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Nonlinear evolution of ion-acoustic waves in unmagnetized plasma
Physics Letters A 305 (2002) 393–398 www.elsevier.com/locate/pla
Nonlinear evolution of ion-acoustic waves in unmagnetized plasma Nikhil Chakrabarti ∗ , M.S. Janaki Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700 064, India Received 4 July 2002; received in revised form 25 October 2002; accepted 28 October 2002 Communicated by F. Porcelli
Abstract In a fluid description large amplitude electrostatic ion-acoustic waves have been studied in an unmagnetized plasma using Lagrangian variables. We obtained solutions for ion-acoustic waves with nontrivial space and time dependence. The nondispersive solutions demonstrate that under well defined initial and boundary conditions the amplitude of the solutions decreases indicating a new class of nonlinear solutions that lead to short lived structures. 2002 Elsevier Science B.V. All rights reserved.
The problem of nonlinear electron plasma oscillations was deeply investigated by various authors independently some time ago [1–3]. The analysis was based on the introduction of Lagrangian variables which follow the plasma fluid. In this Letter we wish to apply the same technique to solve nonlinear acoustic like system of equations in an unmagnetized plasma. A particular example of such Lagrangian to Eulerian inversion has been given explicitly in Davidson’s book [4]. There are lot of works on acoustic like waves existing in the literature, specially nonlinear solution of the relevant equations. Most of the well known nonlinear solutions have, however, been obtained either in small amplitude limit or by assuming
* Corresponding author.
E-mail address: [email protected] (N. Chakrabarti).
stationarity in the moving frame [5]. That is, though the original equation involved x and t dependence, these solutions are functions of x − ut only, where u is the constant velocity of the moving frame. This is rather a drastic assumption and if found to be stable, the solutions can be candidates to represent the final state reached by a plasma evolving from some unstable but simpler state. Thus such solutions could well arise in many experimental situations. However, the method can say nothing about the ‘dynamics’, that is the time evolution of the system. In particular, it can give no information about the initial state from which the stationary mode evolved. Thus any method which allows one to study the time evolution would be extremely useful. One such method is based on the introduction of Lagrangian variables [6,7]. In this Letter we present the analytic time dependent solution to the relevant fully nonlinear ion-acoustic wave equations. An explicit solution for the number density as a function of
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 1 5 0 1 - 3
394
N. Chakrabarti, M.S. Janaki / Physics Letters A 305 (2002) 393–398
space and time is obtained in Eulerian variables also. It is generally believed that nonlinearity leads to unlimited steepening of an initial disturbance until breaking occurs [8]. When nonlinearity keeps transferring energy to these nonlinear waves the spatial scale becomes very short and therefore it is not justified to neglect the spatial gradient, i.e., the dispersive effects. It is the dispersive effect which may eventually limit the nonlinear steepening by building up short scale disturbance. In contrary to the usual belief we find that in the absence of linear dispersion and presence of nonlinear effects there is no evidence of amplification of the initial disturbance but the solutions exhibit a strong dispersive character. We speculate that nonlinearity can act like a source of dispersion for such class of nonlinear solutions. Unfortunately, though extremely powerful in giving explicit solutions to the time dependent problem, it can only be used in few simple cases. According to our knowledge we are applying this method for the first time in case of ion acoustic waves. The availability of high powered laser and various ion accelerator concepts based on laser plasma interaction has got much attention in the study of strongly nonlinear waves [9]. Particularly experimentalists are interested to know whether wave breaking/collapse is expected and if so, what are the critical amplitudes for the density, electric field etc., in the particular experimental set up. Lagrangian method is sometimes very useful to analyze such situations [10]. Another important point is that the method we adopted may be applicable to a general class of nonlinear system of equations. For example, we apply the same technique in dusty plasma physics, especially, nonlinear dust-acoustic system of equations [11]. This method is extremely useful in beam physics particularly in the formation of nonuniform structures in charged particle beams as studied recently [12]. Development of instability and wave breaking limit was also studied by this method [13]. Thus our motivation is manifold both theoretically and experimentally. The basic equations describing electrostatic ionacoustic disturbance in an unmagnetized plasma having warm electrons and cold ions may be written as ∂ ∂n + (nv) = 0, ∂t ∂x ∂v ∂v ∂φ +v =− , ∂t ∂x ∂x
(1)
(2)
2
∂ 2φ φ = e −n . 2 ∂x
(3)
The above three equations are the continuity, momentum and Poisson’s equations; = λD /L where λD is the Debye length (λ2D = Te /4πn0 e2 ). The electron inertia effects are neglected and the isothermal equation of state is adopted in the equation of motion for electron giving Boltzmann relation which is used in Poisson’s equation (Eq. (3)). These equations are all in terms of dimensionless variables such that density is normalized by constant density amplitude n0 . The length scale is the arbitrary length L, and the time scale is inverse of L/Cs (Cs = (Te /mi )1/2 , is the sound speed), the electrostatic potential is normalized by (Te /e). Now we proceed to find an exact solution of Eqs. (1), (2) and (3) in Lagrangian variables. In solving these equations we transform from Eulerian variables (x, t) to Lagrangian variables (ξ, τ ) (such that ξ = x at t = 0) where τ τ ≡ t,
ξ ≡x−
dτ v ξ, τ ,
(4)
0
so that ξ is a function of both x and t, but ξ and τ are treated as independent variables. In terms of these new variables the convective derivative ∂/∂t + v∂/∂x becomes ∂/∂τ and following Ref. [2], we find from Eq. (1) n(ξ, τ ) = n0 (ξ, 0) 1 + ⇒
τ
∂ vˆ ξ, τ dτ ∂ξ
−1
∂ξ n(ξ, τ ) = , n0 (ξ, 0) ∂x
(5)
where n0 (ξ, 0) represents the initial (τ = 0) density distribution in space and the corresponding fluid equations are n2 ∂v ∂n + = 0, ∂τ n0 (ξ, 0) ∂ξ
(6)
∂v n ∂φ + = 0, ∂τ n0 (ξ, 0) ∂ξ
(7)
∂ ∂φ n n = eφ − n . n0 (ξ, 0) ∂ξ n0 (ξ, 0) ∂ξ
(8)
2
N. Chakrabarti, M.S. Janaki / Physics Letters A 305 (2002) 393–398
The above three nonlinear differential equations may be combined to a single differential equation for n ∂2 1 ∂τ 2 n ∂ 1 =− n0 (ξ, 0) ∂ξ
∂ n ∂2 1 ln n − 2 n 2 . × (9) n0 (ξ, 0) ∂ξ ∂τ n The nonlinear equations in three variables are reduced to a single nonlinear equation obviously vindicating the introduction of the Lagrangian scheme. However, the rather complex full form of the above equation is undoubtedly difficult to solve analytically. Fortunately, however, in absence of linear dispersion ( = 0) the equation may be solved analytically. This physically means that we are considering the spatial scale length to be larger than the electron Debye length. In this limit the plasma we are dealing with is totally quasineutral. Before discussing the solution of Eq. (9), we analyze its rich content in various limits. The first interesting limit is reached when kλD 1; we then find ion plasma waves oscillating at the ion plasma frequency. This is because the wavelength is short compared to electron Debye length, the electrons are incapable of shielding, and we have ions oscillating in a uniform background of negative charge. This is quite analogous to electron plasma oscillation in a uniform background positive charge as studied by Davidson and Schram [2]. In the limit kλD 1 since L is smaller than λD , is large and consequently Eq. (8) may be approximated as n ∂ n ∂φ 2 (10) ≈ (1 − n). n0 (ξ, 0) ∂ξ n0 (ξ, 0) ∂ξ The density equation in this case has a simplified form ∂2 1 1 2 − 1 + ω − 1 = 0, (11) pi ∂τ 2 n n where ωpi = (4πn0 e2 /mi )1/2 is the ion plasma frequency. Following Davidson and Schram [2] similar analysis may be carried out for ion plasma oscillation as has been done for electron plasma oscillation. Secondly, as it is well known that weakly nonlinear ion sound wave in an unmagnetized homogeneous
395
plasma background can be described by Korteweg– de Vries (KdV) equation, the same should be recovered from the most general equation (Eq. (9)). It is indeed the case and we are able to extract the KdV equation in homogeneous equilibrium [n0 (ξ, 0) = 1] and small amplitude limit [n = 1 + n], ˜ n˜ 1. In the following only up to second order terms are considered so that we do not loose the leading order nonlinear effect which leads to the steepening of the initial disturbance. We believe that the approximate method outlined below to obtain KdV equation is much more straightforward than reductive perturbative method which is described in most of the textbooks [14]. Using ln n ≈ n˜ in Eq. (9) and keeping up to second order term we have 2 ∂2 ∂ n˜ − ∂τ 2 ∂ξ 2 ∂ ∂ 2 ∂ 2 n˜ ∂ n˜ = (12) . n˜ + 2 2 ∂ξ ∂ξ ∂ξ ∂τ 2 In absence of nonlinearity and dispersion the above equation represents the linear ion-acoustic waves moving in the positive and negative x direction. Now if we are interested in waves moving in positive x direction then for these waves ∂ n/∂ξ ˜ = −∂ n/∂τ ˜ . This relationship is approximately valid for weakly nonlinear case [15]. Replacing ∂/∂τ by −∂/∂ξ and making further transformation ξ¯ = −1 (ξ − τ ), τ¯ = (2 )−1 τ , we recover KdV equation exactly: ∂ n˜ ∂ 2 n˜ ∂ n˜ + n˜ + 2 = 0. ∂ τ¯ ∂ ξ¯ ∂ ξ¯
(13)
The above equation has well known soliton solutions. The general equation (Eq. (9)) given in this Letter not only reproduces the old results, but it can go beyond and give a novel solution for a nondispersive full nonlinear equation. To demonstrate this we take exact neutrality condition, i.e., we drop the dispersion term proportional to 2 in Eq. (9). Therefore the nonlinear equation we have to solve is
∂2 1 1 ∂ n ∂ ln n . (14) = − ∂τ 2 n n0 (ξ, 0) ∂ξ n0 (ξ, 0) ∂ξ We may now present the solutions for Eq. (14) which can be solved analytically by the method of separation
396
N. Chakrabarti, M.S. Janaki / Physics Letters A 305 (2002) 393–398
of variables. Proposing n(ξ, τ ) = P (τ )Q(ξ ),
(15)
we may substitute n(ξ, τ ) in Eq. (14) and separate it into two equations in space and time variables as d2 ln Q = −Γ 2 , dξ 2
(16)
1 d2 1 = Γ 2, P dτ 2 P
(17)
where Γ is an arbitrary separation constant, P = P (τ )/P (0) and we have used the relation n(ξ, 0) = P (0)Q(ξ ). From Eq. (16) together with the boundary condition, let us say at ξ = 0, Q(0) = 1/P (0) and (dQ/dξ )ξ =0 = 0. These conditions determine the spatial profile of the density as Γ 2ξ 2 1 . exp − Q(ξ ) = (18) P (0) 2 To obtain the evolution of this profile, next we solve Eq. (17). Before solving Eq. (17) let us examine whether the solution obtained in this method is consistent with the conservation properties. Since we do not have source and sink in the continuity equation, density should be conserved. If we take solution of Eq. (17) to be some unknown function of τ , say f (τ ), then density evolution solution may be written as Γ 2ξ 2 , n(ξ, τ ) = n0 f (τ ) exp − (19) 2 where n0 is the constant background density amplitude. For the present one dimensional
∞ case, conservation of density may be written as −∞ n(x, t) dx = n0 . Substituting the solution from Eq. (19) and performing the spatial integration the condition on f (τ ) shows that it exactly √ satisfies the differential equation (17) provided Γ = 2π. Therefore, the solutions are consistent with the conservation properties. This gives us confidence to find out the explicit temporal solution of Eq. (17). However, we can solve this nonlinear differential equation analytically exactly with the initial /dτ )0 = 0. The solution conditions P (0) = 1 and (d P is obtained in terms of error functions and is given by Erf ln P = 2iτ. (20)
We can plot this transcendental equation numerically which shows that P (τ ) decays with time. However, to give a closed form of the solution we use the approximation ln(1/P ) 1/P , the approximate solution of Eq. (17) is given by P (τ ) ≈
P (0) √ . 1+2 π τ
(21)
We also find that this approximate solution fairly matches with the exact solution given in Eq. (20). Therefore, the complete solution for the density may be written as n0 √ exp −πξ 2 , n(ξ, τ ) = (22) 1+2 π τ where n0 is a constant number. From the above solution we notice immediately that the spatial density distribution is Gaussian in nature and undergoes amplitude decay in time. This implies that as the density diminishes the scale in x expands such that the mass conservation holds. For τ → ∞ the exact solution becomes self-similar as we have seen from the conservation property before. To get some insight of the obtained solution: in the absence of the nonlinearity we expect the density disturbance in ion-acoustic wave to propagate as a wave in both positive and negative ξ directions with amplitude finally becoming small everywhere. Due to the presence of nonlinear effects we see in our solution that there is no evidence of propagation in ξ because the disturbance continues to be in the same region of ξ but its amplitude decays from the maximum value. We speculate that this is a kind of nonlinear dispersive ion-acoustic wave in absence of linear dispersion. We must emphasize that the solution (22) is not applicable to very short spatial scales since the quasi-neutrality condition used is then violated. Once density is known we can easily find out the fluid velocity assuming v(ξ, 0) = 0. Now substituting v(ξ, τ ) in Eq. (4) we may find the relation between ξ and x. Thus, the transformation from Lagrangian to Eulerian variables, i.e., determination of ξ as a function of x and t results in the following relation x . ξ= √ √ √ 1 1 + 2 {(1 + 2 π τ ) ln(1 + 2 π τ ) − 2 π τ } (23) In view of the above relation it is easy to express ξ in terms of original variable x. Now we can plot the
N. Chakrabarti, M.S. Janaki / Physics Letters A 305 (2002) 393–398
Fig. 1. Density distribution in Lagrangian space variable with time as a parameter.
397
forward to calculate using the given solution Eq. (22) but this is complicated to warrant writing out. The result indicates that corrections due to linear dispersion only add on to the nonlinear dispersive effect so that we see strong dispersive effect particularly towards short scale-lengths. In conclusion, we emphasize that the analysis of the ion-acoustic wave has been oriented towards simple macroscopic situation in which nonlinear time dependent processes may be described analytically to a large extent. In contrary to the usual belief, the presence of nonlinearity in the equation reflects as a nonlinear dispersive effect rather that wave steepening in the solution. As the problem is too complex in its full generality (for example, in the Poisson’s equation we have ignored linear dispersive term to its full generality and we have not introduced spatial variations in more than one dimension etc.), the time dependent analysis diminishes considerably in mathematical tractability. It should be noted, however, that the large amplitude traveling wave solutions which are distortion-less in some frame of reference are often obtainable from nonlinear one dimension plasma models where the general time dependent solutions are only few. This type of solution represents a class of nonlinear solutions that may arise in many such similar physical situations. To the best of our knowledge such time dependent solutions in ion acoustic wave have not been studied before. Full numerical solution of Eq. (9) still eludes us and we hope to report this in future. We believe that more investigations in this direction will enrich our understanding of 1D plasma dynamics.
Fig. 2. Density distribution in real space variable with time as a parameter.
Acknowledgements solution given by Eq. (22) in terms of x, t, which shows dispersion in the spatial profile as time progresses. In Figs. 1 and 2 we show the spatial density distribution with respect to ξ and x variables for different times. Note here that in this solution we have ignored the linear dispersive effects but still dispersive behavior of the nonlinear solution may be the manifestation of ‘nonlinear dispersion’ in the solution. To see the linear dispersive effects on this solution we solve Eq. (9) perturbatively taking 2 as a smallness parameter. The perturbative solutions are straight-
We are thankful to Profs. P.K. Kaw, H. Ramachandran, P.N. Guzdar, G. Ganguly and Drs. S. Sengupta and S. Maiti for various fruitful discussions.
References [1] [2] [3] [4]
J.M. Dawson, Phys. Rev. 113 (1959) 383. R.C. Davidson, P.P. Schram, Nucl. Fusion 8 (1968) 183. E. Infeld, G. Rowlands, Phys. Rev. Lett. 58 (1987) 2063. R.C. Davidson, Methods in Nonlinear Plasma Theory, Academic Press, New York, 1972.
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[5] [6] [7] [8] [9] [10] [11]
N. Chakrabarti, M.S. Janaki / Physics Letters A 305 (2002) 393–398
H. Washimi, T. Taniuti, Phys. Rev. Lett. 17 (1966) 966. H. Schamel, A.R. Karimov, Phys. Plasmas 7 (2000) 2790. A.R. Karimov, H. Schamel, Phys. Plasmas 8 (2001) 1180. D.C. Montgomery, Phys. Rev. Lett. 19 (1967) 1465. B. Rau, T. Tajima, Phys. Plasmas 5 (1998) 3575. W.L. Kruer, Phys. Fluids 22 (1979) 1111. N.N. Rao, P.K. Shukla, M.Y. Yu, Planet. Space Sci. 38 (1990) 543. [12] A.R. Karimov, Physica D 102 (1997) 328.
[13] A.A. Galeev, R.Z. Sagdeev, V.I. Shevchenko, Sov. Phys. JETP 54 (1981) 306. [14] V.I. Karpman, Nonlinear Waves in Dispersive Media, Pergamon, Oxford, 1975; E. Infeld, G. Rowlands, Nonlinear Waves, Solitons and Chaos, Cambridge Univ. Press, Cambridge, 1990. [15] Yu.A. Berezin, V.I. Karpman, Sov. Phys. JETP 19 (1964) 1265.
Nonlinear evolution of ion-acoustic waves in unmagnetized plasma Nikhil Chakrabarti ∗ , M.S. Janaki Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700 064, India Received 4 July 2002; received in revised form 25 October 2002; accepted 28 October 2002 Communicated by F. Porcelli
Abstract In a fluid description large amplitude electrostatic ion-acoustic waves have been studied in an unmagnetized plasma using Lagrangian variables. We obtained solutions for ion-acoustic waves with nontrivial space and time dependence. The nondispersive solutions demonstrate that under well defined initial and boundary conditions the amplitude of the solutions decreases indicating a new class of nonlinear solutions that lead to short lived structures. 2002 Elsevier Science B.V. All rights reserved.
The problem of nonlinear electron plasma oscillations was deeply investigated by various authors independently some time ago [1–3]. The analysis was based on the introduction of Lagrangian variables which follow the plasma fluid. In this Letter we wish to apply the same technique to solve nonlinear acoustic like system of equations in an unmagnetized plasma. A particular example of such Lagrangian to Eulerian inversion has been given explicitly in Davidson’s book [4]. There are lot of works on acoustic like waves existing in the literature, specially nonlinear solution of the relevant equations. Most of the well known nonlinear solutions have, however, been obtained either in small amplitude limit or by assuming
* Corresponding author.
E-mail address: [email protected] (N. Chakrabarti).
stationarity in the moving frame [5]. That is, though the original equation involved x and t dependence, these solutions are functions of x − ut only, where u is the constant velocity of the moving frame. This is rather a drastic assumption and if found to be stable, the solutions can be candidates to represent the final state reached by a plasma evolving from some unstable but simpler state. Thus such solutions could well arise in many experimental situations. However, the method can say nothing about the ‘dynamics’, that is the time evolution of the system. In particular, it can give no information about the initial state from which the stationary mode evolved. Thus any method which allows one to study the time evolution would be extremely useful. One such method is based on the introduction of Lagrangian variables [6,7]. In this Letter we present the analytic time dependent solution to the relevant fully nonlinear ion-acoustic wave equations. An explicit solution for the number density as a function of
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 1 5 0 1 - 3
394
N. Chakrabarti, M.S. Janaki / Physics Letters A 305 (2002) 393–398
space and time is obtained in Eulerian variables also. It is generally believed that nonlinearity leads to unlimited steepening of an initial disturbance until breaking occurs [8]. When nonlinearity keeps transferring energy to these nonlinear waves the spatial scale becomes very short and therefore it is not justified to neglect the spatial gradient, i.e., the dispersive effects. It is the dispersive effect which may eventually limit the nonlinear steepening by building up short scale disturbance. In contrary to the usual belief we find that in the absence of linear dispersion and presence of nonlinear effects there is no evidence of amplification of the initial disturbance but the solutions exhibit a strong dispersive character. We speculate that nonlinearity can act like a source of dispersion for such class of nonlinear solutions. Unfortunately, though extremely powerful in giving explicit solutions to the time dependent problem, it can only be used in few simple cases. According to our knowledge we are applying this method for the first time in case of ion acoustic waves. The availability of high powered laser and various ion accelerator concepts based on laser plasma interaction has got much attention in the study of strongly nonlinear waves [9]. Particularly experimentalists are interested to know whether wave breaking/collapse is expected and if so, what are the critical amplitudes for the density, electric field etc., in the particular experimental set up. Lagrangian method is sometimes very useful to analyze such situations [10]. Another important point is that the method we adopted may be applicable to a general class of nonlinear system of equations. For example, we apply the same technique in dusty plasma physics, especially, nonlinear dust-acoustic system of equations [11]. This method is extremely useful in beam physics particularly in the formation of nonuniform structures in charged particle beams as studied recently [12]. Development of instability and wave breaking limit was also studied by this method [13]. Thus our motivation is manifold both theoretically and experimentally. The basic equations describing electrostatic ionacoustic disturbance in an unmagnetized plasma having warm electrons and cold ions may be written as ∂ ∂n + (nv) = 0, ∂t ∂x ∂v ∂v ∂φ +v =− , ∂t ∂x ∂x
(1)
(2)
2
∂ 2φ φ = e −n . 2 ∂x
(3)
The above three equations are the continuity, momentum and Poisson’s equations; = λD /L where λD is the Debye length (λ2D = Te /4πn0 e2 ). The electron inertia effects are neglected and the isothermal equation of state is adopted in the equation of motion for electron giving Boltzmann relation which is used in Poisson’s equation (Eq. (3)). These equations are all in terms of dimensionless variables such that density is normalized by constant density amplitude n0 . The length scale is the arbitrary length L, and the time scale is inverse of L/Cs (Cs = (Te /mi )1/2 , is the sound speed), the electrostatic potential is normalized by (Te /e). Now we proceed to find an exact solution of Eqs. (1), (2) and (3) in Lagrangian variables. In solving these equations we transform from Eulerian variables (x, t) to Lagrangian variables (ξ, τ ) (such that ξ = x at t = 0) where τ τ ≡ t,
ξ ≡x−
dτ v ξ, τ ,
(4)
0
so that ξ is a function of both x and t, but ξ and τ are treated as independent variables. In terms of these new variables the convective derivative ∂/∂t + v∂/∂x becomes ∂/∂τ and following Ref. [2], we find from Eq. (1) n(ξ, τ ) = n0 (ξ, 0) 1 + ⇒
τ
∂ vˆ ξ, τ dτ ∂ξ
−1
∂ξ n(ξ, τ ) = , n0 (ξ, 0) ∂x
(5)
where n0 (ξ, 0) represents the initial (τ = 0) density distribution in space and the corresponding fluid equations are n2 ∂v ∂n + = 0, ∂τ n0 (ξ, 0) ∂ξ
(6)
∂v n ∂φ + = 0, ∂τ n0 (ξ, 0) ∂ξ
(7)
∂ ∂φ n n = eφ − n . n0 (ξ, 0) ∂ξ n0 (ξ, 0) ∂ξ
(8)
2
N. Chakrabarti, M.S. Janaki / Physics Letters A 305 (2002) 393–398
The above three nonlinear differential equations may be combined to a single differential equation for n ∂2 1 ∂τ 2 n ∂ 1 =− n0 (ξ, 0) ∂ξ
∂ n ∂2 1 ln n − 2 n 2 . × (9) n0 (ξ, 0) ∂ξ ∂τ n The nonlinear equations in three variables are reduced to a single nonlinear equation obviously vindicating the introduction of the Lagrangian scheme. However, the rather complex full form of the above equation is undoubtedly difficult to solve analytically. Fortunately, however, in absence of linear dispersion ( = 0) the equation may be solved analytically. This physically means that we are considering the spatial scale length to be larger than the electron Debye length. In this limit the plasma we are dealing with is totally quasineutral. Before discussing the solution of Eq. (9), we analyze its rich content in various limits. The first interesting limit is reached when kλD 1; we then find ion plasma waves oscillating at the ion plasma frequency. This is because the wavelength is short compared to electron Debye length, the electrons are incapable of shielding, and we have ions oscillating in a uniform background of negative charge. This is quite analogous to electron plasma oscillation in a uniform background positive charge as studied by Davidson and Schram [2]. In the limit kλD 1 since L is smaller than λD , is large and consequently Eq. (8) may be approximated as n ∂ n ∂φ 2 (10) ≈ (1 − n). n0 (ξ, 0) ∂ξ n0 (ξ, 0) ∂ξ The density equation in this case has a simplified form ∂2 1 1 2 − 1 + ω − 1 = 0, (11) pi ∂τ 2 n n where ωpi = (4πn0 e2 /mi )1/2 is the ion plasma frequency. Following Davidson and Schram [2] similar analysis may be carried out for ion plasma oscillation as has been done for electron plasma oscillation. Secondly, as it is well known that weakly nonlinear ion sound wave in an unmagnetized homogeneous
395
plasma background can be described by Korteweg– de Vries (KdV) equation, the same should be recovered from the most general equation (Eq. (9)). It is indeed the case and we are able to extract the KdV equation in homogeneous equilibrium [n0 (ξ, 0) = 1] and small amplitude limit [n = 1 + n], ˜ n˜ 1. In the following only up to second order terms are considered so that we do not loose the leading order nonlinear effect which leads to the steepening of the initial disturbance. We believe that the approximate method outlined below to obtain KdV equation is much more straightforward than reductive perturbative method which is described in most of the textbooks [14]. Using ln n ≈ n˜ in Eq. (9) and keeping up to second order term we have 2 ∂2 ∂ n˜ − ∂τ 2 ∂ξ 2 ∂ ∂ 2 ∂ 2 n˜ ∂ n˜ = (12) . n˜ + 2 2 ∂ξ ∂ξ ∂ξ ∂τ 2 In absence of nonlinearity and dispersion the above equation represents the linear ion-acoustic waves moving in the positive and negative x direction. Now if we are interested in waves moving in positive x direction then for these waves ∂ n/∂ξ ˜ = −∂ n/∂τ ˜ . This relationship is approximately valid for weakly nonlinear case [15]. Replacing ∂/∂τ by −∂/∂ξ and making further transformation ξ¯ = −1 (ξ − τ ), τ¯ = (2 )−1 τ , we recover KdV equation exactly: ∂ n˜ ∂ 2 n˜ ∂ n˜ + n˜ + 2 = 0. ∂ τ¯ ∂ ξ¯ ∂ ξ¯
(13)
The above equation has well known soliton solutions. The general equation (Eq. (9)) given in this Letter not only reproduces the old results, but it can go beyond and give a novel solution for a nondispersive full nonlinear equation. To demonstrate this we take exact neutrality condition, i.e., we drop the dispersion term proportional to 2 in Eq. (9). Therefore the nonlinear equation we have to solve is
∂2 1 1 ∂ n ∂ ln n . (14) = − ∂τ 2 n n0 (ξ, 0) ∂ξ n0 (ξ, 0) ∂ξ We may now present the solutions for Eq. (14) which can be solved analytically by the method of separation
396
N. Chakrabarti, M.S. Janaki / Physics Letters A 305 (2002) 393–398
of variables. Proposing n(ξ, τ ) = P (τ )Q(ξ ),
(15)
we may substitute n(ξ, τ ) in Eq. (14) and separate it into two equations in space and time variables as d2 ln Q = −Γ 2 , dξ 2
(16)
1 d2 1 = Γ 2, P dτ 2 P
(17)
where Γ is an arbitrary separation constant, P = P (τ )/P (0) and we have used the relation n(ξ, 0) = P (0)Q(ξ ). From Eq. (16) together with the boundary condition, let us say at ξ = 0, Q(0) = 1/P (0) and (dQ/dξ )ξ =0 = 0. These conditions determine the spatial profile of the density as Γ 2ξ 2 1 . exp − Q(ξ ) = (18) P (0) 2 To obtain the evolution of this profile, next we solve Eq. (17). Before solving Eq. (17) let us examine whether the solution obtained in this method is consistent with the conservation properties. Since we do not have source and sink in the continuity equation, density should be conserved. If we take solution of Eq. (17) to be some unknown function of τ , say f (τ ), then density evolution solution may be written as Γ 2ξ 2 , n(ξ, τ ) = n0 f (τ ) exp − (19) 2 where n0 is the constant background density amplitude. For the present one dimensional
∞ case, conservation of density may be written as −∞ n(x, t) dx = n0 . Substituting the solution from Eq. (19) and performing the spatial integration the condition on f (τ ) shows that it exactly √ satisfies the differential equation (17) provided Γ = 2π. Therefore, the solutions are consistent with the conservation properties. This gives us confidence to find out the explicit temporal solution of Eq. (17). However, we can solve this nonlinear differential equation analytically exactly with the initial /dτ )0 = 0. The solution conditions P (0) = 1 and (d P is obtained in terms of error functions and is given by Erf ln P = 2iτ. (20)
We can plot this transcendental equation numerically which shows that P (τ ) decays with time. However, to give a closed form of the solution we use the approximation ln(1/P ) 1/P , the approximate solution of Eq. (17) is given by P (τ ) ≈
P (0) √ . 1+2 π τ
(21)
We also find that this approximate solution fairly matches with the exact solution given in Eq. (20). Therefore, the complete solution for the density may be written as n0 √ exp −πξ 2 , n(ξ, τ ) = (22) 1+2 π τ where n0 is a constant number. From the above solution we notice immediately that the spatial density distribution is Gaussian in nature and undergoes amplitude decay in time. This implies that as the density diminishes the scale in x expands such that the mass conservation holds. For τ → ∞ the exact solution becomes self-similar as we have seen from the conservation property before. To get some insight of the obtained solution: in the absence of the nonlinearity we expect the density disturbance in ion-acoustic wave to propagate as a wave in both positive and negative ξ directions with amplitude finally becoming small everywhere. Due to the presence of nonlinear effects we see in our solution that there is no evidence of propagation in ξ because the disturbance continues to be in the same region of ξ but its amplitude decays from the maximum value. We speculate that this is a kind of nonlinear dispersive ion-acoustic wave in absence of linear dispersion. We must emphasize that the solution (22) is not applicable to very short spatial scales since the quasi-neutrality condition used is then violated. Once density is known we can easily find out the fluid velocity assuming v(ξ, 0) = 0. Now substituting v(ξ, τ ) in Eq. (4) we may find the relation between ξ and x. Thus, the transformation from Lagrangian to Eulerian variables, i.e., determination of ξ as a function of x and t results in the following relation x . ξ= √ √ √ 1 1 + 2 {(1 + 2 π τ ) ln(1 + 2 π τ ) − 2 π τ } (23) In view of the above relation it is easy to express ξ in terms of original variable x. Now we can plot the
N. Chakrabarti, M.S. Janaki / Physics Letters A 305 (2002) 393–398
Fig. 1. Density distribution in Lagrangian space variable with time as a parameter.
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forward to calculate using the given solution Eq. (22) but this is complicated to warrant writing out. The result indicates that corrections due to linear dispersion only add on to the nonlinear dispersive effect so that we see strong dispersive effect particularly towards short scale-lengths. In conclusion, we emphasize that the analysis of the ion-acoustic wave has been oriented towards simple macroscopic situation in which nonlinear time dependent processes may be described analytically to a large extent. In contrary to the usual belief, the presence of nonlinearity in the equation reflects as a nonlinear dispersive effect rather that wave steepening in the solution. As the problem is too complex in its full generality (for example, in the Poisson’s equation we have ignored linear dispersive term to its full generality and we have not introduced spatial variations in more than one dimension etc.), the time dependent analysis diminishes considerably in mathematical tractability. It should be noted, however, that the large amplitude traveling wave solutions which are distortion-less in some frame of reference are often obtainable from nonlinear one dimension plasma models where the general time dependent solutions are only few. This type of solution represents a class of nonlinear solutions that may arise in many such similar physical situations. To the best of our knowledge such time dependent solutions in ion acoustic wave have not been studied before. Full numerical solution of Eq. (9) still eludes us and we hope to report this in future. We believe that more investigations in this direction will enrich our understanding of 1D plasma dynamics.
Fig. 2. Density distribution in real space variable with time as a parameter.
Acknowledgements solution given by Eq. (22) in terms of x, t, which shows dispersion in the spatial profile as time progresses. In Figs. 1 and 2 we show the spatial density distribution with respect to ξ and x variables for different times. Note here that in this solution we have ignored the linear dispersive effects but still dispersive behavior of the nonlinear solution may be the manifestation of ‘nonlinear dispersion’ in the solution. To see the linear dispersive effects on this solution we solve Eq. (9) perturbatively taking 2 as a smallness parameter. The perturbative solutions are straight-
We are thankful to Profs. P.K. Kaw, H. Ramachandran, P.N. Guzdar, G. Ganguly and Drs. S. Sengupta and S. Maiti for various fruitful discussions.
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