#### Abstract

Two-dimensional three-component plasma system consisting of nonextensive electrons, positrons, and relativistic thermal ions is considered. The well-known Kadomtsev-Petviashvili-Burgers and Kadomtsev-Petviashvili equations are derived to study the basic characteristics of small but finite amplitude ion acoustic waves of the plasmas by using the reductive perturbation method. The influences of positron concentration, electron-positron and ion-electron temperature ratios, strength of electron and positrons nonextensivity, and relativistic streaming factor on the propagation of ion acoustic waves in the plasmas are investigated. It is revealed that the electrostatic compressive and rarefactive ion acoustic waves are obtained for superthermal electrons and positrons, but only compressive ion acoustic waves are found and the potential profiles become steeper in case of subthermal positrons and electrons.

#### 1. Introduction

Rigorous theoretical and numerical studies on electron-positron-ion () relativistic plasmas are conducted by several researchers [1–19] due to their potential applications and significance in understanding different types of collective processes in astrophysical as well as space [20–36] and laboratory [37–43] plasmas. The experimental observations are also carried out by many experimentalists [37–43] under different plasma conditions in order to uncover the physical processes involved. On the other hand, one-dimensional nonlinear dynamics of the relativistic plasmas are studied by a number of researchers [1–19]. Malik [14] has studied the effect of electron inertia on Kadomtsev-Petviashvili () solitons in relativistic plasmas. Malik [15] has also studied the ion acoustic () solitons in a weakly relativistic magnetized warm plasma. Singh et al. [18, 19] have investigated the effect of electron inertia on one-dimensional evolution of solitons and small amplitude of solitons in two-fluid weakly relativistic plasmas through usual Korteweg–de Vries () equation. Very recently, Hafez and Talukder [11] have studied the weakly relativistic influence on the solitary waves in one-dimensional case by considering nonextensive electrons and isothermal positrons. Hafez et al. [12] have also investigated the one-dimensional weakly relativistic effects on the electrostatic positive as well as negative solitons composing nonextensive electrons and positrons. Only a few works [44, 45] are devoted to studying the two-dimensional nonlinear dynamics of ion acoustic () waves in such plasmas. Han et al. [44] have investigated the existence of solitary waves and their interaction in two-dimensional thermal plasmas considering weakly relativistic hot ions and isothermal electrons by solving two equations for small but finite amplitude solitary waves. They found that only compressive solitons for isothermal electrons and the phase shifts of the colliding solitary waves strongly depend on the colliding angle. Masood and Rizvi [45] have studied the two-dimensional electrostatic shock waves in relativistic unmagnetized plasmas consisting of relativistic thermal ions and Boltzmann distributed electrons and positrons by solving Kadomtsev-Petviashvili-Burgers () equation. They have shown that the ratio of ion to electron temperature, kinematic viscosity, positron concentration, and the relativistic ion streaming velocity significantly modify the structure of the shock waves. However, the effects of superthermal and subthermal electrons as well as positrons can no longer be disregarded in case of two-dimensional relativistic plasmas. It was observed in many space and laboratory plasmas that the distribution of charge particles does not follow the usual Boltzmann-Gibbs statistics which is a well-organized tool for investigating the system when the memories and microscopic interactions are short ranged. Renyi [46] and accordingly Tsallis [47] have noticed the generalization of Boltzmann–Gibbs–Shannon (BGS) entropy for statistical equilibrium having long-range interactions, long-time memories, and dissipation. The rule of composition for the two independent systems and can be represented [48] as , where the parameters denote the degree of correlation of the system under consideration. One can be used for and for all cases , , and correspond to superthermality, isothermality, and subthermality, respectively. El-Tantawy et al. [49] have revealed that both the supersonic and subsonic electrostatic waves may exist in the nonextensive plasmas. To find the influence of charge particles nonextensitivity, the one-dimensional -distribution function for species can be defined [50] as , where is the normalization constant and is the strength of nonextensivity. The nonextensive density of electron and positrons can be obtained by integrating over the velocity space as and , respectively. Therefore, the study of nonextensive relativistic plasma has received a great deal of interest from the plasma physics researchers due to its wide relevance in astrophysical and cosmological scenarios like protoneutron stars [23], stellar polytropes [24], hadronic matter and quark-gluon plasma [25], the laser-plasma interaction [31], positron plasma wakefield accelerator [33], the inner region of accretion disc in the vicinity of black holes [34], and so on.

Being motivated by the above facts, this work is carried out to investigate the basic properties of waves in a fully ionized unmagnetized relativistic plasma system consisting of relativistic hot ions, -distributed electrons, and positrons in a two-dimensional planar geometry. The plasma system is assumed to be in equilibrium with relativistic thermal ions streaming in the -direction having free electrons and positrons, which is already justified by many authors [6, 7, 11, 12, 44, 45]. The influences of nonextensivity parameter, positron concentration, electron-positron temperature ratio, ion-electron temperature ratio, and relativistic streaming factor on the nonlinear propagation of waves in unmagnetized plasmas are studied.

#### 2. Theoretical Model Equations for Two-Dimensional Plasmas

The two-dimensional nonlinear propagation of a fully ionized unmagnetized three-component plasma system consisting of nonextensive electrons, positrons, and relativistic hot ions is considered. The charge neutrality equilibrium condition can be assumed as , where , and represent the unperturbed electrons, positrons, and ions concentrations, respectively. The concentrations of electrons and positrons are assumed to obey equilibrium -distribution function. The normalized nonextensive concentrations of electron and positron can be obtained [50, 51] aswhere and . It is noted that (1) and (2) are used, for superthermal, for subthermal, and for isothermal electrons and positrons, respectively. Thus, the normalized basic equations governing the nonlinear dynamics of waves, where the phase velocity of waves is assumed to be smaller than the superthermal and subthermal electrons and positron velocities, can be written in the following forms: Here is the ion concentration normalized by unperturbed electron concentration (), and , respectively, are the ion fluid velocities in the direction of and normalized by , is the pressure of ions normalized by , is the electrostatic potential normalized by , and are space variables normalized by the electron Debye radius , and is the time variable normalized by the period of ion plasma , respectively, where , , and , , , , and are the speed, ion mass, electrostatic potential, electronic charge, and electron and ion temperatures, respectively. The relativistic effect of ions is considered to be weak and can be expanded as . The dissipation in such collisionless plasmas occurs due to influence of ion kinematic viscosity (). It is well known that the ion kinematic viscosity for collisionless plasmas depends on the ion temperature, ion gyrofrequency, ion-ion collision time, and ion mass and widely applies in many astrophysical issues, especially in the solar wind [52]. The symbol is the normalized viscosity coefficient normalized by .

#### 3. Formation of and Equations and Analytical Solutions

To obtain a nonlinear dynamical equation in investigating the small but finite amplitude weakly relativistic waves in the plasma system, the stretched variables can be considered [10, 45] aswhere is the linear phase velocity and is a small expansion parameter measuring the weakness of dispersion and nonlinearity. Introducing (4), (3) can be converted as follows:Appling the reductive perturbation technique, the perturbed quantities , , , , and taking into account the charge neutrality equilibrium condition can be expanded [45, 53] as Setting (6) into (5), composed of various powers of , the lowest power in gives where , with . Simplifying (7), the phase velocity is obtained asIt is seen from (8) that the phase velocity of such waves strongly depends on the related plasma parameters, but not on ion kinematic viscosity coefficient. However, collecting the next higher-order terms of yields a system of nonlinear partial differential equations () as where . Eliminating , , , and from the above system of , we obtain the following nonlinear evolution equation ():The above equation is the two-dimensional famous equation, which is very useful to investigate the nonlinear propagation of weakly relativistic shock structures in the plasma system considered. The nonlinearity (), dispersion (), dissipation (), and weakly transverse dispersion () coefficients of (10) are obtained in the following forms:where .

To obtain the analytical solution to the of (10), the stretched variables , , and may be combined as , , where is the constant speed of the reference frame normalized by and setting the appropriate boundary conditions , , , and as for localized perturbations. Equation (10) can be converted to the following form: Integrating (12) once with respect to and using boundary conditions for localized perturbations, one obtainsApplying the Bernoulli equation method [54], the traveling wave solutions to (13) can be written aswhere is a solution to the Bernoulli equation The symbols , , , and are constants to be evaluated later. Substituting (14) along with the derivatives of into (13) and collecting all the terms with the same degree of , one can obtain a system of algebraic equations for , , , , and as follows: Evaluating the system of algebraic equations (15), the parameters , , , and can be obtained as Using (14) and (16), the (10) has the following soliton solution:where . It is seen that the height of the shock structure depends on the nonlinear coefficient (), dispersion coefficient (), dissipative coefficient (), and speed of the reference frame (), but the steepness of the shock structure depends only on and . To identify the formation of stable shock, linearizing and integrating (13), one obtainsThe above equation is a higher-order homogeneous linear ordinary differential equation with constant coefficients. It is well known that the solutions of (18) are proportional to , where . It is remarkable to be noted that there will be a shock formation if ; otherwise there will be oscillatory solitons. Substituting [55] into the stability condition yields . The shock structures may be formed only for . If , (10) can be reduced as This is the well-known two-dimensional equation, which is responsible for solitary wave or hump-shaped structures in plasmas. Therefore the solution of equation can be written aswhere is the amplitude of the shock structures and is the width of shock waves with .

Again, solving the algebraic equations (15) by setting , one can determine the values of the parameters , , , and asUsing (14) and (21) and setting , the stationary solution to the equation can be obtained in the following form:where , is the amplitude of solitary waves, and is the width of solitary waves. It should be mentioned here that there will be hump-shaped solitons if ; otherwise it will yield oscillatory waves.

#### 4. Results and Discussion

The effects of unperturbed positron-electron density ratio (), electron-positron temperature ratio (, ion-electron temperature ratio (), normalized ion kinematic viscosity coefficient (, relativistic streaming factor (), and strength of electron and positrons nonextensivity () on the nonlinear propagation of electrostatic waves in such relativistic plasmas have been studied numerically. The following observations are made from this investigation:(1)The finite amplitude shock structures are obtained due to the involvement of dissipative term, that is, ion kinematic viscosity effect, but the finite amplitude solitary waves are obtained in the absence of the dissipative media. It is provided that the basic features of relativistic nonextensive plasmas support finite amplitude waves and strongly depend on the related plasma parameters.(2)The influence of enhanced relativistic streaming factor (i.e., for ) on the nonlinear propagation of in the moving frame for both cases of thermality is displayed in Figures 1(a) and 1(b). It is observed that, with the increase of relativistic streaming factor, the magnitude of electrostatic potential of shock waves increases for thermal electrons and positrons, but the width of the shock waves becomes steeper in the case of subthermality.(3)The effect of for superthermal and subthermal particles on the spatial electrostatic potential profiles of shock waves is shown in Figures 2(a) and 2(b), respectively. It is seen that the amplitude of the shock waves monotonically is increased with the increase of .(4)The variations of spatial electrostatic potential profiles of waves with regard to and for both cases of thermality taking the remaining parameters constants are presented in Figures 3(a), 3(b), and 3(c). Figures 3(a) and 3(c) show that the amplitudes of the shock waves decrease for lower temperature ratio of superthermal and subthermal electrons and positrons while Figure 3(b) shows that the amplitude of shock waves increases for high electron to positron temperature ratio in the case of superthermality.(5)The effect of on spatial electrostatic potential of shock waves for both cases of thermality keeping the remaining parameters constants is presented in Figures 4(a), 4(b), and 4(c). The figures indicate that the shock amplitudes increase with the increase of for superthermal as well as for subthermal electrons and positrons but sharply decrease for superthermal case. It is also found that the rarefactive shock structures are obtained for small values of and negative values of with the considered values of the other parameters.(6)Figures 5(a) and 5(b) represent the influence of on the spatial electrostatic potential profiles of shock waves for different values of taking the constant values of the remaining parameters as mentioned in the figures, respectively. It is observed that, with the increase of , the amplitude of electrostatic shock waves decreases for superthermality but increases for subthermality conditions.(7)The initial electrostatic potential profiles of solitary waves are displayed in Figures 6(a), 6(b), 6(c), and 6(d) considering the different values of the plasma parameters indicated in the figure caption. Visual inspection of Figures 6(a)–6(c) indicates that the amplitude as well as the width of the solitons decreases not only for superthermal case but also for subthermal electrons and positrons as and increase for low electron to positron temperature ratio. The shape of the solitary waves becomes hump, but the amplitude becomes narrower (Figure 6(d)) for high electron to positron temperature ratio and superthermal elections and positrons.(8)Figures 7(a)–7(c) present the variation of spatial electrostatic potential profiles of solitary waves with respect to , whereas Figure 7(d) presents the variation of electrostatic solitary waves for different values of considering the other parameters constants. It is observed that the amplitude of the solitary waves increases for , whereas it decreases for . It is also observed that the solitary waves behave as pulse-like solitons due to the variation of time.On the other hand, the nonlinearity, dispersion, and dissipation coefficients of equation of this manuscript are equivalent to the nonlinearity, dispersion, and dissipation coefficients presented by Masood and Rizvi [45] when . Thus the results of this investigation are also in good agreement with the results provided in [45] for isothermal electrons and positrons on the nonlinear propagation of shock waves in the plasmas considered. Furthermore, we have also studied the nonlinear propagation of solitary waves using the stationary solution of the two-dimensional equation. It is seen that the compressive and rarefactive shock and solitary waves can be obtained for superthermal electrons and positrons, while only compressive waves can be found in case of subthermal electrons and positrons.

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#### 5. Conclusion

Two types of are derived to study the two-dimensional nonlinear electrostatic structures considering the unmagnetized plasma system consisting of relativistic ions, nonextensive electrons, and positrons that exist in many astrophysical and cosmological environments. The solutions of the equations are determined in terms of positron concentration, electron-positron and ion-electron temperature ratios, ion kinematic viscosity, strength of electron and positrons nonextensivity, and weakly relativistic streaming factor by employing the Bernoulli equation method. The fact that the phase velocity of waves strongly depends on the related plasma parameters, but not on ion kinematic viscosity coefficient, is investigated. The amplitudes of the shock waves are monotonically increasing with the increase of viscosity coefficient for both cases of thermality. The solitary waves become hump- and pulse-shaped due to relativistic effect in the absence of the dissipative media, that is, viscosity coefficient. The compressive and rarefactive shock and solitary waves are obtained for superthermal electrons and positrons, while only compressive waves are found in case of subthermal electrons and positrons. Finally, this work may be helpful for understanding the structures of waves for laboratory and astrophysical plasmas. This work is done for nonlinear shock and solitary propagation in our plasma model in case of planar geometry which is only valid for small but finite amplitude ion acoustic structures. But still there are so many possibilities to make further investigation on nonlinear analysis of shock waves, solitary waves, vortices, solitons, double layers, and so forth, with nonextensive electrons, positrons, and relativistic thermal ions using this same plasma model. These are also the problems of great importance in plasma physics for better understanding the characteristics of astrophysical compact objects but beyond the scope of the present work.

#### Competing Interests

The authors declare that they have no competing interests.