Dust Ion Acoustic Solitary Waves in Multi-Ion Dusty Plasma System with Adiabatic Thermal Change

This theoretical work has done on the behavior of dust ion acoustic (DIA) solitary waves (SWs) in an adiabatic plasma system consisting of inertial positive and negative ions, Maxwell’s electrons, and arbitrary charged stationary dust. The dust particles have been regarded as either positively or negatively charged in order to perceive the effects of dust polarity on the DIA SWs. Through this work the changes in the main properties of these waves with adiabatic state have been observed. At first, a detail mathematical derivation has done on the linear properties as well as the dispersion relation in the multi-ion dusty plasma system. In order to perceive the properties of SWs two different approaches Korteweg-de Vries (K-dV) and mixed K-dV (mK-dV) has been made. Here reductive perturbation slant has been employed in all these approaches. First K-dV equation has been derived which let to analyze both bright and dark solitons but for a very limited region. Then mK-dV equation has been derived that let analyze bright soliton for a large region.


INTRODUCTION
Dusty plasma is normal electron-ion plasma with an added highly charged element of small micron or submicron sized extremely massive charged gritty (dust grains). Shukla and Silin (1992) have theoretically shown the low-frequency dust-ion-acoustic waves in a dusty plasma system. Barkan et al., (1995) have experimentally verified the existence of dust-ionacoustic wave in dusty plasma. These waves differ from usual ion-acoustic waves (Lonngren, 1983) due to the conservation of equilibrium charge density n e0 e+n d0 Z de −n i0 e = 0, and the strong inequality, n e0 ≪ n i0 , where n s0 is the particle number density of the species s with s = e (i) d for electrons (ions) dust, Z d is the figure of electrons residing onto the dust grain side, and e is the magnitude of an electronic charge. DIA wave's linear properties are now prudent understood and Shukla and Rosenberg, 1999). The nonlinear structures related with the DIA waves are particularly solitary waves (Bharuthram and Shukla, 1992;Nakamura and Sharma, 2001); shock waves (Nakamura et al., 1999;Luo, 2000;, etc. These waves have also had a great deal of interest to understand the localized electrostatic perturbations in galactic space (Geortz, 1989;Fortov, 2005), and laboratory dusty plasmas Dusty plasmas create a fully modern interdisciplinary area with direct link to astrophysics, nanoscience, fluid mechanics, and material science as specified through experimental, theoretical, analytical, and arithmetical studies. All of these works Bharuthram and Shukla, 1992;Nakamura and Sharma, 2001;Nakamura et al., 1999;Luo, 1995;Mamun, 2009) are limited to planar (1D) geometry and are subjected to some critical value. A few works have also been done on finite amplitude DIA solitons and shock structures (Luo, 1995), where K-dV or Burgers equations are used, which are not valid because, the latter gives infinitely large amplitude structures which break down the validity of the reductive perturbation method) for a parametric regime corresponding to A = 0 or A ∼ 0 (where A is the coefficient of the nonlinear term of the K-dV or Burgers equation) (Luo, 1995). Here, A ∼ 0 means A is not equal to 0, but A is around 0. In our present work, we have been able to show the bright and dark solitons for a large region of multi-ion dusty plasma system in an adiabatic state.
The manuscript is prepared as follows; the model equations are given in Sec. 2, the K-dV equation is derived in Sec. 3, the mK-dV equation is derived in Sec. 4, then results and discussion are given in Sec. 5, and conclusion is given in Sec. 6.

Model Equations
The dynamics of the one-dimensional multi-ion DIA waves are governed by: ∂u n /∂t + u n ∂u n /∂x = 1/µ(∂ψ/∂x − δ n /n n ∂p n /∂x), ∂p s /∂t + u s ∂p s /∂x + γp s ∂u s /∂x = 0, where n s is the number density with s = n(i)e(d) of negative ion (positive ion) electron (stationary dust), u s is the fluid speed of s, m j is the positive (when j = i) or negative (when j = n) ion mass, Z d is the number of electron occupy on the dust grain side, e = magnitude of the electron-charge (q), ϕ is the electrostatic wave potential; n s0 , (n j0 ), and n d0 are the equilibrium value of n s , (n j ), and n d respectively i.e. n s , (n j ), and n d are the number density normalized by n s0 , (n j0 ), and n d0 respectively, p i is the pressure of species i, γ is an adiabatic index, x is the space variable, and t is the time variable.

K-dV Equation
For the DIA K-dV equation we introduce the stretched coordinates: Where, V p is the wave phase speed (ω/k), and ϵ is a smallness parameter (0 < ϵ < 1). To get the dispersion relation, we expand n s , u s , p s , and ϕ with s be the charged species like positive and negative ion, electron in power series of ϵ, to their equilibrium and perturbed parts, Where n s (1) , u s (1) , p s (1) , and ψ (1) are the perturbed part of n s , u s , p s , and ψ respectively.

mK-dV Equation
For the third order calculation a new set of stretched coordinates is applied: Using (15) we can find the same values of n i (1) , n n (1) ,n e , u i , u e , u n , p i , p e , p n (1) , and V p as like as that in K-dV.

RESULTS AND DISCUSSION
Dust ion acoustic K-dV and mK-dV solitons have been investigated in a multi-ion dusty plasma system in adiabatic change that gives; 1. The positive and negative K-dV solitons are observed. 2. The width and amplitude of the K-dV solitons varies with polarity changes. 3. Amplitude of the positive and negative soliton increases with the increasing mass number density for positive dust but decreases for negative dust. 4. Existence of positive mK-dV solitons is observed. 5. The width of the positive mK-dV soliton decreases with adiabatic index for positive dust but both increases for negative dust.

CONCLUSION
Present investigation is valid for tiny amplitude DIA K-dV solitons. Though we have considered positive and negative ions, Maxwell's electrons, and arbitrarily charged stationary dust, our model is applicable for small amplitude waves only, and the similar experimental setups of  or (Nakamura and Sharma, 2001) can be used to observe the solitons. Plasma with dust is violently known as dusty plasma (Bliokh et al.,1995;Verheest, 2002). A numerical approach allowing calculation of the grain charge while including peripheral electron emission and the major role of this approach to the lunar condition is provided. In conclusion, we propose that a new experiment may be performed based on our results to observe such waves and the effects of nonlinearity on these waves in both laboratory and space dusty plasma system.