Collisional positron acoustic soliton and double layer in an unmagnetized plasma having multi-species

This paper explores the head-on collision between two-counter propagating positron acoustic solitons and double layers (DLs) in an unmagnetized collisionless plasma having mobile cold positrons fluid, immobile positive ions and (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r,\;q$$\end{document}r,q)-distributed hot positrons, and hot electrons. By employing the extended Poincaré–Lighthill–Kuo method, the coupled Korteweg–de Vries (KdV), modified KdV (mKdV) and Gardner equations are derived to archive this goal. The effect of dimensionless parameters on the propagation characteristics of interacting KdV solitons (KdVSs), mKdV solitons (mKdVSs), Gardner solitons (GSs) and DLs are examined in detail by considering the limiting cases of (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r,\;q$$\end{document}r,q)-distribution. It is noted that the interaction of GSs and DLs are reported for the first time. The outcomes might be comprehended and beneficial not only in space and astrophysical environments but also in laboratory studies.

www.nature.com/scientificreports/ equations in an unmagnetized plasma consisting of IMPIs, MCPs, and double spectral index non-Maxwellian velocity distributed HPs as well as HEs. The effect of plasma parameters on the collisional PA two-counter propagating KdV solitons (KdVSs), mKdV solitons (mKdVSs), Gardner solitons (GSs) and Gardner DLs (DLs) are described along with graphical representation.

Model equations
An unmagnetized collisonless mult-species plasma system is considered by the mixture of MCPs with mass m cp , IMPIs, HPs and HEs along with N e0 = N cp0 + N hp0 + N i0 , where N i0 , N e0 , N cp0 and N hp0 are the unperturbed IMPIs, HEs, MCP and HPs number densities, respectively. It is noted that HPs and HEs are assumed to follow the generalized non-Maxwellian velocity distribution. Because the existence of flat top or shoulders would not be considered as model by either Maxwellian or kappa distribution as observed in some SAEs [29][30][31][32][33] . As a result, one can use the following double index non-Maxwellian velocity distribution 18,32,33 : where where q and r are the real parameters that measuring the superthermality on the tail of the velocity curve and the high energy particles on a broad shoulder of the velocity curve, respectively. It is noted that such parameters indicate the departure from Maxwellian and kappa equilibrium. One can easily recover the Maxwellian and kappa distribution by setting the limit r = 0, q → ∞ and r = 0, q → κ + 1 , respectively. One can also use such distribution as the physically meaningful distributions by considering q > 1 and q(r + 1) > 5/2 18,33 . By integrating f rq (V e ) with the help of cylindrical coordinates, Ulla et al. 18 have defined the total number HE density as where where φ is the normalized electrostatic potential. Since positron is the opposite charge particle of electron, the total number HPs density can be written as N hp = N hp0 1 − � 1 φ + � 2 φ 2 − � 3 φ 3 + · · · . To study the collisional PA wave phenomena, the hydrodynamic fluid equations in the dimensionless forms are obtained as below. where To formulate the dimensionless equations as in Eqs. (1)- (3), the characteristic scale of used variables and parameters are considered as (1) the number density of MCPs ( N cp ) is normalized by its equilibrium value N cp0 , (2) MCPs fluid speed normalized by k B T ef /m cp ( k B is the Boltzmann constant, T ef is the effective temperature www.nature.com/scientificreports/ and T e (T hp ) is the HEs (HPs) temperature, (3) the electrostatic potential ( φ ) is normalized by k B T ef /e , (4) time ( t ) is normalized by ω −1 pc = m cp /4πN cp0 e 2 and (v) length (z) is normalized by Dm = k B T ef /4π N cp0 e 2 .

Mathematical analysis
To examine the collisional wave phenomena between two-counter propagating soliton and their corresponding phase shift, one can use the following starching coordinates [39][40][41]49 : where ξ and η are the trajectories of soliton moving headed for each other, and p is the phase speed of PAWs and 0 < ε < 1 . Also, the perturbed variables can expand based on the concept of ePLK method 3,13,17 as Implementing Eqs. (4) and (5)  l (ξ , τ ) ≈ φ (2) l and φ (2) r (η, τ ) ≈ φ (2) r , one determines It is noted that one may consider N  www.nature.com/scientificreports/ Coupled mKdV equations and its stationary solutions. For case (2), which yields the critical value (say K v ) of any specific parameter (say N hc ) as In order to examine the collisional wave phenomena between two soliton and phase shift around K v , one may insert φ (11) is obtained as Based on the case (2), the O ε 3 equations are obtained as below.
where Equations (22)-(24) can then be converted with the help of (21) to where (19) 6 is thickness of the DL. Also, the phase functions can be obtained by the following: www.nature.com/scientificreports/ velocity is strongly dominated by the superthermality on the tail of the velocity curve and the high energy particles on a broad shoulder of the velocity rather than the density and temperature ratios. It is also found that p is increased exponentially (red curve of Fig. 1a) with the increase of superthermal parameter of HEs and HPs ( r = 0 ), which is in good agreement with the Ref. 12 . In Fig. 2, the HOC between two equal amplitude PA KdVSs by varying q and r along with the constant parametric values of other parameters are displayed. This figure shows that both of compressive and rarefactive collisional PA KdVSs are supported in which the amplitudes and widths of PA KdVSs are decreasing (monotonically decreasing) with the increase of r ( q ). Whereas, the influences of time ( τ ), δ , N ec and σ on the HOC between two equal amplitude PA KdVSs are demonstrated in Figs. 3 and 4, respectively with the presence of Kappa and flattopped distributions. These figures exhibit that the amplitudes and widths of interacting PA KdVSs are decreasing (monotonically increasing) with the increase of N ec and δ ( σ ). Besides, the interacting PA KdVSs becomes pulse like shaped with the changes of time. It is investigated from the above figures that the colliding PA KdVSs lose their energies monotonically with the increase of both superthermality on the tail of the velocity curve and the high energy particles on a broad shoulder of the velocity. But, the colliding PA KdVSs lose their energies without monotonically with the increase of the population of superthermal HEs and HPs, which is also in good agreement with the previous investigations 12, 17 . In the physical point of view, the HEs and HPs are dynamically interacted with the MCPs with the losses of electron energy and MCPs density. As a result, the contribution of restoring force that provided by HPs and HEs pressure is decreased for the production of electrostatic resonances. However, the driving force (provided by M CPs inertia) is decreased because the growth of ion density is only interpreted as the depopulation of ions from the plasma system. It is also found from the above figures that the right to left propagating KdVSs is initially at ξ = 0, η → −∞ , left to right propagating KdVSs is initially at η = 0, ξ → +∞ and after that they are asymptotically far away from each other. When τ → ±∞ , the reverse situations are arisen, as it is expected. Later than competition of the processes of HOC between KdVSs, the stationary merged coherent structures are formed within −∞ < t < +∞ . Due to the collision of two counter propagating solitons along the trajectories, they are deviated far from their initial position. As a result, the time delays (phase shifts) are generated. To understand the influence of Kappa and flat-topped distributions on phase shift, the variation of phase shift with regards to q and r by considering the remaining parameters constant is displayed in Fig. 5. This figure shows that the phase shift is remarkably increased with the increase of r , while slightly increased up to q → 7 and then almost remain unchanged with the increase of q . It is also found that the maximum growth rate are occurred with the absence of flat-topped parameter, that is r = 0 . This means that the maximum growth of phase shift is produced with the presence of only super-thermal HEs and HPs.
On the other hand, the coupled KdV equations are failed to address the collisional PA wave phenomena based on case (2). Because, the interacting KdVSs amplitude and their corresponding phase shifts are approaching to infinity at K v for any related parameters. To overcome such complicatedness, the coupled mKdV equations involving of higher order nonlinearity are derived based on case (2). By the useful solutions of such equations, the HOC between two-equal amplitude PA mKdVSs and their corresponding phase shifts around K v 's with the presence of Kappa and flat-topped distributions are displayed in Fig. 6. These figures show that the interacting mKdVSs modify their polarities in the neighborhood of K v because such interacting solitons take up maximum energy with the change of maximum phase shifts. In addition, Fig. 6d is shown that the colliding solitons described by the coupled mKdV equations gain much more energies with the presence of only superthermal HEs and HPs, like the colliding solitons described by the coupled KdV equations. But, the amplitude of colliding mKdVSs and their corresponding phase shifts are comparatively higher rather than the colliding KdVSs and their corresponding phase shifts, which is in good agreement with the experimental and numerical investigations 42,50 . It is also observed that the interaction of mKdVSs takes lay only around K v which yields only the compressive two-counter propagating solitons with their shifting phases. However, the mKdV equations are failed to examine the nature of interacting wave phenomena at K v and any value less than K v because the nonlinear coefficient of mKdV equations are becomes complex at these points. In these situations, one needs to derive another coupled NLEEs to overcome such difficulty. Thus, the coupled Gardner equations are first time derived to study the nature of resonance wave phenomena not only around either less or greater than K v but also at K v . It is observed that the coupled Gardner equations are divulged both of collisional PA solitons and DLs. Figure 7a and b show the electrostatic resonance potential due to HOC between two equal amplitude PA GSs propagating towards each other for N hc = 0.1 > K v and N hc = 0.001 < K v , respectively. Whereas, Figs. 8 and 9 show the interaction of left to right ( φ (1) l (ξ , τ ) ) and right to left ( φ (1) r (η, τ ) ) propagating DLs and the electrostatic resonance potential due to HOC between two equal amplitude PA DLs for N hc = 0.1 > K v and N hc = 0.001 < K v , respectively. These figures clearly indicate that both of compressive and rarefactive collitional PA GSs and DLs are produced in the considered plasmas. It is obviously observed that the right to left propagating PA GSs and DLs is initially at ξ = 0, η → −∞ , left to right propagating PA GSs and DLs is initially at η = 0, ξ → +∞ and after that they are asymptotically far away from each other. When τ → ±∞ , the reverse situations are arisen, as it is expected. Later than competition of the processes of HOC between PA GSs and DLs, the stationary merged coherent structures are formed within −∞ < t < +∞ . Figures 7, 8, and 9 also indicate that the maximum amplitudes of the colliding GSs and DLs are occurred not only around the critical but also at the critical number density ratios with the presence of Kappa and flat-topped distributions. It is predicted from the above discussion that the coupled Gardner equations are very useful rather that mKdV equations for describing the nature of both collisional PA solitons and DLs not only around the critical values but also at the critical values. It would be concluded from the above discussions that the theoretical outcomes might be very useful in understanding the nature of nonlinear propagation PA resonance solitons and PA resonance DLs in many SAEs, especially, in auroral acceleration regions, cosmic rays, solar wind, pulsar magmetosphere, and so on and in laboratory plasmas. www.nature.com/scientificreports/  with different values of τ around (a) K v = 0.006,538,446,158 (N hc = 0.08) and (b) K v = 0.006,538,446,158 (N hc = 0.1) with N ec = 0.5, δ = 0.5, σ = 1, q = 3, r = 0 and U 0 = 0.0075; (c) their phase shift ΔQ 0 due to the collision between two PA mKdVSs with N hc = 0.08, N ec = 0.5, δ = 0.5, σ = 1, q = 3, r = 0, ε = 0.1 and U 0 = 0.0075; and (d) effect of r (q = 3.5) on collisional PA mKdVSs with N hc = 0.08, N ec = 0.5, δ = 0.5, σ = 1, and U 0 = 0.0075. www.nature.com/scientificreports/ www.nature.com/scientificreports/