Dusty plasma is normal electron-ion plasma with an added highly charged element of small micron or sub-micron 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-ion-acoustic wave in dusty plasma. These waves differ from usual ion-acoustic waves (Lonngren, 1983) due to the conservation of equilibrium charge density ne0e+nd0Zde−ni0e = 0, and the strong inequality, ne0 ≪ ni0, where ns0 is the particle number density of the species s with s = e (i) d for electrons (ions) dust, Zd is the figure of electrons residing onto the dust grain side, and e is the magnitude of an electronic charge. DIA waves linear properties are now prudent understood (Shukla and Mamun, 2002; Barkan et al., 1996; 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; and Mamun and Shukla, 2002), 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 (Shukla and Mamun, 2002; Nakamura and Sharma, 2001, and Barkan et al., 1996). 

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 (Shukla and Mamun, 2002; 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:

∂ns/∂t + ∂nsus /∂x = 0,                              (1)

∂ui/∂t + ui∂ui /∂x = − ∂ψ /∂x – (δi/ni) (∂pi /∂x),        (2)

∂un/∂t + un∂un/∂x = 1/µ(∂ψ/∂x − δn/nn ∂pn/∂x),        (3)

e∂ψ/∂x – 1/ne ∂pe/∂x = 0,                    (4)

∂ps/∂t + us∂ps/∂x + γps ∂us/∂x = 0,                    (5)

∂2ψ/∂x2 = [1−µn + µd] expψ +µnnn −µd −ni,              (6)

where ns is the number density with s = n(i)e(d) of negative ion (positive ion) electron (stationary dust), us is the fluid speed of s, mj is the positive (when j = i) or negative (when j = n) ion mass, Zd is the number of electron occupy  on the dust grain side, e = magnitude of the electron-charge (q), ϕ is the electrostatic wave potential; ns0, (nj0), and nd0 are the equilibrium value of ns, (nj), and nd respectively i.e. ns, (nj), and nd  are the number density normalized by ns0, (nj0), and nd0 respectively, pi 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:

ζ = ϵ1/2 (x−Vpt), and τ = ϵ3/2 t,               (7)

Where, Vp is the wave phase speed (ω/k), and ϵ is a smallness parameter (0 < ϵ < 1). 

To get the dispersion relation, we expand ns, us, ps, 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,

ns = 1 + ϵns(1) + ϵ2ns(2) + ϵ3ns(3) +•••,                 (8)

us = 0 + ϵus(1) + ϵ2us(2) + ϵ3us(3) +•••,              (9)

ps = 0 + ϵps(1) + ϵ2ps(2) + ϵ3ps(3) +•••,            (10)

ψ = 0 + ϵψ(1) + ϵ2ψ(2) + ϵ3ψ(3) +••,            (11)

Where ns(1) , us(1) , ps(1) , and ψ(1) are the perturbed part of ns, us, ps, and ψ respectively.

Combining above equations, we get -

Vp = (−b±√(b2 −4ac)/2a) ( 1/2),                         (12)

Where,

α = (1−µn + µd), a = µα, b = µn + µ−αγδn −αµγδi, and c = αγ2δnδi−γδn −γδiµn.

Equation (12) represents linear dispersion relation. 

The next higher order of ϵ can be simplified as an equation of the form:

∂ψ/∂τ + Aψ ∂ψ/∂ζ + β∂3ψ/∂3ζ = 0,                       (13)

Where,

A = Y/X,

β = 1/ X,

X = (2Vp/d22) (2Vpµµn/d1),

Y = −c1/d23 + µnc2/d13 + µnc3/d13,

c1 = Vp4 + 2 Vp2γδi −3Vp2 + γ2δi 2−2γδi + γ2δi,

c2 = Vp4µ2 −2µ Vp2γδn + 3µVp2,

c3 = γ2δn2 + 2γδn −γ2δn. 

Equation (13) is known as K-dV equation. We get stationary localized solution of (13) by introducing a transformation ξ = ζ −U0τ; 

ψ = ψmsech2 [(ζ −U0τ)/δ],          (14)

Where the amplitude ψm and the width δ are given by ψm = 3U0/A, and δ =√4β/U0, respectively.

Fig 1: Bright and dark K-dV solitons.

Equation (14) is the solution of K-dV equation. This represents a solitary wave. We observed that the Fig 1 shows the existence of bright and dark K-dV solitons with mass number density (µ).

mK-dV Equation

For the third order calculation a new set of stretched coordinates is applied:

ζ = ϵ(x−Vpt), τ = ϵ3t,                                     (15)

Using (15) we can find the same values of ni (1), nn(1), ne(1), u i(1), ue(1), un(1), pi(1), pe(1), pn(1), and Vp as like as that in K-dV.

To the next order approximation of ϵ, we obtain a set of equations, which, after using the values n i(1), nn(1), ne(1), and Vp, can be simplified and applying the condition, ψ ≠ 0 (so, its coefficient is zero), we get,

½ {A(ψ(1))2} = 0                       (16)

For the next higher order of ϵ, we obtain an equation: 

∂ψ/∂τ + αβψ2∂ψ/∂ζ + β∂3ψ/∂ζ3= 0,                    (17)

Where, 

α = F (−a12  + 15/2 − 21γδn/2a1 − 5γ2δn/2a1 −3γ3δ2n/a12)−F(3γ2δn2/a12) + G(a22 −15/2 −21γδi/2a2)−G(5γ2δi/2a2−3γ3δi2/a22 −3γ2δi2 /a22),

β =Vpa12a22 /(−2µµna22 Vp2 −2γδia12),         

Where,

F = µn/a13, G = 1/a23, a1 = (γδn −µVp2), and 

a2 = (Vp2 −γδi).

Equation (17) is known as mK-dV equation. The stationary localized solution of (17), obtained by introducing a transformation ξ = ζ −U0τ, is, therefore, directly given by 

ψ = ψmsech [ξ/∆],                       (18)

Where the amplitude ψm and the width δ are given by ψm =√6U0/αβ, and δ = 1/ψm√γ. where the amplitude ψm and the width ∆ are given by ψm =√6U0/αβ, ∆ = 1/(√γψm), and γ = α/6.

Fig 2: Bright mK-dV soliton.

Fig 2 gives the existence of mK-dV solitons. We get only one type of solitons, bright solitons.

Dust ion acoustic K-dV and mK-dVsolitons have been investigated in a multi-ion dusty plasma system where we observed;

1. The positive and negative K-dVsolitons are observed.

2. The width and amplitude of the K-dVsolitons varies with polaritychanges.

3. Existence of positive mK-dVsolitons is observed.

Present investigation are valid for tiny amplitude DIA K-dV, and mK-dV solitons. The first K-dV equation has been derived which gave both bright and dark solitons but for a very limited region. Though we have considered positive and negative ions, Maxwells electrons, and arbitrarily charged stationary dust, our model is applicable for small amplitude waves only.

We are so much acknowledged to Prof. Dr. Md. Kamal-Al- Hassan and Asst. Prof. Dr. Farah Deeba for their guidance to this research.

The authors declare that they have no competing interests with respect to the research.