © 2010 Elsevier B.V. All rights reserved.
1. Introduction There has been a great deal of interest in numerous collective processes in dusty plasmas, i.e., plasmas with extremely massive and negatively charged dust grains. Such plasmas occur in laboratory, astrophysical and space environments, such as cometary tails, planetary rings, interstellar medium, the earth’s environment, etc. [1–4]. Due to the presence of the charged dust grains in plasmas, different types of collective processes exist and very rich wave modes can be excited in dusty plasmas [5–8]. One of these is the low frequency dust-acoustic mode [9,10] in an unmagnetized dusty plasma whose constituents are an inertial charge dust fluid and Boltzmann distributed electrons and ions. Thus, in the dust-acoustic waves the dust particle mass provides the inertia and the thermal pressure from the electrons and ions give rise to the restoring force. Rao et al. [9] were the first to report theoretically the existence of dust-acoustic solitary waves by using the reductive perturbation method which is only valid for small but finite amplitude limit. The predictions of Rao et al. [9] were conclusively verified by the laboratory experiment of Barkan [10]. On the other hand, numerical simulation studies [11] on linear and nonlinear dust-acoustic waves exhibit a significant amount of ion trapping in the wave potential. Clearly, there is a departure from the Boltzmann ion distribution and one encounters the
*
Address for correspondence: Department of Basic Science, Babol University of Technology, Babol 47148-71167, Iran. Tel.: +98 111 3234203; fax: +98 111 3234201. E-mail address: [email protected]. 0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.02.047
vortex-like ion distribution in the phase space. It is well known that such ion behavior drastically modifies the conditions for the existence of nonlinear structures such as solitons and shocks, which are not observed in dusty plasma with isothermal ions [12–17]. Most of the studies discussed up to now are restricted to theoretical investigations on soliton dynamics in a dusty plasma with cold dust grains in the frame work of the modified KdV equation using the reductive perturbation method. However the perturbation method is mainly valid for the small amplitude solitary waves, Sagdeev potential approach [18] is a powerful tool for studying large amplitude solitary structures. Adopting this approach, Roychoudhury et al. [19] investigated the large amplitude solitary waves in the finite temperature dusty plasma with isothermal ions. They showed that the dust temperature can restrict the region of existence of dust-acoustic solitary waves. Akhtar et al. [20] studied the nonlinear dynamics of dust-acoustic waves in unmagnetized multicomponent plasmas with hot and cold dust species. They investigated the effects of different masses, charges and concentration ratios of the dust particles on the shape of solitary structures. Mahmood et al. [21] examined the influence of dust temperature on nonlinear dust-acoustic waves in a magnetized plasma. They also showed that the dust thermal energy reduces the wave amplitude in a magnetized plasma, which has the same behavior as in an unmagnetized case. A little attention has been paid to study the effects of dust temperature and deviations from isothermality of ions on the propagation of large amplitude solitary waves in dusty plasma. For example, Mendoza et al. [22] investigated the effects of dust fluid temperature and fast or non-thermal ions on arbitrary ampli-
1856
H. Alinejad / Physics Letters A 374 (2010) 1855–1859
tude dust-acoustic solitary structures. They showed that a suitable distribution of non-thermal ions in a dusty plasma system may change the nature of the dust-acoustic solitary waves and may support the coexistence of large amplitude compressive and rarefactive solitary waves. Also, Das et al. [23] investigated arbitrary amplitude dust-acoustic solitary waves and double layers in nonthermal plasma with the effect of dust temperature. They showed that the amplitude of the positive potential double layer decreases with increasing the dust temperature. To our knowledge, large amplitude dust-acoustic solitary waves including the effects of dust temperature and trapped ions has not been investigated before. Hence the main motivation of the present work is to study the effects of dust temperature and higher-order nonlinearities on the dust-acoustic solitary waves in a dusty plasma with ions following a vortex-like distribution. The trapped ions are included in this model as a result of nonlinear resonant interaction of the localized electrostatic wave potential with ions during its evolution. The electron inertia is neglected and Boltzmann distribution for the electron density is assumed which implies isothermality. To study the regions of existence of arbitrary amplitude solitary waves an energy integral equation involving a Sagdeev potential is derived. Moreover, to confirm the possibility of the dust-acoustic soliton, we investigate higher-order nonlinear and dispersive effects of the electric potential in the expansion of the Sagdeev potential. We also investigate two asymptotic cases of the stationary solution which are related to the contribution of the resonant ions. 2. Basic equations and localized waves We consider a collisionless, unmagnetized three component dusty plasma consisting of extremely massive warm dust grains, Boltzmann distributed electrons and hot ions obeying a vortex-like distribution. Although the size and the charge of the dust grains varies from one grain to another, we assume for simplicity that all grains have the same negative charge, qd = − Z d e, where Z d is the number of charges residing on the surface of the dust grains. At equilibrium, we have ni0 = Z d nd0 + ne0 , where ni0 , nd0 and ne0 are the unperturbed ion, dust and electron number densities, respectively. The dynamics of the nonlinear dust-acoustic waves in a three component dusty plasma system with a constant dust grain charge is thus governed by [9]
∂ ∂N + ( N U ) = 0, ∂t ∂x ∂U ∂U σd ∂ P ∂ψ +U + = , ∂t ∂x N ∂x ∂x ∂P ∂P ∂U +U + 3P = 0, ∂t ∂x ∂x ∂ 2ψ = N − μ1 N i + μ0 e σ ψ , ∂ x2
(1)
perature in the wave potential. The electron inertia is neglected. This expression is obtained by the consideration that a thermal electron moves with a speed much higher than the ion thermal speed. In this case, ions could interact with the wave potential during its evolution, and therefore can be trapped in the wave potential. Then, to model an ion distribution with trapped particles, we employ a vortex-like ion distribution function of Schamel [24,25], which solves the ion Vlasov equation. In such environment, the ion number density can be expressed as
e −α ψ N i = I (−ψ) + √ erf ( −α ψ ), where
I (−ψ) = 1 − erf ( −ψ ) e −ψ , √
2 erf ( −α ψ ) = √
(3) (4)
where N is the dust particle number density normalized to its equilibrium value nd0 , N i is the ion number density normalized to ni0 , U is the dust fluid velocity normalized to the dustacoustic speed C d = ( Z d T i /md )1/2 , and ψ is the electrostatic wave potential normalized to T i /e, and P is the dust pressure normalized to nd0 T d , where T d is the dust fluid temperature and md is the mass of negatively charged dust particulates. σ = T i / T e , with T i ( T e ) being the ion (electron) temperature, μ0 = β/(1 − β) and μ1 = 1/(1 − β), where β = ne0 /ni0 . The time and space variables are given in the units of the dust plasma 1 period ω− = (md /4π Z d2 ndo e 2 )1/2 and the Debye length λ Dd = pd
( T i /4π Z d ndo e 2 )1/2 , respectively. The model used here is based on the assumption of neglecting some of dust grains which might be trapped due to the finite tem-
−α ψ
e −t dt .
π
2
0
Here α = T i / T it (ratio of the free ion temperature to trapped ion temperature) is the trapping parameter describing the temperature of the trapped ions. Positive values of α , which we are interested in here, lead to a vortex-like distribution for electrons. Note that if we neglect the resonant effects, N i reduces to the Maxwellian ion distribution. To obtain stationary localized solution from the basic equations, we make all the dependent variables depend only on a single variable ξ = x − Mt, where ξ is normalized to λ Dd and M is the Mach number of the solitary waves with respect to the dust-acoustic wave frame for the system. Then, using the steady state condition (∂/∂ t = 0) and the appropriate boundary conditions for localized perturbations, namely, N → 1, U → 0, ψ → 0, P → 1 and dψ/dξ → 0 at ξ → ±∞, in Eqs. (1)–(3), one can express the dust particle number density N as
N=
√
2M M 2 + 3σd + 2ψ
+
M 2 + 3σd + 2ψ
2
− 12σd M 2
− 12
(6)
.
Now using (6), the qualitative nature of the solutions of Eq. (4) is most easily seen by introducing the Sagdeev potential [18]. Therefore, Poisson’s equation (4) reduces to the form
1 2
(2)
(5)
|α |
dψ
2 + V (ψ) = 0,
dξ
(7)
where the Sagdeev potential V (ψ) is defined as
2μ1 μ0 1 σ ψ + μ1 1 − I (−ψ) − √ −ψ V (ψ) = 1−e 1− σ α π μ1 e−α ψ erf ( −α ψ ) − √ α3 √
−
3γ
√
+
1 −1 e 2 cos h (χ /γ ) γ 2 + χ χ − 3/2
2M
1 −1 e 2 cos h (η/γ ) γ 2 + η η − 3/2
2M
3γ
χ2 − γ 2
η2 − γ 2 ,
(8)
with
χ = M 2 + 3σd + 2ψ,
(9)
γ = 2M 3σd ,
(10)
2
(11)
η = M + 3σd .
We note that Eq. (7) can be regarded as an “energy integral” of an oscillating particle of unit mass, with pseudo-speed dψ/dξ ,
H. Alinejad / Physics Letters A 374 (2010) 1855–1859
Fig. 1. Behavior of the Sagdeev potential V (ψ) against the potential ψ and β for σd = 0.01, α = 0.7, σ = 0.05, and M = 1.32.
pseudo-position ψ , pseudo-time ξ and pseudo-potential V (ψ). It is also important to mention that in our √ study the condition for the dust density to be real ψ > −( M − 3σd )2 /2 must always be satisfied. On the other hand, the form of the pseudo-potential could determine whether solitary wave solutions of (7) exist or not. It is clear that V (ψ) = 0 and dV /dξ = 0 at ψ = 0. In this case, solitary wave structure exists if d2 V /dψ 2 0 at ψ = 0, so that the zero as a fixed point is unstable. All the specified conditions are satisfied. Besides that V (ψ) should be negative between ψ = 0 and ψm , where ψm is maximum (or minimum) value of ψ . To investigate the existence regions and nature of the solitary wave excitation in plasma, we have numerically analyzed the general expression (7) for V (ψ) using typical parameters. The results are displayed in Figs. 1–4. These figurers show that the plasma model under consideration permits only rarefactive soliton. In Fig. 1, the dependence of the solitary waves on β have been investigated for σd = 0.01, σ = 0.05, α = 0.7 and M = 1.32. It is obvious that the amplitude increases with increasing the amount of the equilibrium electron density ne0 . This is in good agreement with our earlier work [17] for the dynamics of large amplitude dust-acoustic solitary waves in dusty plasmas whose constituents are cold charged dust fluid, free electrons and ions with excavated trapped distribution. On the other hand, to see what happens when the dust temperature is increased, we study the behavior of the Sagdeev potential and show the effect of dust temperature on the formation of soliton in the case of β = 0.05, σ = 0.05, α = 0.5 and M = 1.34. From Fig. 2, we observe that both the depth of Sagdeev potential curves and amplitude increase remarkably as the dust temperature decreases. This shows that the dust temperature is destructive for the formation of electrostatic localized waves. Similar behavior is also observed in magnetized and unmagnetized dusty plasma with isothermal ions, as shown in Refs. [19,21]. Fig. 3 clearly shows contrast between two types of plasmas in which ions follow the Boltzmann (α = 1) and vortex-like (α = 0.6) distributions. For both the plasmas, solitary waves can exist but the potential wells represented by V (ψ) profiles are of different depths. In the isothermal case, because of the greater depth of Sagdeev potential, the amplitude of solitary wave is larger compared to that for vortex-like ion distributions. It follows that trapped ions introduce some kind of inertia on the propagation of the localized dust-acoustic waves. This effect has also been observed in the study of ion-acoustic solitary waves with trapped electrons, as shown in Refs. [24,25]. Fig. 4 exhibits the effect of trapped ions on the formation of soliton. It is clear that an increase of the trapping parameter α leads to an increase of the potential depth when the Mach number, equilibrium free electron concentration and dust temperature are held constant. This confirms completely the result observed in Fig. 3.
1857
Fig. 2. Behavior of the Sagdeev potential V (ψ) against the potential ψ and dust temperature σd for β = 0.05, α = 0.5, σ = 0.05 and M = 1.34.
Fig. 3. Contrast between the Boltzmann (solid curves) and vortex-like (dotted curves) ion distributions relative to the profile of V (ψ) for M = 1.2, β = 0.04, σ = 0.05, α = 0.6 and σd = 0.02.
Fig. 4. Behavior of the Sagdeev potential V (ψ) against the potential ψ and the trapping parameter α for σd = 0.05, β = 0.05, σ = 0.05 and M = 1.33.
To confirm the possibility of the localized dust-acoustic waves in our plasma model, we investigate the effects of higher-order nonlinearities of the electric potential in the expansion of the Sagdeev potential. The specific results can be obtained by expanding V (ψ) in powers of ψ and keeping up to third-order terms. Accordingly, under the assumption ψ 1, Eq. (7) takes the form
dψ
2
5
= γ1 ψ 2 + γ2 (−ψ) 2 + γ3 ψ 3 ,
dξ
(12)
where
γ1 = μ1 + μ0 σ − γ2 =
16 15
1 M2
√ (1 − α ),
π
− 3σd
,
(13) (14)
1858
H. Alinejad / Physics Letters A 374 (2010) 1855–1859
electron density, as shown in Fig. 1. In contrast to the amplitude variation, Fig. 6 shows that the width of solitary waves increases with the increasing of the dust temperature whereas decreases along with the increase in the concentration of free electrons. This also confirms what has been found in the large amplitude approximation. We now switch our attention on (16) and express two asymptotic cases of stationary solution which are related to the contribution of the resonant ions. We first consider a situation that the coefficient γ2 and γ3 of the nonlinear terms of the electric potential in the expansion of Sagdeev potential (12) are such that γ2 γ3 . Considering this assumption and expanding the righthand side of (16) in powers of 1/γ2 , the stationary solution can be expressed as
Fig. 5. Variation of the maximum amplitude ψm against the dust temperature for α = 0.5, σ = 0.05, M = 1.2, β = 0.06 (dotted) and β = 0.04 (solid).
σd
ψ =−
γ12 2γ3 ξ ξ 1 − 2 cos h sec h2 λ 2λ γ22 γ2
sec h4
ξ . 2λ
(17)
It is clear that the amplitude of the solitary wave modifies when the number of trapped ions increases. In particular case, as the contribution of the resonant ions becomes larger and larger such that we can ignore γ3 in comparison to the resonant term γ2 , Eq. (17) reduces to
ψ = −ψm sec h4
Fig. 6. Variation of the width λ against the dust temperature 0.05, M = 1.2, β = 0.04 (solid) and β = 0.06 (dotted).
γ3 =
M 2 + σd
( M 2 − 3σd )3
+
1 3
μ0 σ 2 − μ1 .
σd for α = 0.5, σ =
(15)
(16)
where λ = 4/γ1 measures the width of dust-acoustic solitary waves. The soliton solution (16) is similar to the exact stationary solution of the generalized KdV equation with the mixed nonlinearity of the electric potential [26]. This solution clearly indicates the existence of solitary waves only with negative potential (rarefactive solitary waves), corresponding to a hump in the dust density. For the stationary solution (16) obtained in the small amplitude limit, variations of the amplitude ψm = ψ (ξ = 0), and the width λ with the dust temperature and the densities ratio of electrons and ions in equilibrium β , have been depicted in Figs. 5 and 6. It can be seen from Fig. 4 that the amplitude decreases with the dust temperature. This confirms completely results observed in Fig. 2 which are obtained for the large amplitude limit. This is also similar to prior results with isothermal ion distribution in magnetized and unmagnetized dusty plasmas [19,21]. The amplitude can be observed that increases with the increase in equilibrium free
(18)
where ψm = (γ1 /γ2 )2 is the amplitude of the solitary wave. The localized solution (18) is similar to the stationary solution of the modified KdV equation for a three components dusty plasma system with cold dust grains, as shown in Ref. [12]. This clearly indicates that the small amplitude solitary wave exists only with negative potential. This also confirms the prior results which are based on the large amplitude limit. On the other hand, for the case that the coefficient γ3 of the nonlinear term is dominant over γ2 in Eq. (12), the expansion of stationary solution (16) in powers of 1/γ3 gives
Here γ2 and γ3 are respectively the coefficients of nonlinear terms of the KdV and modified KdV equations, which can be obtained from the basic equations (1)–(4) using the reductive perturbation theory. Integration of the resulting energy equation (12) with respect ξ along with the appropriate boundary conditions, gives stationary soliton solution as
−2 γ22 γ3 ξ −γ2 ψ =− + + cos h , 2γ 1 γ1 λ 4γ12
ξ , 2λ
γ1 15 2 3 −1/2 ξ 1− γ γ γ3 sec h γ3 8 2 1 λ
γ2 ξ ξ . − 2 1 − 3 sec h2 sec h2 4γ 3 λ λ
ψ =−
(19)
The solution (19) is obtained when the effect of nonlinear interaction of ions with the electrostatic wave potential during its evolution is small. This also indicates the modified amplitude and width of the solitary wave profile from the case that the coefficients γ2 and γ3 are comparable in (16). By neglecting the resonant effect γ2 , we observe that γ3 is similar coefficient of nonlinear term of the KdV equation, which can be obtained from the basic equations by using the reductive perturbation method. In this case, ions follow the isothermal distribution function and this procedure reduces solitary wave solutions as
ψ = −ψm sec h2
ξ , λ
(20)
where the maximum amplitude of solitary waves is ψm = γ1 /γ3 , and the width λ has the same form as before. The solution (12) is similar to the stationary soliton-like solution of the KdV equation for a three-component dusty plasma system with cold dust grains, as shown in Ref. [27]. This also indicates the existence of the small amplitude solitary structure with negative potential. In summary, we have investigated the effects of dust temperature and higher-order nonlinearities on the nonlinear propagation of dust-acoustic waves in three-component dusty plasma whose constituents are warm dust grains, Boltzmann distributed electrons and hot ions obeying vortex-like distribution. Both highly
H. Alinejad / Physics Letters A 374 (2010) 1855–1859
and weakly nonlinear analysis are examined by deriving an energy integral equation involving Sagdeev potential. The effects of the dust temperature, equilibrium free electron density and trapped ions are found to significantly change the regions of the existence of arbitrary amplitude solitary waves. In particular, both of the depth of Sagdeev potential curves and amplitude decrease as the dust temperature increases. If we compare our results with that of isothermal case, we find that the amplitude of soliton in the presence of isothermal ions is larger compared to that for vortexlike ion distributions. It follows that trapped ions introduce some kind of inertia on the propagation of localized electrostatic waves. To examine the possibility of dust-acoustic solitary waves in the small amplitude limit, we have investigated the effects of higherorder nonlinearities of the electric potential in the expansion of the Sagdeev potential and integrated the resulting energy equation with the proper boundary condition required for a solitary wave structure. The dependence of the width and amplitude of solitary wave solution on the dust temperature and trapped ions are studied. We have also investigated two asymptotic cases of the stationary soliton-like solution which are related to the population of resonant particles present. Finally, the results yield an improved understanding of the propagation of dust-acoustic waves in an unmagnetized warm dusty plasma. It is shown from both highly and weakly nonlinear analysis that the plasma system under consideration supports only rarefactive solitary waves. The results of the present investigation should also help us to explain the basic features of localized dust-acoustic perturbations propagating in the space and laboratory dusty plasmas, where negatively charged warm dust particulates, thermally distributed electrons and resonant ions are the major plasma species.
1859
Acknowledgement This work has been supported financially by Research Institute for Astronomy and Astrophysics of Maragha. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
M. Horanyi, D.A. Mendis, Astrophys. J. 307 (1986) 800. T.G. Northrop, Phys. Scripta 45 (1992) 475. D.A. Mendis, M. Rosenberg, Rev. Astron. Astrophys. 32 (1994) 419. F. Verheest, Space Sci. Rev. 77 (1996) 267. W.F. El-Taibany, R. Sabry, Phys. Plasmas 12 (2005) 082302. W.M. Moslem, W.F. El-Taibany, Phys. Plasmas 12 (2005) 122309. A.A. Mamun, Phys. Lett. A 372 (2008) 4610. M. Tribeche, G. Boumezound, Phys. Plasmas 15 (2008) 053702. N.N. Rao, P.K. Shukla, M.Y. Yu, Planet. Space Sci. 38 (1990) 543. A. Barkan, R.L. Merlino, D. D’Angelo, Phys. Plasmas 2 (1995) 3563. D. Winske, S.P. Gary, M.E. Jones, M. Rosenberg, D.A. Mendis, V.W. Chow, Geophys. Res. Lett. 22 (1995) 2069. A.A. Mamun, J. Plasma Phys. 59 (1998) 575. A.A. Mamun, R.A. Cairns, P.K. Shukla, Phys. Plasmas 3 (1996) 2610. W.S. Duan, Phys. Lett. A 317 (2003) 275. A.A. Mamun, M.A. Hassann, J. Plasma Phys. 63 (2000) 191. M. Kakti, K.S. Goswami, Phys. Plasmas 5 (1998) 4508. H. Alinejad, Phys. Lett. A 37 (2009) 2935. R.Z. Sagdeev, in: Review of Plasma Physics, vol. 4, Consultants Bureau, New York, 1966, p. 23. R. Roychoudhury, S. Mukherjee, Phys. Plasmas 4 (1997) 2305. N. Akhtar, S. Mahmood, H. Saleem, Phys. Lett. A 361 (2007) 126. S. Mahmood, N. Akhtar, H. Saleem, Phys. Plasmas 14 (2007) 074507. A. Mendoza, S.M. Russel, A.A. Mamun, Planet. Space Sci. 48 (2000) 599. A. Das, B. Bandyopadhyay, K.P. Das, Phys. Plasmas 16 (2009) 073703. H. Schamel, J. Plasma Phys. 14 (1972) 905. H. Schamel, J. Plasma Phys. 9 (1973) 377. S.G. Tagare, A. Chakrabarti, Phys. Fluids 17 (1974) 1331. A.A. Mamun, Astrophys. Space Sci. 268 (1999) 448.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Effects of dust temperature and trapped ions on the formation of dust-acoustic solitary waves H. Alinejad a,b,∗ a b
Department of Basic Science, Babol University of Technology, Babol 47148-71167, Iran Research Institute for Astronomy and Astrophysics of Maragha, P.O. Box 55134-441, Maragha, Iran
a r t i c l e
i n f o
Article history: Received 14 December 2009 Received in revised form 7 February 2010 Accepted 18 February 2010 Available online 21 February 2010 Communicated by F. Porcelli Keywords: Dust temperature Trapped ions Higher-order nonlinearities Solitary waves
a b s t r a c t The effects of dust temperature and trapped ions are incorporated in the study of dust-acoustic solitary waves. An energy integral equation involving the Sagdeev potential is derived, and the basic properties of large amplitude solitary structures are investigated. It is shown that the effects of dust temperature, resonant ions and equilibrium free electron density significantly change the regions of the existence of large amplitude solitary waves. Expanding the Sagdeev potential to include higher-order nonlinearities of electric potential, an exact steady state solution is also obtained which confirms the possibility of dust-acoustic soliton in the small amplitude limit. Furthermore, two asymptotic cases of the stationary solution are found which are related to the contribution of trapped ions. © 2010 Elsevier B.V. All rights reserved.
1. Introduction There has been a great deal of interest in numerous collective processes in dusty plasmas, i.e., plasmas with extremely massive and negatively charged dust grains. Such plasmas occur in laboratory, astrophysical and space environments, such as cometary tails, planetary rings, interstellar medium, the earth’s environment, etc. [1–4]. Due to the presence of the charged dust grains in plasmas, different types of collective processes exist and very rich wave modes can be excited in dusty plasmas [5–8]. One of these is the low frequency dust-acoustic mode [9,10] in an unmagnetized dusty plasma whose constituents are an inertial charge dust fluid and Boltzmann distributed electrons and ions. Thus, in the dust-acoustic waves the dust particle mass provides the inertia and the thermal pressure from the electrons and ions give rise to the restoring force. Rao et al. [9] were the first to report theoretically the existence of dust-acoustic solitary waves by using the reductive perturbation method which is only valid for small but finite amplitude limit. The predictions of Rao et al. [9] were conclusively verified by the laboratory experiment of Barkan [10]. On the other hand, numerical simulation studies [11] on linear and nonlinear dust-acoustic waves exhibit a significant amount of ion trapping in the wave potential. Clearly, there is a departure from the Boltzmann ion distribution and one encounters the
*
Address for correspondence: Department of Basic Science, Babol University of Technology, Babol 47148-71167, Iran. Tel.: +98 111 3234203; fax: +98 111 3234201. E-mail address: [email protected]. 0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.02.047
vortex-like ion distribution in the phase space. It is well known that such ion behavior drastically modifies the conditions for the existence of nonlinear structures such as solitons and shocks, which are not observed in dusty plasma with isothermal ions [12–17]. Most of the studies discussed up to now are restricted to theoretical investigations on soliton dynamics in a dusty plasma with cold dust grains in the frame work of the modified KdV equation using the reductive perturbation method. However the perturbation method is mainly valid for the small amplitude solitary waves, Sagdeev potential approach [18] is a powerful tool for studying large amplitude solitary structures. Adopting this approach, Roychoudhury et al. [19] investigated the large amplitude solitary waves in the finite temperature dusty plasma with isothermal ions. They showed that the dust temperature can restrict the region of existence of dust-acoustic solitary waves. Akhtar et al. [20] studied the nonlinear dynamics of dust-acoustic waves in unmagnetized multicomponent plasmas with hot and cold dust species. They investigated the effects of different masses, charges and concentration ratios of the dust particles on the shape of solitary structures. Mahmood et al. [21] examined the influence of dust temperature on nonlinear dust-acoustic waves in a magnetized plasma. They also showed that the dust thermal energy reduces the wave amplitude in a magnetized plasma, which has the same behavior as in an unmagnetized case. A little attention has been paid to study the effects of dust temperature and deviations from isothermality of ions on the propagation of large amplitude solitary waves in dusty plasma. For example, Mendoza et al. [22] investigated the effects of dust fluid temperature and fast or non-thermal ions on arbitrary ampli-
1856
H. Alinejad / Physics Letters A 374 (2010) 1855–1859
tude dust-acoustic solitary structures. They showed that a suitable distribution of non-thermal ions in a dusty plasma system may change the nature of the dust-acoustic solitary waves and may support the coexistence of large amplitude compressive and rarefactive solitary waves. Also, Das et al. [23] investigated arbitrary amplitude dust-acoustic solitary waves and double layers in nonthermal plasma with the effect of dust temperature. They showed that the amplitude of the positive potential double layer decreases with increasing the dust temperature. To our knowledge, large amplitude dust-acoustic solitary waves including the effects of dust temperature and trapped ions has not been investigated before. Hence the main motivation of the present work is to study the effects of dust temperature and higher-order nonlinearities on the dust-acoustic solitary waves in a dusty plasma with ions following a vortex-like distribution. The trapped ions are included in this model as a result of nonlinear resonant interaction of the localized electrostatic wave potential with ions during its evolution. The electron inertia is neglected and Boltzmann distribution for the electron density is assumed which implies isothermality. To study the regions of existence of arbitrary amplitude solitary waves an energy integral equation involving a Sagdeev potential is derived. Moreover, to confirm the possibility of the dust-acoustic soliton, we investigate higher-order nonlinear and dispersive effects of the electric potential in the expansion of the Sagdeev potential. We also investigate two asymptotic cases of the stationary solution which are related to the contribution of the resonant ions. 2. Basic equations and localized waves We consider a collisionless, unmagnetized three component dusty plasma consisting of extremely massive warm dust grains, Boltzmann distributed electrons and hot ions obeying a vortex-like distribution. Although the size and the charge of the dust grains varies from one grain to another, we assume for simplicity that all grains have the same negative charge, qd = − Z d e, where Z d is the number of charges residing on the surface of the dust grains. At equilibrium, we have ni0 = Z d nd0 + ne0 , where ni0 , nd0 and ne0 are the unperturbed ion, dust and electron number densities, respectively. The dynamics of the nonlinear dust-acoustic waves in a three component dusty plasma system with a constant dust grain charge is thus governed by [9]
∂ ∂N + ( N U ) = 0, ∂t ∂x ∂U ∂U σd ∂ P ∂ψ +U + = , ∂t ∂x N ∂x ∂x ∂P ∂P ∂U +U + 3P = 0, ∂t ∂x ∂x ∂ 2ψ = N − μ1 N i + μ0 e σ ψ , ∂ x2
(1)
perature in the wave potential. The electron inertia is neglected. This expression is obtained by the consideration that a thermal electron moves with a speed much higher than the ion thermal speed. In this case, ions could interact with the wave potential during its evolution, and therefore can be trapped in the wave potential. Then, to model an ion distribution with trapped particles, we employ a vortex-like ion distribution function of Schamel [24,25], which solves the ion Vlasov equation. In such environment, the ion number density can be expressed as
e −α ψ N i = I (−ψ) + √ erf ( −α ψ ), where
I (−ψ) = 1 − erf ( −ψ ) e −ψ , √
2 erf ( −α ψ ) = √
(3) (4)
where N is the dust particle number density normalized to its equilibrium value nd0 , N i is the ion number density normalized to ni0 , U is the dust fluid velocity normalized to the dustacoustic speed C d = ( Z d T i /md )1/2 , and ψ is the electrostatic wave potential normalized to T i /e, and P is the dust pressure normalized to nd0 T d , where T d is the dust fluid temperature and md is the mass of negatively charged dust particulates. σ = T i / T e , with T i ( T e ) being the ion (electron) temperature, μ0 = β/(1 − β) and μ1 = 1/(1 − β), where β = ne0 /ni0 . The time and space variables are given in the units of the dust plasma 1 period ω− = (md /4π Z d2 ndo e 2 )1/2 and the Debye length λ Dd = pd
( T i /4π Z d ndo e 2 )1/2 , respectively. The model used here is based on the assumption of neglecting some of dust grains which might be trapped due to the finite tem-
−α ψ
e −t dt .
π
2
0
Here α = T i / T it (ratio of the free ion temperature to trapped ion temperature) is the trapping parameter describing the temperature of the trapped ions. Positive values of α , which we are interested in here, lead to a vortex-like distribution for electrons. Note that if we neglect the resonant effects, N i reduces to the Maxwellian ion distribution. To obtain stationary localized solution from the basic equations, we make all the dependent variables depend only on a single variable ξ = x − Mt, where ξ is normalized to λ Dd and M is the Mach number of the solitary waves with respect to the dust-acoustic wave frame for the system. Then, using the steady state condition (∂/∂ t = 0) and the appropriate boundary conditions for localized perturbations, namely, N → 1, U → 0, ψ → 0, P → 1 and dψ/dξ → 0 at ξ → ±∞, in Eqs. (1)–(3), one can express the dust particle number density N as
N=
√
2M M 2 + 3σd + 2ψ
+
M 2 + 3σd + 2ψ
2
− 12σd M 2
− 12
(6)
.
Now using (6), the qualitative nature of the solutions of Eq. (4) is most easily seen by introducing the Sagdeev potential [18]. Therefore, Poisson’s equation (4) reduces to the form
1 2
(2)
(5)
|α |
dψ
2 + V (ψ) = 0,
dξ
(7)
where the Sagdeev potential V (ψ) is defined as
2μ1 μ0 1 σ ψ + μ1 1 − I (−ψ) − √ −ψ V (ψ) = 1−e 1− σ α π μ1 e−α ψ erf ( −α ψ ) − √ α3 √
−
3γ
√
+
1 −1 e 2 cos h (χ /γ ) γ 2 + χ χ − 3/2
2M
1 −1 e 2 cos h (η/γ ) γ 2 + η η − 3/2
2M
3γ
χ2 − γ 2
η2 − γ 2 ,
(8)
with
χ = M 2 + 3σd + 2ψ,
(9)
γ = 2M 3σd ,
(10)
2
(11)
η = M + 3σd .
We note that Eq. (7) can be regarded as an “energy integral” of an oscillating particle of unit mass, with pseudo-speed dψ/dξ ,
H. Alinejad / Physics Letters A 374 (2010) 1855–1859
Fig. 1. Behavior of the Sagdeev potential V (ψ) against the potential ψ and β for σd = 0.01, α = 0.7, σ = 0.05, and M = 1.32.
pseudo-position ψ , pseudo-time ξ and pseudo-potential V (ψ). It is also important to mention that in our √ study the condition for the dust density to be real ψ > −( M − 3σd )2 /2 must always be satisfied. On the other hand, the form of the pseudo-potential could determine whether solitary wave solutions of (7) exist or not. It is clear that V (ψ) = 0 and dV /dξ = 0 at ψ = 0. In this case, solitary wave structure exists if d2 V /dψ 2 0 at ψ = 0, so that the zero as a fixed point is unstable. All the specified conditions are satisfied. Besides that V (ψ) should be negative between ψ = 0 and ψm , where ψm is maximum (or minimum) value of ψ . To investigate the existence regions and nature of the solitary wave excitation in plasma, we have numerically analyzed the general expression (7) for V (ψ) using typical parameters. The results are displayed in Figs. 1–4. These figurers show that the plasma model under consideration permits only rarefactive soliton. In Fig. 1, the dependence of the solitary waves on β have been investigated for σd = 0.01, σ = 0.05, α = 0.7 and M = 1.32. It is obvious that the amplitude increases with increasing the amount of the equilibrium electron density ne0 . This is in good agreement with our earlier work [17] for the dynamics of large amplitude dust-acoustic solitary waves in dusty plasmas whose constituents are cold charged dust fluid, free electrons and ions with excavated trapped distribution. On the other hand, to see what happens when the dust temperature is increased, we study the behavior of the Sagdeev potential and show the effect of dust temperature on the formation of soliton in the case of β = 0.05, σ = 0.05, α = 0.5 and M = 1.34. From Fig. 2, we observe that both the depth of Sagdeev potential curves and amplitude increase remarkably as the dust temperature decreases. This shows that the dust temperature is destructive for the formation of electrostatic localized waves. Similar behavior is also observed in magnetized and unmagnetized dusty plasma with isothermal ions, as shown in Refs. [19,21]. Fig. 3 clearly shows contrast between two types of plasmas in which ions follow the Boltzmann (α = 1) and vortex-like (α = 0.6) distributions. For both the plasmas, solitary waves can exist but the potential wells represented by V (ψ) profiles are of different depths. In the isothermal case, because of the greater depth of Sagdeev potential, the amplitude of solitary wave is larger compared to that for vortex-like ion distributions. It follows that trapped ions introduce some kind of inertia on the propagation of the localized dust-acoustic waves. This effect has also been observed in the study of ion-acoustic solitary waves with trapped electrons, as shown in Refs. [24,25]. Fig. 4 exhibits the effect of trapped ions on the formation of soliton. It is clear that an increase of the trapping parameter α leads to an increase of the potential depth when the Mach number, equilibrium free electron concentration and dust temperature are held constant. This confirms completely the result observed in Fig. 3.
1857
Fig. 2. Behavior of the Sagdeev potential V (ψ) against the potential ψ and dust temperature σd for β = 0.05, α = 0.5, σ = 0.05 and M = 1.34.
Fig. 3. Contrast between the Boltzmann (solid curves) and vortex-like (dotted curves) ion distributions relative to the profile of V (ψ) for M = 1.2, β = 0.04, σ = 0.05, α = 0.6 and σd = 0.02.
Fig. 4. Behavior of the Sagdeev potential V (ψ) against the potential ψ and the trapping parameter α for σd = 0.05, β = 0.05, σ = 0.05 and M = 1.33.
To confirm the possibility of the localized dust-acoustic waves in our plasma model, we investigate the effects of higher-order nonlinearities of the electric potential in the expansion of the Sagdeev potential. The specific results can be obtained by expanding V (ψ) in powers of ψ and keeping up to third-order terms. Accordingly, under the assumption ψ 1, Eq. (7) takes the form
dψ
2
5
= γ1 ψ 2 + γ2 (−ψ) 2 + γ3 ψ 3 ,
dξ
(12)
where
γ1 = μ1 + μ0 σ − γ2 =
16 15
1 M2
√ (1 − α ),
π
− 3σd
,
(13) (14)
1858
H. Alinejad / Physics Letters A 374 (2010) 1855–1859
electron density, as shown in Fig. 1. In contrast to the amplitude variation, Fig. 6 shows that the width of solitary waves increases with the increasing of the dust temperature whereas decreases along with the increase in the concentration of free electrons. This also confirms what has been found in the large amplitude approximation. We now switch our attention on (16) and express two asymptotic cases of stationary solution which are related to the contribution of the resonant ions. We first consider a situation that the coefficient γ2 and γ3 of the nonlinear terms of the electric potential in the expansion of Sagdeev potential (12) are such that γ2 γ3 . Considering this assumption and expanding the righthand side of (16) in powers of 1/γ2 , the stationary solution can be expressed as
Fig. 5. Variation of the maximum amplitude ψm against the dust temperature for α = 0.5, σ = 0.05, M = 1.2, β = 0.06 (dotted) and β = 0.04 (solid).
σd
ψ =−
γ12 2γ3 ξ ξ 1 − 2 cos h sec h2 λ 2λ γ22 γ2
sec h4
ξ . 2λ
(17)
It is clear that the amplitude of the solitary wave modifies when the number of trapped ions increases. In particular case, as the contribution of the resonant ions becomes larger and larger such that we can ignore γ3 in comparison to the resonant term γ2 , Eq. (17) reduces to
ψ = −ψm sec h4
Fig. 6. Variation of the width λ against the dust temperature 0.05, M = 1.2, β = 0.04 (solid) and β = 0.06 (dotted).
γ3 =
M 2 + σd
( M 2 − 3σd )3
+
1 3
μ0 σ 2 − μ1 .
σd for α = 0.5, σ =
(15)
(16)
where λ = 4/γ1 measures the width of dust-acoustic solitary waves. The soliton solution (16) is similar to the exact stationary solution of the generalized KdV equation with the mixed nonlinearity of the electric potential [26]. This solution clearly indicates the existence of solitary waves only with negative potential (rarefactive solitary waves), corresponding to a hump in the dust density. For the stationary solution (16) obtained in the small amplitude limit, variations of the amplitude ψm = ψ (ξ = 0), and the width λ with the dust temperature and the densities ratio of electrons and ions in equilibrium β , have been depicted in Figs. 5 and 6. It can be seen from Fig. 4 that the amplitude decreases with the dust temperature. This confirms completely results observed in Fig. 2 which are obtained for the large amplitude limit. This is also similar to prior results with isothermal ion distribution in magnetized and unmagnetized dusty plasmas [19,21]. The amplitude can be observed that increases with the increase in equilibrium free
(18)
where ψm = (γ1 /γ2 )2 is the amplitude of the solitary wave. The localized solution (18) is similar to the stationary solution of the modified KdV equation for a three components dusty plasma system with cold dust grains, as shown in Ref. [12]. This clearly indicates that the small amplitude solitary wave exists only with negative potential. This also confirms the prior results which are based on the large amplitude limit. On the other hand, for the case that the coefficient γ3 of the nonlinear term is dominant over γ2 in Eq. (12), the expansion of stationary solution (16) in powers of 1/γ3 gives
Here γ2 and γ3 are respectively the coefficients of nonlinear terms of the KdV and modified KdV equations, which can be obtained from the basic equations (1)–(4) using the reductive perturbation theory. Integration of the resulting energy equation (12) with respect ξ along with the appropriate boundary conditions, gives stationary soliton solution as
−2 γ22 γ3 ξ −γ2 ψ =− + + cos h , 2γ 1 γ1 λ 4γ12
ξ , 2λ
γ1 15 2 3 −1/2 ξ 1− γ γ γ3 sec h γ3 8 2 1 λ
γ2 ξ ξ . − 2 1 − 3 sec h2 sec h2 4γ 3 λ λ
ψ =−
(19)
The solution (19) is obtained when the effect of nonlinear interaction of ions with the electrostatic wave potential during its evolution is small. This also indicates the modified amplitude and width of the solitary wave profile from the case that the coefficients γ2 and γ3 are comparable in (16). By neglecting the resonant effect γ2 , we observe that γ3 is similar coefficient of nonlinear term of the KdV equation, which can be obtained from the basic equations by using the reductive perturbation method. In this case, ions follow the isothermal distribution function and this procedure reduces solitary wave solutions as
ψ = −ψm sec h2
ξ , λ
(20)
where the maximum amplitude of solitary waves is ψm = γ1 /γ3 , and the width λ has the same form as before. The solution (12) is similar to the stationary soliton-like solution of the KdV equation for a three-component dusty plasma system with cold dust grains, as shown in Ref. [27]. This also indicates the existence of the small amplitude solitary structure with negative potential. In summary, we have investigated the effects of dust temperature and higher-order nonlinearities on the nonlinear propagation of dust-acoustic waves in three-component dusty plasma whose constituents are warm dust grains, Boltzmann distributed electrons and hot ions obeying vortex-like distribution. Both highly
H. Alinejad / Physics Letters A 374 (2010) 1855–1859
and weakly nonlinear analysis are examined by deriving an energy integral equation involving Sagdeev potential. The effects of the dust temperature, equilibrium free electron density and trapped ions are found to significantly change the regions of the existence of arbitrary amplitude solitary waves. In particular, both of the depth of Sagdeev potential curves and amplitude decrease as the dust temperature increases. If we compare our results with that of isothermal case, we find that the amplitude of soliton in the presence of isothermal ions is larger compared to that for vortexlike ion distributions. It follows that trapped ions introduce some kind of inertia on the propagation of localized electrostatic waves. To examine the possibility of dust-acoustic solitary waves in the small amplitude limit, we have investigated the effects of higherorder nonlinearities of the electric potential in the expansion of the Sagdeev potential and integrated the resulting energy equation with the proper boundary condition required for a solitary wave structure. The dependence of the width and amplitude of solitary wave solution on the dust temperature and trapped ions are studied. We have also investigated two asymptotic cases of the stationary soliton-like solution which are related to the population of resonant particles present. Finally, the results yield an improved understanding of the propagation of dust-acoustic waves in an unmagnetized warm dusty plasma. It is shown from both highly and weakly nonlinear analysis that the plasma system under consideration supports only rarefactive solitary waves. The results of the present investigation should also help us to explain the basic features of localized dust-acoustic perturbations propagating in the space and laboratory dusty plasmas, where negatively charged warm dust particulates, thermally distributed electrons and resonant ions are the major plasma species.
1859
Acknowledgement This work has been supported financially by Research Institute for Astronomy and Astrophysics of Maragha. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
M. Horanyi, D.A. Mendis, Astrophys. J. 307 (1986) 800. T.G. Northrop, Phys. Scripta 45 (1992) 475. D.A. Mendis, M. Rosenberg, Rev. Astron. Astrophys. 32 (1994) 419. F. Verheest, Space Sci. Rev. 77 (1996) 267. W.F. El-Taibany, R. Sabry, Phys. Plasmas 12 (2005) 082302. W.M. Moslem, W.F. El-Taibany, Phys. Plasmas 12 (2005) 122309. A.A. Mamun, Phys. Lett. A 372 (2008) 4610. M. Tribeche, G. Boumezound, Phys. Plasmas 15 (2008) 053702. N.N. Rao, P.K. Shukla, M.Y. Yu, Planet. Space Sci. 38 (1990) 543. A. Barkan, R.L. Merlino, D. D’Angelo, Phys. Plasmas 2 (1995) 3563. D. Winske, S.P. Gary, M.E. Jones, M. Rosenberg, D.A. Mendis, V.W. Chow, Geophys. Res. Lett. 22 (1995) 2069. A.A. Mamun, J. Plasma Phys. 59 (1998) 575. A.A. Mamun, R.A. Cairns, P.K. Shukla, Phys. Plasmas 3 (1996) 2610. W.S. Duan, Phys. Lett. A 317 (2003) 275. A.A. Mamun, M.A. Hassann, J. Plasma Phys. 63 (2000) 191. M. Kakti, K.S. Goswami, Phys. Plasmas 5 (1998) 4508. H. Alinejad, Phys. Lett. A 37 (2009) 2935. R.Z. Sagdeev, in: Review of Plasma Physics, vol. 4, Consultants Bureau, New York, 1966, p. 23. R. Roychoudhury, S. Mukherjee, Phys. Plasmas 4 (1997) 2305. N. Akhtar, S. Mahmood, H. Saleem, Phys. Lett. A 361 (2007) 126. S. Mahmood, N. Akhtar, H. Saleem, Phys. Plasmas 14 (2007) 074507. A. Mendoza, S.M. Russel, A.A. Mamun, Planet. Space Sci. 48 (2000) 599. A. Das, B. Bandyopadhyay, K.P. Das, Phys. Plasmas 16 (2009) 073703. H. Schamel, J. Plasma Phys. 14 (1972) 905. H. Schamel, J. Plasma Phys. 9 (1973) 377. S.G. Tagare, A. Chakrabarti, Phys. Fluids 17 (1974) 1331. A.A. Mamun, Astrophys. Space Sci. 268 (1999) 448.