Properties of Damped Cylindrical Solitons in Nonextensive Plasmas

1 Introduction

Collisions in complex dusty plasma in real practical experiments, space and astrophysics containing fluid of ions, neutrals, electrons and liquid or solid grains are established [1], [2], [3], [4], [5], [6], [7], [8], [9]. New acoustic modes have been introduced by charged grains such as dust acoustic and dust ion acoustic (DIA) waves in plasma [10], [11], [12]. These modes have been confirmed by many experimental investigations [2], [13]. Also, in dusty plasma laboratories plasmas may contain neutrals and wave features affected by dusty-neutral collisions [6], [7], [8], [9], [14], [15]. Popel and Yu discussed the progress of DIA waves by neglecting grains absorption and electron-ion scattering [16]. Mahanta and Goswami studied the dusty-neutral collisions effect on the plasma wave properties [17]. The anomalous dissipation for shock waves in supernova is taken into account [18]. Other investigations studied damping collisional rate and electron trapping in dust plasma by introducing ion recombination and ion collision frequencies [19], [20], [21], [22]. Bulanov et al. reported that under some resonance conditions the nonlinearity oscillator can introduce uniform and chaotic oscillations with a maximal restoring force. For chaotic conditions, the oscillatory amplitude can grow in time as a cubic root which gives an indication for the diffusion trajectory in phase space [23]. Chatterjee et al. investigated damped solitary solution in collisional dusty superthermal plasmas. The effects of frequency, strength of an external form of periodic force and superthermal parameter on the damped DIA solitons have been obtained [24]. There are many research articles that deduced not only negatively charged dust but also negative-positive dust grains applications in experimental laboratories and space [25], [26], [27], [28], [29]. On the other hand, in most space plasmas and experimental investigations the distribution of charged particles diverges from thermodynamic state of equilibrium [30], [31], [32]. Behery used nonextensive parameters to study the phase shift produced by two collisional solitary waves in dust having different grain shape [32]. A theoretical investigation of heavy ion and strong coupling effects on both linear and nonlinear shock propagation in nonextensive plasmas has been introduced [33]. Also, the normalised viscosity coefficient and nonextensive influences on the properties of shock formation have been examined. It was founded that the extension of shock wave decreases with nonextensive q parameter and the shock width decreases with viscosity coefficient. Accordingly, many theoretical studies that introduced nonextensive distribution of particles reported an agreement with experiments and space observations data [34], [35], [36], [37]. This distribution has found considerable plasma applications in many astrophysical matters such as solar wind, magnetosphere of the earth, Saturn rings, hadronic matters and quark plasma [38], [39], [40]. Bains et al. studied the electron and ion q-nonextensive effects on the modulation of acoustic wave in dusty fluid [37]. Furthermore, the interest of nonextensivity effects of ion and electron on shock and rogue waves feature has been investigated [39], [40], [41]. Later, anisotropic dusty pressure impacts on two waves of oblique collision in plasma with q-nonextensive particles have been studied [42]. But, many studies on damping rate in dusty plasma are about the unbounded coordinates system. This is not agreeable for many laboratory and space investigations. Accordingly, some studies introduced the non-planar aspect of nonlinear cylindrical geometry [43], [44]. Several theoretical studies on the non-planar nonlinear wave structures in dust plasmas have been explored [45], [46], [47], [48]. Ema et al. studied the non-planar heavy ion acoustic (IA) shock and soliton waves in strongly coupled multicomponents plasma [48]. The physical effects of non-planar factor, heavy ions and time parameter on the wave picture have been reported and they noted that both shock and soliton amplitude rises with increasing factor of non-planar and decreases with time.

In this study we aim to investigate the characteristics of the damping wave behaviour for the cylindrical damped three-dimensional DIA waves in collisional dusty plasma model containing q-nonextensive distributed electrons, mobile ions and negative-positive micro grains. The organisation of this study is in the following form: In Section 2, we introduce the fluid model. In Section 3, the nonlinear three-dimensional damped Kadomtsev-Petviashvili (3D-CDKP) equation is displayed. The damped solution is given in Section 4. Section 5 is devoted to some discussions.

2 System of Equations

A four-component dusty collisional plasma system contains a mixture of nonextensive distributed electrons, mobile ions and negative-positive dust grains. For mobile components three-dimensional continuity equations are given as

The corresponding momentum equations are

where the two terms νrenir and νiner are approximately given by νre(ni−ni0) and νi(ne−ne0) [18]. These equations are coupled with the Poisson equation

and the nonextensive electrons density n_e is given by

In the earlier equations nj(j=i,n,p,e) are the perturbed number densities, and n_{i ⁢ 0}, n_{n ⁢ 0}, n_{p ⁢ 0} and n_{e ⁢ 0} are the related equilibrium values. Also, uj(j=i,n,p) are the ion, negative and positive dusty plasma velocities, respectively, normalised by the ion sound velocity (KBTe/mi)1/2; φ is a potential and normalised by (KBTee); time variable t and space coordinate are normalised by the inverse of the plasma frequency ωpe−1=(mi/4πe2ne0)1/2 and electron Debye length λd=(KBTe/4πe2ne0)1/2; and q is a parameter identifying the nonextensivity degree of electrons. Its value is an important indicator of the state of the system; i.e. q = 1 refers to the density reduction of the distribution of Maxwell-Boltzmann, whereas q < 1 refers to superextensive and q > 1 to the subextensive states [49], [50]. νre is the recombination frequency of ions on dusty particles, ν_i is the ionisation frequency of plasma, ν⌢ is a frequency for loss in ion momentum due to the recombination on dusty particles and collisions between ions and grains of dust, and νpn is a frequency for loss in positive dust momentum due to negative-positive dust collisions. The collision frequencies νre, ν_i, ν⌢ and νpn are normalised by ωpe−1, μ=Znmi/mn, and α=Zpmi/mp. Here, K_B and T_e are the Boltzmann constant and temperature of electron, e is the electronic charge, and mj(j=i,n,p) denote ion, negative and positive dust masses, respectively. From the charge neutrality condition, we have

with δi=ni0/ne0, δn=Znnn0/ne0, and δp=Zpnp0/ne0, where Z_n and Z_p are the charge numbers of negative and positive dust, respectively.

3 Nonlinear Calculations

To study DIA wave properties, a reductive perturbation (RP) analysis is used [51]. We introduce the following new independent variables [52], [53]:

where ε is a small parameter that measures the degree of perturbation and the λ is the velocity of wave propagation (we have assumed that νre∼ε3/2νre0, νi∼ε3/2νi0, ν⌢∼ε3/2ν⌢ and νpn∼ε3/2νpn0). All variables in the model are expanded in the powers of ε as

where u_j, v_j and w_j are the ion, negative and positive dust velocities in R, Θ, and Z directions. Using (6) and (7) in (1)–(3), the first orders in ε for ions are

Those for negative dust are given by

And those for positive dust are given by

The Poisson equation leads to the following compatibility condition:

At second order in ε, the nonsecularity condition for second-order quantities leads to the 3D-CDKP equation in the form

where

Equation (12) is a nonlinear evolution of φ(1) with coefficients of nonlinear and dispersion terms A, B and dissipation coefficient C. For neglecting the Z and Θ dependence, (12) is reduced to damped Korteweg-de Vries equation.

4 3D-CDKP Solution

Equation (12) is not exactly solvable in analytic form. However, for a weak dissipation effect (damping due to collisions) and by introducing the transformation

one can obtain an approximate form of analytical solution for (12) [54], [55], [56]:

where the amplitude and width of damped soliton φ1(τ) and L(τ) are given by

where φ0(0) is the amplitude of the single localised soliton in the absence of the damping coefficient (C = 0) and U₀ satisfies the condition U0=λ(1−Lr2)2Lr.

5 Results and Discussion

In this study a four-component nonextensive collisional plasma model is inspected by RP calculations [51]. The plasma model is reduced to a 3D-CDKP (12) which admits the time damping solitons. Now we examine the dependence of damped soliton features on some plasma system parameters using mesospheric information [57], [58]. To characterise the results, the DIA time evolution solitons are plotted in Figure 1. It shows that soliton existence appears in a time of weakly dissipative structures with decreasing amplitude and increasing width. The reliance of both the width and amplitude of the DIA damped solitons on the nonextensive parameter of electrons q is graphically depicted in Figure 2. It shows that the amplitude and width of the damped solitary profile decrease with q. Moreover, the main aim of this work is to examine the effects of some collisional parameters ν_{i ⁢ 0}, νre0, ν⌢ and νpn0 on plasma wave propagation. The effects of these frequencies are studied for the damped soliton profile in Figures 3–5. It is established that both ν⌢ and νre0 frequencies increase the damped solitary width and decrease its amplitude as shown in Figures 3 and 4. The same influence is expected for the frequency νpn0. In contrast, Figure 5 specifies that by increasing the frequency ν_{i ⁢ 0} the damped solitonic amplitude φ1(τ) is growing and its width L(τ) is reduced. On the geometric point of view, the dependence of the solution (15) for damped soliton profile in the geometric coordinates Θ, R, Z and τ are depicted in Figures 6 and 7. It is shown that the soliton form not only tends to deflect towards the positive radial coordinate but also decays with time as shown in Figure 6. Finally, the variation of DIA damped profile with τ and Θ coordinates at different values of R is shown in Figure 7. It is noted that there is an intense deviation and rapid damping produces sharp geometric soliton distortion as depicted in Figure 7. It is important to indicate the plasma range of parameters (see the table) that are applied in this study which correspond both in laboratory and space [21], [25], [49], [57], [58].

Figure 1:

Variation of φ(χ,τ) with χ and τ for δn=1.3, δp=1.5, α = 0.002, μ = 0.005, νre0=0.5, ν⌢=0.3, νi0=0.2, νpn0=0.3 and q = 1.

Figure 2:

Graph of φ(χ,τ) versus time χ and q for α = 0.002, μ = 0.005, Lr=0.5, δn=1.3, δp=1.5, νre0=0.5, ν⌢=0.3, νi0=0.2, τ = 0.5 and νpn0=0.3.

Figure 3:

Graph of φ(χ,τ) versus time χ and ν⌢ for α = 0.002, μ = 0.005, Lr=0.5, δn=1.3, δp=1.5, νre0=0.2, νi0=0.5, νpn0=0.3, q = 1 and τ = 1.4.

Figure 4:

Graph of φ(χ,τ) versus time χ and νre0 for α = 0.002, μ = 0.005, Lr=0.5, δn=1.3, δp=1.5, ν⌢=0.4, νi0=0.5, νpn0=0.3, q = 1 and τ = 1.4.

Figure 5:

Graph of φ(χ,τ) versus time χ and ν_{i ⁢ 0} for α = 0.002, μ = 0.005, Lr=0.5, δn=1.3, δp=1.5, ν⌢=0.4, νre0=0.2, νpn0=0.3, q = 1 and τ = 1.4.

Figure 6:

3D profile φ(χ,τ) against R and Θ for Z = 0.5 and for Lr=0.5, α = 0.002, μ = 0.005, δn=1.3, δp=1.5, νre0=0.5, ν⌢=0.3, νi0=0.2, νpn0=0.3 and q = 1 for variable τ.

Figure 7:

3D profile φ(χ,τ) against R, Θ and τ for Z = 0.5 and for Lr=0.5, α = 0.002, μ = 0.005, δn=1.3, δp=1.5, νre0=0.5, ν⌢=0.3, νi0=0.2, νpn0=0.3, q=1. For variable R.

To summarise, we have investigated the nonlinearity effects of the damping structure of DIA solitons in multicomponent nonextensive, collisional, unmagnetised, dusty fluid plasmas having nonextensive electrons, mobile cold ions and positive and negative grains in cylindrical coordinates. The angular, radial and physical quantities of contributions have been discussed. The effects of some physical indicators such as nonextensive parameter of electrons q, system collisional parameters ν_{i ⁢ 0}, νre0, ν⌢, νpn0 and geometric coordinates (Θ, R and τ) on the fundamental properties of the damped solitary behaviour have been studied numerically using recorded mesospheric parameter values [57], [58]. This investigation is different from that in [48], [59]; Ema et al. inspected the influences of nonextensive, non-planar and heavy ion effects on the heavy IA shock and soliton formation in strongly coupled plasmas with Maxwellian light ions [48]. El-Labany et al. examined plasma instability, and the existence of enveloped huge pulses are examined in mesospheric plasma fluid containing isothermal electrons, ions and duple polarity dust grains [59]. In this study, we reported graphically the importance of collisional and nonextensive parameters in the characteristics, formation and existence of damped solitonic form. Furthermore, the damped soliton coexistence with geometrical distortion regions is explained. Accordingly, our nonlinear study on the formation of damped soliton could be helpful for nonlinear damped wave features achievement on multicomponent nonextensive plasmas in both laboratory and space astrophysical mediums (e.g. in laboratory and mesospheric regions [8], [25], [49], [57], [58]).