Time-Dependent Nonplanar Dust-Ion-Acoustic Gardner Double Layers

A theoretical investigation has been made on the nonplanar (cylindrical and spherical) dust-ion-acoustic (DIA) double layers (DLs) in a dusty plasma system, containing inertial ions, Boltzmann electrons, and negatively charged stationary dust. In this investigation, in order to analyze the time dependent nonplanar DIA DLs, we have used the modified Gardner equation, which has been obtained by employing the reductive perturbation method. It has been found that the behaviors of DIA DLs have been significantly modified by the time period and the nonplanar geometry. The nonplanar DIA DLs has been found to be similar with planar DIA DLs only at large time scale and the cylindrical DIA DLs have been found to be smaller than the spherical DLs, but larger than the planar DLs.


Introduction
The existence of novel dust-ion-acoustic (DIA) waves was first predicted by Shukla and Silin [1] about twenty years ago.Nearly four years later, the prediction of Shukla and Silin [1] was conclusively verified by a laboratory experiment of Barkan et al. [2].Then the linear features of the DIA waves have been rigorously investigated by a number of authors [1,3,4].The linear properties of the DIA waves in dusty plasmas are now well understood from both theoretical and experimental points of view [1][2][3][4][5].
On the other hand, to the best of our knowledge, no attempt has been made on nonplanar DIA DLs.Therefore, in our present work we are going to analyze the nonplanar DIA DLs in dusty plasma system by deriving e modified Gardner (mG) equation.th

Basic Equations
We consider the nonlinear propagation of the DIA waves in an unmagnetized nonplanar (cylindrical and spherical) dusty plasma, consisting of inertial ions, Boltzmann electrons, and negatively charged stationary dust.

Derivation of mG Equation
To study finite amplitude DIA DLs by employing the reductive perturbation method [27,33,34], we first introduce the stretched coordinates: where  is a smallness parameter ( 0 < < 1  ) measuring the weakness of the dispersion, and p V i n (normalized by i ) is the phase speed of the perturbation mode.We then expand all the dependent variables (viz.
To the lowest order in  , one obtains Z n ).To the next higher order in  , we obtain a set of equations, which, after using ( 11) and ( 12), can be simplified as   It is obvious from ( 14) that since = 0 A 0   .Now by solving for = 0 A  , we found A can be zero for both = 1 n (dust number density at equilibrium) will be equal to 0 i (ion number density at equilibrium), which means that there is no electron present in the system, i.e., all electrons will be captured by dust particles.As our model contains all of electron, ion, and dust, this is an invalid condition for our present model.For our system < 1


. Therefore the only valid solution of for = 0 A  is given by  s  for > c   .So, (2)   can be expressed as  


This means that for c    , (2)    must be included in the third order Poisson's equation.To the next higher order in  , we obtain a set of equations: Copyright © 2011 SciRes.
Now, using ( 11)-( 14) and ( 18)-( 20), we finally obtain a nonlinear dynamical equation of the form: where   Equation ( 21) is known as modified Gardner (mG) equation.The modification is due to the extra term, where the pseudo-potential ( ) We note here that 0 U and  are always positive since 1 The conditions ( 26) and ( 27) imply that DL solution of ( 24) exist if and only if where m  is the amplitude of the DLs.The latter can be expressed as m U s (30)  Now, using ( 25) and ( 30) in (24) we have where  is the DL width of the DLs, and is given From ( 32) and (33), it is clear that DLs can be formed in the dusty plasma system if and only if < 0  , is the lower (upper) limit of  above (below) which DLs exist.
In Figures 1 and 2, the variations of m  with 0 U (Figure 1) and e  (Figure 2   DLs are bounded by the lower and upper surface plot of Figure 3, and DLs exist for parameters corresponding to any point in between two ( = 0  ) surface plots.The point to be noted here that if we would keep only the lower order nonlinear term of (21) (viz.the third term of (21) or the term containing 2  ) instead of the higher order nonlinear term (viz.the fourth term of ( 21) or the term containing 3  ), we would obtain the solitary structures.On the other hand, in our present work, we have kept both the terms containing 2   and 3  , and have obtained the DL structures (associated with 2   and 3  ).

Numerical Analysis
Now in order to analyze the nonplanar DIA DLs, we turn to (21) with the term  

Discussion and Conclusions
In this paper we have investigated time-dependent nonplanar dust-ion-acoustic Gardner double layers in a dusty plasma system (composed of inertial ions, Boltzmann electrons, and negatively charged stationary dust), by deriving modified Gardner (sG) equation.The outcomes, which have been obtained from this investigation can be pinpointed as follows: 1) The dusty plasma system under consideration supports both finite amplitude planar and nonplanar DLs, whose basic features (polarity, amplitude, width, etc.) depend on the ion and dust number densities.
2) The DLs having large width exist for 0.74 < < 1  and only have negative potential, i.e., no positive DLs Copyright © 2011 SciRes.OJA  have been formed.
3) The magnitude of the amplitude of the DLs increases with the increase of , but decreases with the increase of 0 U  .
4) The magnitude of the amplitudes of both cylindrical and spherical DLs increase with decrease of  .
5) The spherical DLs have larger amplitude and potential than the cylindrical and planar DLs.
is the number of electrons residing on a dust grain surface; B k is the Boltzmann constant, and e is the magnitude of the electron charge.The time and space variables are in units of the ion plasma period ) represents the linear dispersion relation for the DIA waves propagating in a dusty plasma under consideration.This equation clearly indicates that the DIA wave phase speed ( p V ) increases with the increase of the dust charge density ( 0 d d


For  around its critical value ( c  ), i.e. for = and dimensionless parameter, and can be taken as the expansion parameter  , i.e.
due to the effects of the nonplanar geometry.Because of the existence of both 2  and 3  term, this equation supports both Sws and DLs solutions.We have already mentioned that = 0  corresponds to a 1D planar geometry which reduces (21) to a standard Gardner (sG) equation.We are now going to numerically analyze the mG equation.However, for a better understanding, before going to numerical solutions of mG equation, we first briefly discuss the stationary double layer (DL) solution of this standard Gardner equation [i.e.(21) with = 0  ].The stationary DL solution of the sG equation [i.e.(21   .These boundary conditions for the stationary DL solution allow us to express the sG equation as 2 1 d

Figure 1 .
Figure 1.Showing the variation of the amplitude of DIA DLs with U 0 at μ = 0.75.

Figure 2 .
Figure 2. Showing the variation of the amplitude of DIA DLs with μ at U 0 = 0.1.

Figure 3 .
Figure 3. Showing the parametric regime for the existence of DLs.The upper (lower) surface plot represents the lower (upper) limit of μ below (above) which DLs exist.

2 
  , which is due to the effects of the nonplanar (cylindrical or spherical) geometry.An exact analytic solution of (21) is not possible.Therefore, we have numerically solved(21), and have studied the effects of cylindrical and spherical geometries on time-dependent DIA DLs.The results are depicted in Figures4 and 5.The initial condition, which we have used in numerical analysis, is in the form of the stationary solution of (21) without the term   2    .

Figure 4 2 ) 2  2 
Figure4shows how the effects of a cylindrical geometry modify the DIA DLs, and Figure5shows how the effects of a spherical geometry modify the DIA DLs.From the numerical solutions of (21) (displayed in Figures4 and 5) we may conclude that for a large value of  (e.g.= 40  ), the cylindrical ( = 1 ) and spherical ( = 2 

Figure 4 .
Figure 4. Showing the effects of cylindrical geometry on DIA DLs at μ = 0.75.

Figure 5 .
Figure 5. Showing the effect of spherical geometry on DIA DLs at μ = 0.75.