1. Introduction
In recent times, there has been a significant surge in researchers' interest in investigating positron-acoustic waves (PAWs) in unmagnetized plasma environments and plasmas subjected to magnetic fields (MFs). The PAWs are distinguished by the presence of two separate temperature positron components. The provision of inertia is attributed to the mass of cold positrons, whereas the thermal pressure exerted by hot positrons (electrons) is the restoring force. However, it is assumed that the heavy ions are stationary and only sustain the quasi-neutrality condition at equilibrium. Several studies have explored the nonlinear characteristics of PAWs [1–5]. For instance, Rahman et al [6, 7] and Uddin et al [8, 9] examined the characteristics of nonlinear propagation of nonplanar Gardner solitons, specifically in cylindrical and spherical geometries, associated with PAWs in a four-component plasma system. These investigations have revealed that the properties of nonplanar PAWs, such as amplitude, polarity and speed, differ from those of planar geometry. In addition, the nonlinear positron-acoustic (PA) shock waves within a high-density plasma system consisting of non-relativistic and ultra-relativistic degenerate electrons, cold (hot) positrons and stationary positively charged ions have been examined [10]. Tribeche et al [11] conducted a theoretical investigation in order to demonstrate the presence and potential manifestation of PA solitary waves (PASWs) involving the dynamics of mobile cold positrons. Furthermore, the study examined the presence of PA double layers (DLs) within a four-component plasma model. The findings revealed that the plasma model accommodates compressive and rarefactive PA-DLs. Moreover, the characteristics of nonlinear PA shock waves have been investigated in an unmagnetized plasma composed of cold positrons, immobile positive ions, and electrons with a Boltzmann distribution [12]. The analysis considers both unbounded planar geometry and bounded nonplanar geometry. For this purpose, the authors [12] applied the reductive perturbation method (RPM) to derive the nonplanar Korteweg–de Vries–Burgers equation to investigate the characteristics of PASWs. In a recent study conducted by Saha et al [13], the authors examined the propagation of PAWs in an unmagnetized plasma. The findings of this investigation offer valuable insight into comprehending the intricacies of nonlinear propagation within the auroral area. Furthermore, Saha and his research group [14, 15] have documented the influence of superthermal on PAWs in several plasma models.
In addition to the global Coulomb interaction of all the charged species, the particles, taken two by two, can interact [16, 17]. These direct interactions between particles give rise to transfers of momentum, energy, or even more complex reactions such as ionization, excitation and recombination [18, 19]. They are grouped under the general term of collisions and are characterized at the microscopic scale by collision cross-sections. Two distinct types of collisions exist, overtaking collision (OC) and head-on collision (HOC), which can be differentiated based on their propagation direction. Concerning the OC, the waves propagate in a parallel path but with varying velocities, whereas in the HOC, the colliding waves propagate in opposite directions. The HOC of both solitary and shock waves has been investigated in several plasma models through the application of the Poincaré–Lighthill–Kuo approach [20–25]. On the other hand, Hirota's method [26–29] or the inverse scattering transformation method [30] are considered for investigating the OC. Saha et al [31] used Hirota's bilinear method to investigate the multi-soliton solution in a complex plasma model. Furthermore, they derived the phase shift that arises after the collision of these solitons. Moreover, the latter method was applied to study the OC of two solitons in a non-Maxwellian unmagnetized plasma with nonextensive electrons and Maxwellian positrons. The obtained findings highlight the significant influence of the nonextensive parameter on both the generation of the two solitons and the characteristics of their phase shifts after collision. Moreover, Singh et al [32] examined the OC among multisolitons in a non-Maxwellian electron-positron-ion magnetoplasma with nonthermal electrons and positrons. The OC of nonplanar electron-acoustic solitons has also been investigated in an unmagnetized plasma consisting of inertial cold electrons and inertialess hot superthermal electrons, as well as stationary ions [33]. Despite the fact that soliton collisions are considered to be one of the wonderful and fascinating phenomena of nonlinear wave dynamics, the OC of PAWs is less investigated. We can cite the work of [34], where the author examined the inelastic collision of PAWs in four-component plasmas and found that the plasma physical parameters play a major role in changing the amplitude of PAWs.
Over the decades, most of the studies related to nonlinear PAWs were focused on the Maxwellian electrons and positrons [35, 36]. However, observations of astrophysical plasmas give some information about the nonthermal plasma, which is distinguished by the presence of an elongated tail in the high-energy domain [37–39]. With the help of a weak MF, the solar wind leads to the creation of nonthermal ions in the atmosphere of planets [40–42]. There remains a multitude of other sources that show the existence of fluxes of nonthermal electrons (ions). Moreover, the genesis and mechanism of nonthermal particle production in space plasma have always impressed researchers [43–46]. Hence, it is crucial to examine the influence of nonthermal positrons on the properties of nonlinear phenomena in a plasma with nonthermal positrons.
Recently, the general class of multidimensional soliton solutions, called dromions, has drawn considerable attention in plasma physics [47]. A well-known multidimensional system that admits dromion solutions is the Davey–Stewartson (DS) equation. Many researchers have investigated dromion structures in various plasma models. For example, 2D modulated ion-acoustic waves (IAWs) have been studied in electronegative plasma by Panguetna et al [48]. By applying the RPM, they showed that the basic fluid equations could be transformed into a (2+1)-dimensional DS system, and they obtained several solutions for these equations, namely one- and two-dromion solutions. Modulated IAWs were analyzed through the activation of modulation instability (MI) within the framework of (3+1)-dimensional DS equations in [49] where the negative ion concentration ratio's impact was observed through the growth rate of instability. Note that MI is a nonlinear phenomenon that arises when a small modulating perturbation acting on a plane wave propagating through a nonlinear medium results in exponential wave amplitude growth. Bedi et al [50] derived the DS equations in hot electron plasma, which obeys Kappa distribution. They found the best analogy that exists between the DS equations and the nonlinear Schrödinger equation. Jianping et al [51] used the dynamical system method to obtain periodic, solitary wave solutions, as well as unbounded wave solutions governed by the DS equations.
Nevertheless, the investigation of MI and the OC of dromion-like solitons of PAWs in magnetized plasma with Cairns nonthermal statistic on the framework of DS equations has not been reported so far, even if PAWs are omnipresent in the Universe. Moreover, studying the effects of the nonthermal parameter helps researchers understand non-Maxwellian plasmas, which are common in various natural and laboratory settings. This insight is crucial for advancing our knowledge of plasma physics and for applications such as controlled fusion and astrophysical research. Thus, based on the DS equations, our present paper aims to examine the criteria for the appearance of dromion-like solitons of PAWs by MI and the OC of the dromion structures in this plasma. In addition, the relevant plasma parameters' response to the stability condition and the characteristics of localized structures are reported. Our study will provide new information on the dynamics of dromions for PAW plasmas. Our results will thus contribute to a better understanding of localized multidimensional phenomena as well as their interactions in this specific type of plasma.
The study is extended as follows. In section 2 , we present the model equations. We derive the DS equations in section 3 . In section 4 , the analysis of MI is addressed. The localized structures, such as dromion-like solitons, are discussed in section 5 . A brief discussion is provided in the last section.
2. Theoretical fluid model and equations of motion
Here, we consider a magnetoplasma model having inertialess nonthermal hot electrons and positrons and inertial cold positrons as well as stationary positive ions. The influence of the MF has also been considered. The quasi-neutrality condition can be expressed in the following manner: ncp0 + nhp0 + ni0 = nhe0, with ncp0, nhp0, ni0, and nhe0, which indicates the unperturbed number densities of cold(hot) positrons, stationary ions and hot electrons, respectively. The MF ${\hat{B}}_{0}$ is assumed to lie in the z-direction, i.e, ${\hat{B}}_{0}$=${B}_{0}\hat{z}$ with $\hat{z}$ is a unit vector in the direction of the z-axis. The fluid model of the plasma, consisting of quantities, such as density and average velocity around each position, is described by [2, 3]:
♢ the continuity equation, which expresses the conservation of the number of particles,1 )—(3 ), with the above assumptions one can obtain the dimensionless continuity, momentum and Poisson's equations:
$\begin{eqnarray}\frac{\partial }{\partial t}{n}_{{\rm{cp}}}+{\rm{\nabla }}\left({n}_{{\rm{cp}}}{U}_{{\rm{cp}}}\right)=0,\end{eqnarray}$
♢ the momentum equation which describes the fluid equation of motion, $\begin{eqnarray}\frac{\partial }{\partial t}{U}_{{\rm{cp}}}+({U}_{{\rm{cp}}}{\rm{\nabla }}\,){U}_{{\rm{cp}}}=-{\rm{\nabla }}{\rm{\Phi }}-{U}_{{\rm{cp}}}\times \,{B}_{0}\,\hat{z},\end{eqnarray}$
♢ the Poisson's equation, which highlights the conservation of the electric charges in plasma, $\begin{eqnarray}\begin{array}{rcl}{{\rm{\nabla }}}^{2}{\rm{\Phi }} & = & {\mu }_{{\rm{e}}}\,{n}_{{\rm{he}}}-{\mu }_{{\rm{p}}}\,{n}_{{\rm{hp}}}+\varpi ,\\ \varpi & = & -{n}_{{\rm{cp}}}-{\mu }_{{\rm{e}}}\,+{\mu }_{{\rm{p}}}\,+1.\end{array}\end{eqnarray}$
Here, ncp represents the number density of cold positron scaled by ncp0. ${U}_{{\rm{cp}}}=u\hat{x}+v\hat{y}$, where u and v denote the velocities of the cold positron fluid scaled by the PA velocity ${C}_{{\rm{cp}}}={\left(\frac{{k}_{{\rm{B}}}{T}_{{\rm{he}}}}{{m}_{{\rm{p}}}}\right)}^{1/2}$, (kB indicates the Boltzmann constant, The(Thp) represents the hot electron (positron) temperature, mp being the positron's mass). The electrostatic potential Φ is scaled by the quantity $\frac{{k}_{{\rm{B}}}{T}_{{\rm{he}}}}{e}$, with e the magnitude of the electron charge. The other parameters are given by μe = nhe0/ncp0 and μp = nhp0/ncp0. The time is scaled by the plasma period ${\omega }_{{\rm{cp}}}^{-1}={({m}_{{\rm{p}}}/4\pi {e}^{2}{n}_{{\rm{cp0}}})}^{1/2}$, whereas the space variables are in the unit of the Debye length ${\lambda }_{{\rm{D}}}={\left(\frac{{k}_{{\rm{B}}}{T}_{{\rm{he}}}}{4\pi {e}^{2}{n}_{{\rm{cp0}}}}\right)}^{1/2}$ and ${\omega {}_{{\rm{c}}}}_{1}=\frac{{{\rm{\Omega }}}_{1}}{{m}_{{\rm{p}}}}$ denotes the MF scaled by the gyro-frequency ${{\rm{\Omega }}}_{1}=\frac{| e| {B}_{0}}{{m}_{{\rm{p}}}}$. According to Cairn's nonthermal statistic [52–54], the number densities of hot positrons (via nhp) and hot electrons (via nhe) are given as: $\begin{eqnarray}{n}_{{\rm{hp}}}=\left(\rho \,{\sigma }^{2}{{\rm{\Phi }}}^{2}+{\rm{\Phi }}\,\rho \,\sigma +1\right){{\rm{e}}}^{-\sigma \,{\rm{\Phi }}},\end{eqnarray}$
$\begin{eqnarray}{n}_{{\rm{he}}}=\left(\rho \,{{\rm{\Phi }}}^{2}-{\rm{\Phi }}\,\rho +1\right){{\rm{e}}}^{{\rm{\Phi }}},\end{eqnarray}$
where the temperature's ratio of hot electrons to hot positrons reads $\sigma =\frac{{T}_{{\rm{he}}}}{{T}_{{\rm{hp}}}}$. The quantity $\rho =\frac{4\alpha }{(1+3\alpha )}$ estimates the deviation from the thermalization state. However, the nonthermal parameter ρ is in the range 0 < ρ < 1.3 and (α > 0) in the practical situation [55]. The proportion of the fastest energetic electrons is determined by the parameter α. When ρ → 0, the positron (electron) density reduces to the well-known Boltzmann distribution. From equations ( $\begin{eqnarray}\frac{\partial }{\partial t}n+\frac{\partial }{\partial x}\left(nu\right)+\frac{\partial }{\partial y}\left(nv\right)=0,\end{eqnarray}$
$\begin{eqnarray}\frac{\partial }{\partial t}u+u\frac{\partial }{\partial x}u+v\frac{\partial }{\partial y}u+\frac{\partial }{\partial x}{\rm{\Phi }}-{\omega }_{{{\rm{c}}}_{1}}v=0,\end{eqnarray}$
$\begin{eqnarray}\frac{\partial }{\partial t}v+u\frac{\partial }{\partial x}v+v\frac{\partial }{\partial y}v+\frac{\partial }{\partial y}{\rm{\Phi }}+{\omega }_{{{\rm{c}}}_{1}}u=0,\end{eqnarray}$
$\begin{eqnarray}\frac{{\partial }^{2}}{\partial {x}^{2}}{\rm{\Phi }}+\frac{{\partial }^{2}}{\partial {y}^{2}}{\rm{\Phi }}+={\gamma }_{2}{{\rm{\Phi }}}^{2}+{\gamma }_{1}{\rm{\Phi }}-n+1,\end{eqnarray}$
with
$\begin{eqnarray}\begin{array}{rcl}{\gamma }_{1} & = & \left(1-\rho \right)\left({\mu }_{{\rm{e}}}\sigma +{\mu }_{{\rm{e}}}\right),\\ {\gamma }_{2} & = & \frac{\left({\mu }_{{\rm{e}}}-{\mu }_{{\rm{p}}}{\sigma }^{2}\right)}{2},\\ {\gamma }_{3} & = & \frac{\left(1+3\,\rho \right)\left({\sigma }^{3}{\mu }_{{\rm{p}}}+{\mu }_{{\rm{e}}}\right)}{6}.\end{array}\end{eqnarray}$
Before going to the numerical part of this work, we note that the range of parameters used here are [3, 54, 56, 57]: μe = 1.5 − 3.5; μp = 0.1 − 0.3; 0 < ρ < 1.3 and σ = 2 − 2.6, which satisfies numerous plasma models from the laboratory to the plasma environment such as auroral acceleration regions, solar wind and cosmic rays.3. Formation of the Davey–Stewartson equations
The standard RPM is the most used method of its type in the literature to obtain many nonlinear differential equations that govern particle dynamics in the fluid medium. Thus, it is applied here to derive the DS equations in our fluid model. The following stretching of independent variables is employed with respect to this method [58]: 12 )–(14 ) into equations (6 )–(9 ), one can easily obtain the following algebraic equations: 17 ) are obtained below: appendix . At the same order (second-order), with s = 2 and q = 2, the harmonic is found to be proportional to $| {n}_{1}^{(1)}{| }^{2}$: appendix .
$\begin{eqnarray}\xi =\epsilon \left(x-{v}_{{\rm{g}}}t\right),\quad \,\eta =\epsilon \,y,\quad \,\tau ={\epsilon }^{2}t,\end{eqnarray}$
where ε, (ε ≪ 1) is a small parameter that measures the strength of nonlinearity and vg is the group velocity of the PAWs. This is determined in the next section according to the solvability condition. Moreover, the scalar quantities n(x, y, t), u(x, y, t), v(x, y, t) and Φ(x, y, t) in the above relations can be written as: $\begin{eqnarray}n=1+\displaystyle \sum _{s=1}^{\infty }{\epsilon }^{s}\displaystyle \sum _{q=-\infty }^{\infty }{{n}_{q}}^{(s)}\left(\xi ,\eta ,\tau \right){{S}_{1}\left(x,t\right)}^{q},\end{eqnarray}$
$\begin{eqnarray}U=\displaystyle \sum _{s=1}^{\infty }{\epsilon }^{s}\displaystyle \sum _{q=-\infty }^{\infty }\left(\begin{array}{c}{u}_{q}^{(s)}\left(\xi ,\eta ,\tau \right)\\ {v}_{q}^{(s)}\left(\xi ,\eta ,\tau \right)\end{array}\right){{S}_{1}\left(x,t\right)}^{s},\end{eqnarray}$
$\begin{eqnarray}{\rm{\Phi }}=\displaystyle \sum _{s=1}^{\infty }{\epsilon }^{s}\displaystyle \sum _{q=-\infty }^{\infty }{{\rm{\Phi }}}_{q}^{(s)}\left(\xi ,\eta ,\tau \right){{S}_{1}\left(x,t\right)}^{q},\end{eqnarray}$
where ${S}_{1}{\left(x,t\right)}^{q}={{\rm{e}}}^{{\rm{i}}\,q\left(k\,x-\omega \,t\right)}$ with k and ω being the scaled wavenumber and frequency of the carrier wave, respectively. Thus, the relationship between the quantities ${({{n}_{q}}^{(s)})}^{* }={{n}_{-q}}^{(s)}$, ${({{u}_{q}}^{(s)})}^{* }={{u}_{-q}}^{(s)}$, ${({{v}_{q}}^{(s)})}^{* }={{v}_{-q}}^{(s)}$ and ${({{{\rm{\Phi }}}_{q}}^{(s)})}^{* }={{{\rm{\Phi }}}_{-q}}^{(s)}$ must be satisfied. By substituting equations ( $\begin{eqnarray}\begin{array}{c}\,-{\rm{i}}{n}_{1}^{(1)}\,\omega -{u}_{1}^{(1)}\,k=0,\\ {\rm{i}}{{\rm{\Phi }}}_{1}^{(1)}\,k-{\rm{i}}{u}_{1}^{(1)}\,\omega -\,{\omega }_{{{\rm{c}}}_{1}}\,{v}_{1}^{(1)}=0,\\ \,-\,{\rm{i}}{v}_{1}^{(1)}\,\omega +{\omega }_{{{\rm{c}}}_{1}}{u}_{1}^{(1)}=0,\\ \,-{k}^{2}{{\rm{\Phi }}}_{1}^{(1)}-\,{{\rm{\Phi }}}_{1}^{(1)}\,{\gamma }_{1}+{n}_{1}^{(1)}=0.\end{array}\end{eqnarray}$
Thus, their corresponding solutions are giving as: $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Phi }}}_{1}^{(1)} & = & \frac{{n}_{1}^{(1)}}{{k}^{2}+{\gamma }_{1}},\quad \,{u}_{1}^{(1)}=-\frac{{n}_{1}^{(1)}k\omega }{\left({k}^{2}+{\gamma }_{1}\right)\left(-{\omega }^{2}+{\omega }_{{{\rm{c}}}_{1}}^{2}\right)},\\ {v}_{1}^{(1)} & = & \frac{-{\rm{i}}{n}_{1}^{(1)}{\omega }_{{{\rm{c}}}_{1}}^{2}k}{\left({k}^{2}+{\gamma }_{1}\right)\left({\omega }^{2}-{\omega }_{{{\rm{c}}}_{1}}^{2}\right)}.\end{array}\end{eqnarray}$
Then, the dispersion relation for the PAWs reads: $\begin{eqnarray*}{\omega }^{2}=\frac{{k}^{2}}{{k}^{2}+{\gamma }_{1}}+{\omega }_{{{\rm{c}}}_{1}}^{2}.\end{eqnarray*}$
When the MF is neglected (${\omega }_{{{\rm{c}}}_{1}}=0$), the dispersion relation reaches to the same relation already found in [57]. Now, the second-order mode and first harmonic (s = 2 and q = 1), gives the following relations: $\begin{eqnarray}\begin{array}{l}{({k}^{2}+{\gamma }_{1})}^{2}{n}_{1}^{(2)}-{({k}^{2}+{\gamma }_{1})}^{3}\,{{\rm{\Phi }}}_{1}^{(2)}+2\,{\rm{i}}({k}^{2}+{\gamma }_{1})k\frac{\partial }{\partial \xi }{n}_{1}^{(1)}=0,\\ {\rm{i}}\,({k}^{2}+{\gamma }_{1})\,k\,{\omega }_{{{\rm{c}}}_{1}}\,{v}_{{\rm{g}}}({\omega }^{2}-{\omega }_{{{\rm{c}}}_{1}}^{2})\,\frac{\partial }{\partial \xi }\,{n}_{1}^{(1)}+({\omega }^{4}+{\omega }_{{{\rm{c}}}_{1}}^{4})\\ \times \,({k}^{2}+{\gamma }_{1})\frac{\partial }{\partial \eta }{n}_{1}^{(1)}\\ -{\rm{i}}\omega \,{({k}^{2}+{\gamma }_{1})}^{2}{(\omega -{\omega }_{{{\rm{c}}}_{1}})}^{2}{(\omega +{\omega }_{{{\rm{c}}}_{1}})}^{2}{v}_{1}^{(2)}\\ +{\omega }_{{{\rm{c}}}_{1}}{(\omega -{\omega }_{{{\rm{c}}}_{1}})}^{2}{(\omega +{\omega }_{{{\rm{c}}}_{1}})}^{2}{({k}^{2}+{\gamma }_{1})}^{2}{u}_{1}^{(2)}=0,\\ {\rm{i}}\,k\,{({k}^{2}+{\gamma }_{1})}^{2}{(\omega +{\omega }_{{{\rm{c}}}_{1}})}^{2}{(\omega -{\omega }_{{{\rm{c}}}_{1}})}^{2}\,{{\rm{\Phi }}}_{1}^{(2)}\\ -(k\omega \,{v}_{{\rm{g}}}-{\omega }^{2}+{\omega }_{{{\rm{c}}}_{1}}^{2})({k}^{2}+{\gamma }_{1})({\omega }^{2}-{\omega }_{{{\rm{c}}}_{1}}^{2})\frac{\partial }{\partial \xi }\,{n}_{1}^{(1)}\\ -{\omega }_{{{\rm{c}}}_{1}}{(\omega -{\omega }_{{{\rm{c}}}_{1}})}^{2}{(\omega +{\omega }_{{{\rm{c}}}_{1}})}^{2}\,{({k}^{2}+{\gamma }_{1})}^{2}{v}_{1}^{(2)}\\ -{\rm{i}}\omega \,{({k}^{2}+{\gamma }_{1})}^{2}{(\omega -{\omega }_{{{\rm{c}}}_{1}})}^{2}{(\omega +{\omega }_{{{\rm{c}}}_{1}})}^{2}{u}_{1}^{(2)}=0,\\ -{\rm{i}}({k}^{2}+{\gamma }_{1})\,k\,({\omega }^{2}-{\omega }_{{{\rm{c}}}_{1}}^{2}){u}_{1}^{(2)}+{\rm{i}}({k}^{2}+{\gamma }_{1})\omega \,({\omega }^{2}-{\omega }_{{{\rm{c}}}_{1}}^{2}){n}_{1}^{(2)}\\ +(({k}^{2}+{\gamma }_{1})\,{v}_{{\rm{g}}}({\omega }^{2}-{\omega }_{{{\rm{c}}}_{1}}^{2})-k\omega )\frac{\partial }{\partial \xi }{n}_{1}^{(1)}+{\rm{i}}{\omega }_{{{\rm{c}}}_{1}}\,k\frac{\partial }{\partial \eta }{n}_{1}^{(1)}=0.\end{array}\end{eqnarray}$
The solutions of the system ( $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Phi }}}_{1}^{(2)} & = & \frac{2\,{\rm{i}}k\frac{\partial }{\partial \xi }{n}_{1}^{(1)}}{{\left({k}^{2}+{\gamma }_{1}\right)}^{2}}+\frac{{n}_{1}^{(2)}}{{k}^{2}+{\gamma }_{1}},\\ {u}_{1}^{(2)} & = & -\frac{{\omega }_{{{\rm{c}}}_{1}}\frac{\partial }{\partial \eta }{n}_{1}^{(1)}+{n}_{1}^{(2)}k\omega }{-{k}^{2}{\omega }^{2}+{\omega }_{{{\rm{c}}}_{1}}^{2}{k}^{2}-{\omega }^{2}{\gamma }_{1}+{\gamma }_{1}{\omega }_{{{\rm{c}}}_{1}}^{2}}+\frac{{\rm{i}}\left(\frac{\partial }{\partial \xi }{n}_{1}^{(1)}\right)\left({F}_{1}+{F}_{2}\right)}{F3+F4},\\ {v}_{1}^{(2)} & = & \frac{{\rm{i}}\left(\frac{\partial }{\partial \eta }{n}_{1}^{(1)}\right)\omega +{\rm{i}}k{\omega }_{{{\rm{c}}}_{1}}{n}_{1}^{(2)}}{-{k}^{2}{\omega }^{2}+{\omega }_{{{\rm{c}}}_{1}}^{2}{k}^{2}-{\omega }^{2}{\gamma }_{1}+{\omega }_{{{\rm{c}}}_{1}}^{2}{\gamma }_{1}}+\frac{{\omega }_{{{\rm{c}}}_{1}}\,{F}_{5}\,\frac{\partial }{\partial \xi }{n}_{1}^{(1)}}{{F}_{3}+{F}_{4}}.\end{array}\end{eqnarray}$
The group velocity (vg) of PAWs is obtained through the compatibility condition as follows: $\begin{eqnarray}{v}_{{\rm{g}}}=\frac{-2\,k\omega \,{\gamma }_{1}\left(-{\omega }^{2}+{\omega }_{{{\rm{c}}}_{1}}^{2}\right)}{\left({\omega }^{4}+{\omega }_{{{\rm{c}}}_{1}}^{4}-1\right){k}^{4}+{F}_{6}},\end{eqnarray}$
and the coefficients F1−6 are given in the $\begin{eqnarray}\begin{array}{l}{({k}^{2}+{\gamma }_{1})}^{2}{n}_{2}^{(2)}-(4\,{k}^{2}+{\gamma }_{1})\,{({k}^{2}+{\gamma }_{1})}^{2}\,{{\rm{\Phi }}}_{2}^{(2)}-| {n}_{1}^{(1)}{| }^{2}\,{\gamma }_{2}=0,\\ -2\,{\rm{i}}\omega \,{({k}^{2}+{\gamma }_{1})}^{2}{(\omega -{\omega }_{{{\rm{c}}}_{1}})}^{2}{(\omega +{\omega }_{{{\rm{c}}}_{1}})}^{2}{v}_{2}^{(2)}\\ +| {n}_{1}^{(1)}{| }^{2}\,{\omega }_{{{\rm{c}}}_{1}}{k}^{3}\omega \\ +{\omega }_{{{\rm{c}}}_{1}}{(\omega -{\omega }_{{{\rm{c}}}_{1}})}^{2}{(\omega +{\omega }_{{{\rm{c}}}_{1}})}^{2}{({k}^{2}+{\gamma }_{1})}^{2}{u}_{2}^{(2)}=0,\\ -{\omega }_{{{\rm{c}}}_{1}}{(\omega -{\omega }_{{{\rm{c}}}_{1}})}^{2}{(\omega +{\omega }_{{{\rm{c}}}_{1}})}^{2}{({k}^{2}+{\gamma }_{1})}^{2}{v}_{2}^{(2)}\\ -2\,{\rm{i}}\omega \,{({k}^{2}+{\gamma }_{1})}^{2}{(\omega -{\omega }_{{{\rm{c}}}_{1}})}^{2}{(\omega +{\omega }_{{{\rm{c}}}_{1}})}^{2}{u}_{2}^{(2)}\\ +2\,{\rm{i}}\,k{({k}^{2}+{\gamma }_{1})}^{2}{(\omega +{\omega }_{{{\rm{c}}}_{1}})}^{2}{(\omega -{\omega }_{{{\rm{c}}}_{1}})}^{2}\,{{\rm{\Phi }}}_{2}^{(2)}\\ +{\rm{i}}| {n}_{1}^{(1)}{| }^{2}\,{k}^{3}{\omega }^{2}=0,\end{array}\end{eqnarray}$
and thus, the corresponding solutions read: $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Phi }}}_{2}^{(2)} & = & {C}_{6}{\left|{n}_{1}^{(1)}\right|}^{2},\quad \,{n}_{2}^{(2)}={C}_{4}{\left|{n}_{1}^{(1)}\right|}^{2},\\ {u}_{2}^{(2)} & = & {C}_{5}{\left|{n}_{1}^{(1)}\right|}^{2},\quad \,{v}_{2}^{(2)}={C}_{7}{\left|{n}_{1}^{(1)}\right|}^{2},\end{array}\end{eqnarray}$
and the coefficients C4−7 are given in the Combining the second-order harmonic with s = 2 and q = 0 and the third-order equations, one can obtain the following system: 22 ) and using the above solutions leads to the following amplitude equation: appendix . Equations (23 ) and (25 ) are called DS equations in (2+1)-dimensional form, which were first obtained by Davey and Stewartson to show the behavior of a modulated wave train in shallow water [59].
$\begin{eqnarray}\begin{array}{l}{k}^{2}({k}^{2}+2\,{\gamma }_{1}){n}_{0}^{(2)}-{k}^{2}{\gamma }_{1}\,({k}^{2}+2\,{\gamma }_{1})\,{{\rm{\Phi }}}_{0}^{(2)}-2\,{\left|{n}_{1}^{(1)}\right|}^{2}=0,\\ (-{v}_{{\rm{g}}}{k}^{2}\,{(\omega -{\omega }_{{{\rm{c}}}_{1}})}^{2}\,{(\omega +{\omega }_{{{\rm{c}}}_{1}})}^{2}\,({k}^{2}+2\,{\gamma }_{1})-{(\omega -{\omega }_{{{\rm{c}}}_{1}})}^{2}\,{(\omega +{\omega }_{{{\rm{c}}}_{1}})}^{2}\,{v}_{{\rm{g}}}{{\gamma }_{1}}^{2})\,\frac{\partial }{\partial \xi }{n}_{0}^{(2)}\\ +((({k}^{2}+{\gamma }_{1})\,{\omega }^{2}-{\omega }_{{{\rm{c}}}_{1}}^{2}({k}^{2}+2\,{\gamma }_{1}))(({k}^{2}+{\gamma }_{1})\,{\omega }^{2}-{k}^{2}{\omega }_{{{\rm{c}}}_{1}}^{2})+{{\gamma }_{1}}^{2}{\omega }_{{{\rm{c}}}_{1}}^{2})\,\frac{\partial }{\partial \eta }{v}_{0}^{(2)}\\ +{(\omega -{\omega }_{{{\rm{c}}}_{1}})}^{2}{(\omega +{\omega }_{{{\rm{c}}}_{1}})}^{2}{({k}^{2}+{\gamma }_{1})}^{2}\,\frac{\partial }{\partial \xi }{u}_{0}^{(2)}+2\,k\omega \,({k}^{2}+{\gamma }_{1})({\omega }^{2}-{\omega }_{{{\rm{c}}}_{1}}^{2})\,\frac{\partial }{\partial \xi }{\left|{n}_{1}^{(1)}\right|}^{2}\\ -2\,{\rm{i}}\left({k}^{2}+{\gamma }_{1}\right)k{\omega }_{{{\rm{c}}}_{1}}\left({\omega }^{2}-{\omega }_{{{\rm{c}}}_{1}}^{2}\right)\,\frac{\partial }{\partial \eta }{\left|{n}_{1}^{(1)}\right|}^{2}=0,\\ ({({k}^{2}+{\gamma }_{1})}^{3}{\omega }_{{{\rm{c}}}_{1}}^{6}-3\,(3\,{k}^{2}{{\gamma }_{1}}^{2}+{{\gamma }_{1}}^{3}+(-3\,{k}^{4}{\omega }^{2}\\ +3\,{k}^{4}){\gamma }_{1}+{k}^{6}){\omega }^{2}{\omega }_{{{\rm{c}}}_{1}}^{4}+3\,{\omega }^{4}({k}^{6}+3\,{k}^{2}{{\gamma }_{1}}^{2}+{{\gamma }_{1}}^{3})\,{\omega }_{{{\rm{c}}}_{1}}^{2}-{\omega }^{6}{({k}^{2}+{\gamma }_{1})}^{3})\,\frac{\partial }{\partial \xi }{{\rm{\Phi }}}_{0}^{(2)}\\ +{v}_{{\rm{g}}}{(\omega -{\omega }_{{{\rm{c}}}_{1}})}^{3}{(\omega +{\omega }_{{{\rm{c}}}_{1}})}^{3}{({k}^{2}+{\gamma }_{1})}^{3}\frac{\partial }{\partial \xi }{u}_{0}^{(2)}-{k}^{2}{\omega }^{2}({k}^{2}+{\gamma }_{1})({\omega }^{2}-{\omega }_{{{\rm{c}}}_{1}}^{2})\frac{\partial }{\partial \xi }{\left|{n}_{1}^{(1)}\right|}^{2}\\ +{\rm{i}}({k}^{2}+{\gamma }_{1})\omega \,{\omega }_{{{\rm{c}}}_{1}}{k}^{2}({\omega }^{2}-{\omega }_{{{\rm{c}}}_{1}}^{2})\,\frac{\partial }{\partial \eta }{\left|{n}_{1}^{(1)}\right|}^{2}=0,\\ -{(\omega -{\omega }_{{{\rm{c}}}_{1}})}^{3}{(\omega +{\omega }_{{{\rm{c}}}_{1}})}^{3}{({k}^{2}+{\gamma }_{1})}^{3}\frac{\partial }{\partial \eta }{{\rm{\Phi }}}_{0}^{(2)}+{v}_{{\rm{g}}}{(\omega -{\omega }_{{{\rm{c}}}_{1}})}^{3}{(\omega +{\omega }_{{{\rm{c}}}_{1}})}^{3}{({k}^{2}+{\gamma }_{1})}^{3}\frac{\partial }{\partial \xi }{v}_{0}^{(2)}\\ -{\rm{i}}(k{v}_{{\rm{g}}}-\omega )({k}^{2}+{\gamma }_{1}){\omega }_{{{\rm{c}}}_{1}}{k}^{2}({\omega }^{2}-{\omega }_{{{\rm{c}}}_{1}}^{2})\,\frac{\partial }{\partial \xi }{\left|{n}_{1}^{(1)}\right|}^{2}\\ -{k}^{2}({\omega }^{2}-2\,{\omega }_{{{\rm{c}}}_{1}}^{2})({k}^{2}+{\gamma }_{1})\,({\omega }^{2}-{\omega }_{{{\rm{c}}}_{1}}^{2})\,\frac{\partial }{\partial \eta }{\left|{n}_{1}^{(1)}\right|}^{2}=0.\end{array}\end{eqnarray}$
Solving system ( $\begin{eqnarray}\begin{array}{rcl} & & {\delta }_{1}\frac{{\partial }^{2}}{\partial {\xi }^{2}}{n}_{0}^{(2)}-\frac{{\partial }^{2}}{\partial {\eta }^{2}}{n}_{0}^{(2)}\\ & & \quad -\,{\delta }_{2}\frac{{\partial }^{2}}{\partial {\xi }^{2}}{\left|{n}_{1}^{(1)}\right|}^{2}-{\delta }_{3}\frac{{\partial }^{2}}{\partial {\eta }^{2}}{\left|{n}_{1}^{(1)}\right|}^{2}=0,\end{array}\end{eqnarray}$
with the coefficients, $\begin{eqnarray}\begin{array}{rcl}{\delta }_{1} & = & {\gamma }_{1}{{v}_{{\rm{g}}}}^{2}-1,\\ {\delta }_{2} & = & \frac{{k}^{2}\left(2\,{v}_{{\rm{g}}}{k}^{3}\omega \,{\gamma }_{1}\left({k}^{2}+2\,{\gamma }_{1}\right)+{F}_{7}-2\,{k}^{2}{\gamma }_{2}\right)}{{\left({k}^{2}+{\gamma }_{1}\right)}^{4}{\left(-{\omega }^{2}+{\omega }_{{{\rm{c}}}_{1}}^{2}\right)}^{2}},\\ {\delta }_{3} & = & -\frac{{k}^{2}\left({\omega }_{{{\rm{c}}}_{1}}^{2}{\gamma }_{1}{\left({k}^{2}+{\gamma }_{1}\right)}^{2}-{k}^{2}{\gamma }_{1}\left({k}^{2}+{\gamma }_{1}\right)-2\,{k}^{2}{\gamma }_{2}\right)}{{\left({k}^{2}+{\gamma }_{1}\right)}^{4}{\left(-{\omega }^{2}+{\omega }_{{{\rm{c}}}_{1}}^{2}\right)}^{2}}.\end{array}\end{eqnarray}$
Applying the same method to the third-order algebraic equations and simultaneously using the above solutions and equations, one can obtain the second equation of the DS system: $\begin{eqnarray}\begin{array}{rcl} & & {\rm{i}}\frac{\partial }{\partial \tau }{n}_{1}^{(1)}+{\lambda }_{1}\frac{{\partial }^{2}}{\partial {\xi }^{2}}{n}_{1}^{(1)}+{\lambda }_{2}\frac{{\partial }^{2}}{\partial {\eta }^{2}}{n}_{1}^{(1)}\\ & & \quad +{\lambda }_{3}{\left|{n}_{1}^{(1)}\right|}^{2}{n}_{1}^{(1)}+{\lambda }_{4}{n}_{0}^{(2)}{n}_{1}^{(1)}=0,\end{array}\end{eqnarray}$
with their corresponding coefficients as: $\begin{eqnarray}\begin{array}{rcl}{\lambda }_{1} & = & -\frac{\left({\omega }_{{{\rm{c}}}_{1}}^{2}\left({k}^{2}+{\gamma }_{1}\right)\left(3\,{k}^{2}-{\gamma }_{1}\right)+3\,{k}^{4}\right)\omega \,{\gamma }_{1}}{2\,{\left({k}^{2}+{\gamma }_{1}\right)}^{2}{\left({\omega }_{{{\rm{c}}}_{1}}^{2}{k}^{2}+{\omega }_{{{\rm{c}}}_{1}}^{2}{\gamma }_{1}+{k}^{2}\right)}^{2}},\\ {\lambda }_{2} & = & \frac{\omega \,{\gamma }_{1}}{2\,\left({k}^{2}+{\gamma }_{1}\right)\left({k}^{2}+{\omega }_{{{\rm{c}}}_{1}}^{2}\left({k}^{2}+{\gamma }_{1}\right)\right)},\\ {\lambda }_{3} & = & -\frac{\omega \,\left(A{\omega }_{{c}_{1}}^{4}+B{\omega }_{{{\rm{c}}}_{1}}^{2}+C\right)}{{F}_{8}\,\left({k}^{2}+{\gamma }_{1}\right)\left(\left(4\,{k}^{4}+5\,{k}^{2}{\gamma }_{1}+{{\gamma }_{1}}^{2}\right){\omega }_{{{\rm{c}}}_{1}}^{2}+4\,{k}^{4}\right)},\\ {\lambda }_{4} & = & -\frac{\omega \,{k}^{2}\left({\gamma }_{1}\left({k}^{2}+{\gamma }_{1}\right)-2\,{\gamma }_{2}\right)}{2\,\left({k}^{2}+{\gamma }_{1}\right){\gamma }_{1}\left({k}^{2}+{\omega }_{{{\rm{c}}}_{1}}^{2}\left({k}^{2}+{\gamma }_{1}\right)\right)},\end{array}\end{eqnarray}$
and the coefficients F7, F8, A, B and C are given in the 4. Modulational instability analysis
The emergence of modulated waves, such as solitons, is strongly dependent on the stability criteria. Thereby, in this section, we are going to look for the (in)stability conditions of PAWs, according to the DS equations. We set $H={n}_{1}^{(1)}$ and $S={n}_{0}^{(2)}$, and then equations (23 ) and (25 ) become: 27 ) and (28 ). These equations have trivial homogeneous solutions in the form:
$\begin{eqnarray}{\rm{i}}\frac{\partial H}{\partial \tau }+{\lambda }_{1}\frac{{\partial }^{2}H}{\partial {\xi }^{2}}+{\lambda }_{2}\,\frac{{\partial }^{2}H}{\partial {\eta }^{2}}+{\lambda }_{3}{\left|H\right|}^{2}\,H+{\lambda }_{4}S\,H=0,\end{eqnarray}$
$\begin{eqnarray}{\delta }_{1}\frac{{\partial }^{2}S}{\partial {\xi }^{2}}-\frac{{\partial }^{2}S}{\partial {\eta }^{2}}-{\delta }_{2}\frac{{\partial }^{2}| H{| }^{2}}{\partial {\xi }^{2}}-{\delta }_{3}\frac{{\partial }^{2}| H{| }^{2}}{\partial {\eta }^{2}}=0.\end{eqnarray}$
In this section, we are going to look after the MI of a plane wave solution of equations ( $\begin{eqnarray}\begin{array}{rcl}H & = & {H}_{0}{{\rm{e}}}^{{\rm{i}}\left(-\varpi \,\tau +{a}_{2}\eta +{a}_{1}\xi +\varphi \right)},\\ S & = & {S}_{0},\end{array}\end{eqnarray}$
where the dispersion relation reads: $\begin{eqnarray}\varpi =-{\lambda }_{3}{H}_{0}^{2}+{\lambda }_{1}{a}_{1}^{2}+{\lambda }_{2}{a}_{2}^{2}-{\lambda }_{4}{S}_{0}.\end{eqnarray}$
Furthermore, we insert small perturbations in both the amplitude and phase to perform the stability of the system. $\begin{eqnarray}\begin{array}{rcl}H & = & \left({H}_{0}+{\rm{\Delta }}H\right){{\rm{e}}}^{{\rm{i}}\left(-\varpi \,\tau +{a}_{2}\eta +{a}_{1}\xi +{\rm{\Delta }}\varphi +\varphi \right)},\\ S & = & {S}_{0}+{\rm{\Delta }}S.\end{array}\end{eqnarray}$
Thus, the perturbed solutions are assumed to be: $\begin{eqnarray}\left[\begin{array}{c}{\rm{\Delta }}H\\ {\rm{\Delta }}S\\ {\rm{\Delta }}\varphi \end{array}\right]=\left[\begin{array}{c}\delta H\\ \delta S\\ \delta \varphi \end{array}\right]\,{\rm{Re}}\,({{\rm{e}}}^{{\rm{i}}\left(-{\rm{\Omega }}\,\tau +{\theta }_{2}\eta +{\theta }_{1}\xi \right)}),\end{eqnarray}$
where θ1,2 are the perturbed wavenumbers. Using all the assumptions above and after a whole mathematical calculation, the nonlinear dispersion relation is obtained as: $\begin{eqnarray}{{\rm{\Omega }}}^{2}={\left({\lambda }_{1}{\theta }_{1}^{2}+{\lambda }_{2}{\theta }_{2}^{2}\right)}^{2}\left(1+\frac{2\,{H}_{0}^{2}\left(\frac{{\lambda }_{4}\left({\delta }_{2}{\theta }_{1}^{2}+{\delta }_{3}{\theta }_{2}^{2}\right)}{-{\delta }_{1}{\theta }_{1}^{2}+{\theta }_{2}^{2}}-{\lambda }_{3}\right)}{{\lambda }_{1}{\theta }_{1}^{2}+{\lambda }_{2}{\theta }_{2}^{2}}\right).\end{eqnarray}$
According to the instability condition (Ω2 < 0), the threshold amplitude reads: $\begin{eqnarray}{H}_{0,{\rm{cr}}}^{2}=\frac{\left({\lambda }_{1}{\theta }_{1}^{2}+{\lambda }_{2}{\theta }_{2}^{2}\right)\left(-{\delta }_{1}{\theta }_{1}^{2}+{\theta }_{2}^{2}\right)}{-2\,{\delta }_{1}{\theta }_{1}^{2}+2\,{\theta }_{2}^{2}-2\,{\lambda }_{4}\left({\delta }_{2}{\theta }_{1}^{2}+{\delta }_{3}{\theta }_{2}^{2}\right)}.\end{eqnarray}$
If Ω2 > 0, the perturbed frequency is real and the system is stable. On the other hand, if Ω2 < 0, the perturbed frequency becomes imaginary and the system develops an instability. The MI growth rate generally takes the form ${\rm{\Gamma }}=\sqrt{-{{\rm{\Omega }}}^{2}}$. Thus, figure 1 displays the growth rate of MI with different plasma parameters versus the wavenumber θ2. The red color indicates the stable region whereas the other colors show the unstable region. In the (${\omega }_{{{\rm{c}}}_{1}},{\theta }_{2}$) plane (see figure 1(a)), a single lobe of instability is observed and the maximum MI gain decreases with the increase in the MF parameter. Therefore, modulated PAWs become stable for higher values of ${\omega }_{{{\rm{c}}}_{1}}$. This behavior is also observed in the (μe, θ2) plane (see figure 1(b)) and (μp, θ2) plane (see figure 1(c)). However, in the case of figure 1(b), the MI gain displays three lobes of instability. One can see in figure 1(d), which displays the MI gain in the (ρ, θ2) plane, that the maximum growth rate occurs at higher values of the nonthermal parameter and the instability tends to disappear at lower values of ρ, i.e. when positrons obey Maxwellian distribution.
Figure 1. Growth rate of MI is plotted against θ2 and the other physical parameters: (a) (${\omega }_{{{\rm{c}}}_{1}},{\theta }_{2}$), (b) (μe, θ2), (c) (μp, θ2) and (d) (ρ, θ2). Here, μe = 1.5, ${\omega }_{{{\rm{c}}}_{1}}=0.3$, ρ = 0.5, θ1 = 0.1, μp = 0.3 and σ = 2. |
Figure 2 displays the MI gain versus the wavenumber θ1 with variations in the MF parameter ${\omega }_{{{\rm{c}}}_{1}}$ (figure 2(a)), the density ratio of hot electron and cold positron μe (figure 2(b)), the density ratio of cold positron and hot positron μp (figure 2(c)), and nonthermal parameter ρ (figure 2(d)). One observes here two lobes of instability that are symmetric with respect to the line θ1 = 0 and the maximum MI gain decreases with the increase in the above plasma parameters. When the MF parameter ${\omega }_{{{\rm{c}}}_{1}}$ and the density ratio of cold positron and hot positron μp increase, the MI gain is shifted towards higher values of the wavenumber θ1, as shown in panels (a) and (c) of figure 2, respectively. However, in the (μe, θ2) plane (see figure 2(b)), the MI gain is bounded. In these previous cases, the system is stable for the higher values of different plasma parameters. This is different from the case obtained in figure 2(d) where there is no MI gain both for the lower and higher values of the nonthermal parameter. The MI appears only for the finite values of the nonthermal parameter.
Figure 2. Growth rate of MI is plotted against θ1 and the other physical parameters: (a) (${\omega }_{{{\rm{c}}}_{1}},{\theta }_{1}$), (b) (μe, θ1), (c) (μp, θ1), (d) (ρ, θ1). Here, μe = 1.5, ${\omega }_{{{\rm{c}}}_{1}}=0.3$, ρ = 0.5, μp = 0.3 and σ = 2. |
According to equation (33 ), we can approximate the stability condition through the following function:
$\begin{eqnarray}g(\rho ,k)=\left({\lambda }_{1}+{\lambda }_{2}\right)\left(-{\lambda }_{3}+\displaystyle \frac{{\lambda }_{4}\left({\delta }_{2}+{\delta }_{3}\right)}{1-{\delta }_{1}}\right).\end{eqnarray}$
Then, the stability/instability diagram of parameters ρ and k is plotted in figure 3, which shows the region where the localized structures, such as dromion solitons, appear. This figure shows that the stability/instability zone is very sensitive to the nonthermal parameter. This can help us to choose the appropriate values of parameters k and ρ for which the dromion solitons emerge. The localized structures are generally observed in the unstable region of MI, i.e Ω2 < 0. However, all the values of the parameters (k, ρ) that fall inside the green region can lead to the formation of the dromion solitons.
Figure 3. Diagram of stability/instability versus rho, with μe = 1.5, μp = 0.3, ${\omega }_{{{\rm{c}}}_{1}}=0.3$, ρ = 0.5 and σ = 2. |
Previous MI studies regarding PAWs have only been done in the (1+1) dimension, and the MI in the (2+1)-dimension differs from the (1+1) dimension, both qualitatively and quantitatively. The MI growth rate of PAWs in the (1+1) dimension depends on the product of the coefficients of nonlinear and dispersive terms [3], which are the functions of the physical parameters involved. If the product is positive, the amplitude-modulated envelope is unstable for perturbation wavelengths larger than a critical value, hence the carrier wave leads to the formation of a bright soliton. On the other hand, if the product is negative, the amplitude-modulated envelope will be stable against external perturbations and may propagate in the form of a dark soliton. Moreover, the stability zone is very well separated from the instability zone. From our plasma model of (2+1) PAW, we noticed that the growth rate and instability condition is more complicated, involving longitudinal and transverse components through θ1 and θ2, respectively. In addition, the (in)stability diagram obtained in our study is more complex and we observed an alternation between the stability and instability zones, as demonstrated by figure 3.
5. Dromion structure
In this section, we look for a particular kind of soliton solution of equations (27 ) and (28 ) called the dromion soliton. However, it is necessary to convert these equations into standard DS ones. Furthermore, by introducing a new dependent variable (ζ, ς) and pivoting from 0 to $\frac{\pi }{4}$ the coordinate axis, we obtain [48]: 37 ) and (38 ) reads: 37 ) and (38 ) reads:
$\begin{eqnarray}\begin{array}{rcl}\zeta & = & \frac{\xi }{\sqrt{{\delta }_{1}{\lambda }_{3}+{\delta }_{2}{\lambda }_{4}}},\quad \,\varsigma =\frac{\eta }{\sqrt{-{\delta }_{3}{\lambda }_{4}+{\lambda }_{3}}},\\ X & = & \frac{\left(\zeta +\varsigma \right)}{\sqrt{2}},\quad \,Y=\frac{\left(\zeta -\varsigma \right)}{\sqrt{2}}.\end{array}\end{eqnarray}$
$\begin{eqnarray}{\rm{i}}{H}_{\tau }+a\left({H}_{XX}+{H}_{YY}\right)+b{H}_{YY}+d\,S\,H+c{\left|H\right|}^{2}H=0,\end{eqnarray}$
$\begin{eqnarray}e({S}_{XX}+{S}_{YY})+f{S}_{XY}+2{\left(\left|H\right|\right)}_{XY}^{2}=0,\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}a & = & \frac{{\lambda }_{1}}{2\,{\delta }_{1}{\lambda }_{3}+2\,{\delta }_{2}{\lambda }_{4}}+\frac{{\lambda }_{2}}{-2\,{\delta }_{3}{\lambda }_{4}+2\,{\lambda }_{2}},\\ b & = & \frac{{\lambda }_{1}}{2\,{\delta }_{1}{\lambda }_{3}+2\,{\delta }_{2}{\lambda }_{4}}-\frac{{\lambda }_{2}}{-2\,{\delta }_{3}{\lambda }_{4}+2\,{\lambda }_{2}},\\ c & = & {\lambda }_{3},\quad d={\lambda }_{4},\\ e & = & \frac{{\delta }_{1}}{2\,{\delta }_{1}{\lambda }_{3}+2\,{\delta }_{2}{\lambda }_{4}}-\frac{1}{\left(-2\,{\delta }_{3}{\lambda }_{4}+2\,{\lambda }_{2}\right)},\\ f & = & \frac{{\delta }_{1}}{2\,{\delta }_{1}{\lambda }_{3}+2\,{\delta }_{2}{\lambda }_{4}}+\frac{1}{\left(-2\,{\delta }_{3}{\lambda }_{4}+2\,{\lambda }_{2}\right)}.\end{array}\end{eqnarray}$
Now, we transform them into the bilinear form by assuming that $H=\frac{G}{F}$ and S = c (lnH)XY . Then, the bilinear form of equations ( $\begin{eqnarray}\left[\right.{\rm{i}}\,{D}_{\tau }+a\,({D}_{XX}+{D}_{yy})+b\,{D}_{XY}\left]\right.G.H=0,{D}_{XY}\,H.H=m\,G.{G}^{* },\end{eqnarray}$
where F = 1 + ϵ2 f2 + ϵ4 f4 and G = ϵ g1 + ϵ3 g3. However, according to the Hirota method [60–69] the expressions of f2, f4, g1 and g3 can be easily found. Then, the single dromion solution of DS equations ( $\begin{eqnarray}{H}_{1}=\frac{{h}_{1}{{\rm{e}}}^{{\vartheta }_{1}+{\vartheta }_{2}}}{1+{\chi }_{1}{{\rm{e}}}^{{{\vartheta }_{1}}^{* }+{\vartheta }_{1}}+{\chi }_{2}{{\rm{e}}}^{{{\vartheta }_{2}}^{* }+{\vartheta }_{2}}+{\chi }_{3}{{\rm{e}}}^{{{\vartheta }_{1}}^{* }+{{\vartheta }_{2}}^{* }+{\vartheta }_{1}+{\vartheta }_{2}}},\end{eqnarray}$
with the quantities: $\begin{eqnarray}\begin{array}{rcl}{\vartheta }_{1} & = & {\rm{i}}{\omega }_{1}t+{r}_{1}X+{b}_{1},\\ {\vartheta }_{2} & = & {\rm{i}}{\omega }_{2}t+{r}_{2}Y+{b}_{2}.\end{array}\end{eqnarray}$
Here, ϑ* is the conjugate of ϑ, ${\omega }_{1}={\rm{i}}\,a\,{r}_{1}^{2}$ and ${\omega }_{2}={\rm{i}}\,a\,{r}_{2}^{2}$, along with h1, ${({\chi }_{i})}_{i=1,2,3}$ and ${({r}_{i})}_{i=1,2}$ are all the real constants. By a similar procedure, the two-dromion solution of DS is given as: $\begin{eqnarray}{H}_{2}=\frac{{l}_{1}{{\rm{e}}}^{{\vartheta }_{1}+{\vartheta }_{3}}+{l}_{2}{{\rm{e}}}^{{\vartheta }_{2}+{\vartheta }_{3}}+{l}_{3}{{\rm{e}}}^{{{\vartheta }_{1}}^{* }+{\vartheta }_{1}+{\vartheta }_{2}+{\vartheta }_{3}}+{l}_{4}{{\rm{e}}}^{{{\vartheta }_{2}}^{* }+{\vartheta }_{1}+{\vartheta }_{2}+{\vartheta }_{3}}}{{{\rm{\Gamma }}}_{1}+{{\rm{\Gamma }}}_{2}+{{\rm{\Gamma }}}_{3}},\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}_{1} & = & 1+{L}_{1}{{\rm{e}}}^{{{\vartheta }_{1}}^{* }+{\vartheta }_{1}}+{L}_{2}{{\rm{e}}}^{{{\vartheta }_{2}}^{* }+{\vartheta }_{2}}+{L}_{3}{{\rm{e}}}^{{{\vartheta }_{3}}^{* }+{\vartheta }_{3}}+{L}_{4}\left({{\rm{e}}}^{{{\vartheta }_{2}}^{* }+{\vartheta }_{1}}+{{\rm{e}}}^{{{\vartheta }_{1}}^{* }+{\vartheta }_{2}}\right),\\ {{\rm{\Gamma }}}_{2} & = & {L}_{5}\left({{\rm{e}}}^{{{\vartheta }_{2}}^{* }+{{\vartheta }_{3}}^{* }+{\vartheta }_{1}+{\vartheta }_{3}}+{{\rm{e}}}^{{{\vartheta }_{1}}^{* }+{{\vartheta }_{3}}^{* }+{\vartheta }_{2}+{\vartheta }_{3}}\right)+{L}_{6}{{\rm{e}}}^{{{\vartheta }_{1}}^{* }+{{\vartheta }_{2}}^{2}+{\vartheta }_{1}+{\vartheta }_{2}},\\ {{\rm{\Gamma }}}_{3} & = & {L}_{7}{{\rm{e}}}^{{{\vartheta }_{2}}^{* }+{{\vartheta }_{3}}^{* }+{\vartheta }_{2}+{\vartheta }_{3}}+{L}_{8}{{\rm{e}}}^{{{\vartheta }_{1}}^{* }+{{\vartheta }_{3}}^{* }+{\vartheta }_{1}+{\vartheta }_{3}}+{L}_{9}{{\rm{e}}}^{{{\vartheta }_{1}}^{* }+{{\vartheta }_{2}}^{* }+{{\vartheta }_{3}}^{* }+{\vartheta }_{1}+{\vartheta }_{2}+{\vartheta }_{3}},\end{array}\end{eqnarray}$
where ${({l}_{i})}_{i=1,2,3,4}$, ${({L}_{i})}_{i=1-9}$, ${({b}_{i})}_{i=1,2,3}$ and ${({r}_{i})}_{i=1,2,3}$ are all the real constants: $\begin{eqnarray}\begin{array}{rcl}{\vartheta }_{1} & = & {\rm{i}}{\omega }_{1}t+{r}_{1}X+{b}_{1},\\ {\vartheta }_{2} & = & {\rm{i}}{\omega }_{2}t+{r}_{2}X+{b}_{2},\\ {\vartheta }_{3} & = & {\rm{i}}{\omega }_{3}t+{r}_{3}Y+{b}_{3}.\end{array}\end{eqnarray}$
The one-dromion soliton is depicted in figure 4, which shows the impact of the nonthermal parameter ρ and the ratio of hot electron to cold positron μe. The impact of ρ on the amplitude of this wave is shown in figures 4(a–c) and one can see that the amplitude of the one-dromion soliton increases with the increase in the value of ρ. Thus, increasing the nonthermal parameter would lead to an enhancement in the energy in the system, which causes an increase in nonlinearity. Therefore, the amplitude becomes larger, indicating that the nonthermal parameter is pumping more energy towards the background waves, which are sucked in to create the acoustic positron waves. The response of the one-dromion soliton due to μe is shown in figures 4(d–f). Here, one can observe that the amplitude of this wave drops with the increase in the value of μe. Certainly, increasing μe diminishes the nonlinearity of the system, and consequently diminishes the amplitude of the wave.
Figure 4. Plots of one-dromion solitons are plotted in the (x, y)-plane and for different values of (ρ, μe): (a) (ρ, μe)=(0.3, 1.5), (b) (ρ, μe)=(0.5, 1.5), (c) (ρ, μe)=(0.7, 1.5), (d) (ρ, μe)=(0.5, 2), (e) (ρ, μe)=(0.5, 2.5), (f) (ρ, μe)=(0.5, 3). Here, μp = 0.3, ${\omega }_{{{\rm{c}}}_{1}}=0.3$, σ = 2, t = 0, h1 = 1, r1 = r2 = b1 = 1 and ${({\chi }_{i})}_{i=1,2,3}=1$. |
A comparison of the above results with those of (1+1)-dimensional PAW plasma models is also enlightening. It is interesting to note that the DS equation supports only the compressive (positive) solitons. This behavior is different from that obtained in the 1D case. In the latter case, using a PAW model, compressive (positive) and rarefactive (negative) solitons were studied within the framework of the Korteweg–de Vries equation with emphasis on the effect of the nonthermal parameter [44]. It was found that the nonthermal parameter increases the amplitude and width of the compressive and rarefactive solitary waves.
The interactions of the two dromion solitons are depicted in Figures 5–7. In general, the interactions between two solitons are elastic and inelastic collisions. Figure 5 displays the elastic collision between two dromion solitons at time t = 8 due to the effect of the nonthermal parameter (Panel (5a) with ρ = 0.3; panel 5(b) with ρ = 0.5; panel 5(c) with ρ = 0.65; panel 5(d) with ρ = 0.7 and panel 5(e) with ρ = 0.8). Here, we observe a remarkable effect of the ρ parameter during the interaction between these two solitons. Indeed, when ρ increases, the amplitude of the two waves increases (see figures 5(a), 5(b)). For a particular value of ρ, the two dromions collide to form a single structure (see figure 5(c). As the value of ρ increases, the two dromions dissociate (see figure 5(d)) and gradually move away from each other (see figure 5(e)) with an amplitude gain due to the increase in ρ. In this context, we can confirm that the nonthermal parameter favors the phenomenon of elastic collision between two dromions in a magnetized plasma model where the dynamics of positrons is considered.
Figure 5. Profile of elastic collision between two dromion solitons is plotted against ρ: (a) ρ = 0.3, (b) ρ = 0.5, (c) ρ = 0.65, (d) ρ = 0.7, (e) (a) ρ = 0.8. Here, μe = 1.5, μp = 0.3, ${\omega }_{{{\rm{c}}}_{1}}=0.3$, t = 8, ρ = 0.5, ${({l}_{i})}_{i=1,2,3,4}=1$, ${({L}_{i})}_{i=1-9}=1$, b1 = b3 = 1, b2 = 2, r1 = r3 = 1, r2 = − 1 and σ = 2. |
Figure 6. Profile of elastic collision between two dromion solitons is plotted against different values of t: (a) t = − 10, (b) t = 0, (c) t = 3, (d) t = 5, (e) t = 10. Here, μe = 1.5, μp = 0.3, ${\omega }_{{{\rm{c}}}_{1}}=0.3$, ρ = 0.5, σ = 2, ${({l}_{i})}_{i=1,2,3,4}=1$, ${({L}_{i})}_{i=1-9}=1$, b1 = b3 = 1, b2 = 2, r1 = r3 = 1 and r2 = − 1. |
Figure 7. Profile of inelastic collision between two dromion solitons is plotted against different values of t: (a) t = − 10, (b) t = 0, (c) t = 3, (d) t = 5, (e) t = 10. Here, μe = 1.5, μp = 0.3, ${\omega }_{{{\rm{c}}}_{1}}=0.3$, ρ = 0.5, σ = 2, ${({L}_{i})}_{i=1-9}=1$, b1 = b3 = 1, b2 = 2, r1 = r3 = 1, r2 = − 1, ${({l}_{i})}_{i=1,2,3}=1$ and l4 = 1.5. |
Figure 6 illustrates the elastic collision between two dromion solitons at different times. Figure 6(a) displays two dromion solitons that come from far away with equal amplitude at time t = − 10, and collide at t = 0 (see figure 6(b)), then become unique at t = 3 with higher amplitude (see figure (6c)). After collision, each soliton keeps its characteristics constant. These kinds of interactions were obtained by Tabi et al [49] in electronegative plasma. Figure 7 displays the inelastic collision between two dromion solitons at different times. One can observe from figure 7(a) that, at t = − 10, two dromion solitons come from far away with different amplitudes and collide (see figure (7b)) at t = 0, then become a single dromion at t = 3 (see figure (7c)). The remarkable and rare phenomenon observed here is that of energy transfer during the collision. Thus, this phenomenon is justified by the amplitudes of the solitons of figures 7(d), (e) after collision. The tallest has transferred an amount of energy to the shortest. The amplitude of the tallest decreases to compensate for that of the shortest. These elastic and inelastic collision phenomena are observed between energetic particles in confined plasma, during the production of energy by thermonuclear fusion in a magnetized enclosure called a tokamak [70].
6. Conclusion
In this study, we have investigated the MI and OC between dromion structures of the PAWs in magnetized plasma with nonthermal distribution. The (2+1)-dimensional DS equations have been derived according to the standard RPM. Through the stability criteria, we have determined the region where the localized structures, such as dromion solitons, appear. In that respect, a comprehensive analysis of wave instability has been conducted, where regions of stability/instability have been revealed to be very sensitive to the nonthermal parameter, the strength of MF, the density ratio of hot electron and cold positron, and the density ratio of cold positron and hot positron. Using the Hirota method, we have shown that the plasma model supports some exact solutions such as one- and two-dromion solutions. We have discussed the elastic and inelastic collisions between two dromions. We have noticed that the characteristics of dromion solutions are very sensitive to the change of some plasma parameters. In particular, we have shown that the nonthermal parameter favors the phenomenon of elastic collision between two dromions, whereas, during the inelastic collision, the remarkable and rare phenomenon observed here is that of energy transfer during the collision. The relevance of this study is that it can help to explain the main mechanisms leading to the generation of plasma particles, through energy recombination among the available dynamical modes when the positrons and electrons interact. This insight is crucial for advancing our knowledge of plasma physics and for applications such as controlled fusion and astrophysical research.
Acknowledgments
The authors extend their appreciation to the Deanship of Scientific Research and Libraries in Princess Nourah bint Abdulrahman University for funding this research work through the Research Group project under Grant No. (RG-1445-0005).
Conflict of interest
The authors declare that they have no conflicts of interest.
Author contributions
All authors contributed equally and approved the final version of the current manuscript.
Use of AI tools declaration
The authors declare that they have not used artificial intelligence (AI) tools in the creation of this article.
Appendix. Expression of the coefficients
The coefficients of equations (18 ) and (19 ) are:
$\begin{eqnarray}\begin{array}{rcl}{F}_{1} & = & {v}_{{\rm{g}}}{k}^{3}{\omega }^{2}+{v}_{g}{\omega }_{{{\rm{c}}}_{1}}^{2}{k}^{3}+{k}^{2}{\omega }^{3}-{\omega }_{{{\rm{c}}}_{1}}^{2}{k}^{2}\omega ,\\ {F}_{2} & = & {v}_{{\rm{g}}}k{\omega }^{2}{\gamma }_{1}+k{\gamma }_{1}{\omega }_{{{\rm{c}}}_{1}}^{2}{v}_{{\rm{g}}}-{\omega }^{3}{\gamma }_{1}+\omega \,{\gamma }_{1}{\omega }_{{{\rm{c}}}_{1}}^{2},\\ {F}_{3} & = & {k}^{4}{\omega }^{4}-2\,{k}^{4}{\omega }^{2}{\omega }_{{{\rm{c}}}_{1}}^{2}+{k}^{4}{\omega }_{{{\rm{c}}}_{1}}^{4}+2\,{k}^{2}{\omega }^{4}{\gamma }_{1}-4\,{k}^{2}{\omega }^{2}{\gamma }_{1}{\omega }_{{{\rm{c}}}_{1}}^{2},\\ {F}_{4} & = & 2\,{k}^{2}{\gamma }_{1}{\omega }_{{{\rm{c}}}_{1}}^{4}+{\omega }^{4}{{\gamma }_{1}}^{2}-2\,{\omega }^{2}{{\gamma }_{1}}^{2}{\omega }_{{{\rm{c}}}_{1}}^{2}+{{\gamma }_{1}}^{2}{\omega }_{{{\rm{c}}}_{1}}^{4},\\ {F}_{5} & = & 2\,{v}_{{\rm{g}}}{k}^{3}\omega +{k}^{2}{\omega }^{2}-{\omega }_{{{\rm{c}}}_{1}}^{2}{k}^{2}+2\,{v}_{g}k\omega \,{\gamma }_{1}-{\omega }^{2}{\gamma }_{1}+{\omega }_{{{\rm{c}}}_{1}}^{2}{\gamma }_{1},\\ {F}_{6} & = & \left(-2\,{\omega }^{4}+2\,{\omega }_{{{\rm{c}}}_{1}}^{4}\right){\gamma }_{1}{k}^{2}+\left(-{\omega }^{4}+{\omega }_{{{\rm{c}}}_{1}}^{4}\right){{\gamma }_{1}}^{2}.\end{array}\end{eqnarray}$
The coefficients of equation (21 ) are:
$\begin{eqnarray}\begin{array}{rcl}{C}_{4} & = & -\frac{\left({F}_{9}-5\,{\omega }_{{{\rm{c}}}_{1}}^{2}{\omega }^{2}{\gamma }_{1}+{\omega }_{{{\rm{c}}}_{1}}^{4}{\gamma }_{1}+2\,{k}^{2}{\omega }^{2}+{\omega }_{{{\rm{c}}}_{1}}^{2}{k}^{2}\right){k}^{2}}{\left(-4\,{\omega }^{2}+{\omega }_{{{\rm{c}}}_{1}}^{2}\right){\left(-{k}^{2}{\omega }^{2}+{\omega }_{{{\rm{c}}}_{1}}^{2}{k}^{2}-{\omega }^{2}{\gamma }_{1}+{\omega }_{{{\rm{c}}}_{1}}^{2}{\gamma }_{1}\right)}^{2}}-\,\frac{4{C}_{6}{k}^{2}}{-4\,{\omega }^{2}+{\omega }_{{{\rm{c}}}_{1}}^{2}},\\ {F}_{9} & = & 4\,{k}^{2}{\omega }^{4}-5\,{\omega }_{{{\rm{c}}}_{1}}^{2}{k}^{2}{\omega }^{2}+{\omega }_{{{\rm{c}}}_{1}}^{4}{k}^{2}+4\,{\omega }^{4}{\gamma }_{1},\\ {C}_{6} & = & \frac{4\,{k}^{4}{\omega }^{4}-5\,{\omega }_{{{\rm{c}}}_{1}}^{2}{k}^{4}{\omega }^{2}+{\omega }_{{{\rm{c}}}_{1}}^{4}{k}^{4}+4\,{k}^{2}{\omega }^{4}{\gamma }_{1}-5\,{\omega }_{{{\rm{c}}}_{1}}^{2}{k}^{2}{\omega }^{2}{\gamma }_{1}+{\omega }_{{{\rm{c}}}_{1}}^{4}{k}^{2}{\gamma }_{1}}{{\left({k}^{2}+{\gamma }_{1}\right)}^{2}\left(16\,{k}^{2}{\omega }^{2}-4\,{\omega }_{{{\rm{c}}}_{1}}^{2}{k}^{2}+4\,{\omega }^{2}{\gamma }_{1}-{\gamma }_{1}{\omega }_{{{\rm{c}}}_{1}}^{2}-4\,{k}^{2}\right){\left(-{\omega }^{2}+{\omega }_{{{\rm{c}}}_{1}}^{2}\right)}^{2}}\\ & & +\frac{-4\,{\omega }^{6}{\gamma }_{2}+9\,{\omega }_{{{\rm{c}}}_{1}}^{2}{\omega }^{4}{\gamma }_{2}-6\,{\omega }_{{{\rm{c}}}_{1}}^{4}{\omega }^{2}{\gamma }_{2}+{\omega }_{{{\rm{c}}}_{1}}^{6}{\gamma }_{2}+2\,{k}^{4}{\omega }^{2}+{\omega }_{{{\rm{c}}}_{1}}^{2}{k}^{4}}{{\left({k}^{2}+{\gamma }_{1}\right)}^{2}\left(16\,{k}^{2}{\omega }^{2}-4\,{\omega }_{{{\rm{c}}}_{1}}^{2}{k}^{2}+4\,{\omega }^{2}{\gamma }_{1}-{\gamma }_{1}{\omega }_{{{\rm{c}}}_{1}}^{2}-4\,{k}^{2}\right){\left(-{\omega }^{2}+{\omega }_{{{\rm{c}}}_{1}}^{2}\right)}^{2}},\\ {C}_{5} & = & \frac{-4\,{C}_{6}k\omega }{-4\,{\omega }^{2}+{\omega }_{{{\rm{c}}}_{1}}^{2}}+\frac{{k}^{3}\left(2\,{\omega }^{2}+{\omega }_{{{\rm{c}}}_{1}}^{2}\right)\omega }{\left(2\,\omega -{\omega }_{{{\rm{c}}}_{1}}\right)\left(2\,\omega +{\omega }_{{{\rm{c}}}_{1}}\right){\left(\omega -{\omega }_{{{\rm{c}}}_{1}}\right)}^{2}{\left(\omega +{\omega }_{{{\rm{c}}}_{1}}\right)}^{2}{\left({k}^{2}+{\gamma }_{1}\right)}^{2}},\\ {C}_{8} & = & \frac{{\rm{i}}{\omega }_{{{\rm{c}}}_{1}}k\left(2\,{C}_{6}{\left(\omega -{\omega }_{{{\rm{c}}}_{1}}\right)}^{2}{\left(\omega +{\omega }_{{{\rm{c}}}_{1}}\right)}^{2}{\left({k}^{2}+{\gamma }_{1}\right)}^{2}+3\,{k}^{2}{\omega }^{2}\right)}{{\left(\omega -{\omega }_{{{\rm{c}}}_{1}}\right)}^{2}{\left(\omega +{\omega }_{{{\rm{c}}}_{1}}\right)}^{2}{\left({k}^{2}+{\gamma }_{1}\right)}^{2}\left(-4\,{\omega }^{2}+{\omega }_{{{\rm{c}}}_{1}}^{2}\right)}.\end{array}\end{eqnarray}$
The coefficients of equations (24 ) and (26 ) are:
$\begin{eqnarray}\begin{array}{rcl}{F}_{7} & = & {\gamma }_{1}\left({\left({k}^{2}+{\gamma }_{1}\right)}^{2}{\omega }_{{{\rm{c}}}_{1}}^{2}+2\,{v}_{{\rm{g}}}k\omega \,{{\gamma }_{1}}^{2}\right)+{k}^{2}{\gamma }_{1}\left({k}^{2}+{\gamma }_{1}\right),\\ {F}_{8} & = & 6\,{\gamma }_{1}(({k}^{6}+3\,{k}^{4}{\gamma }_{1}+3\,{k}^{2}{{\gamma }_{1}}^{2}+{{\gamma }_{1}}^{3}){\omega }_{{{\rm{c}}}_{1}}^{2}+{k}^{6}+2\,{k}^{4}{\gamma }_{1}+{k}^{2}{{\gamma }_{1}}^{2}),\\ A & = & 24\,{k}^{12}{\gamma }_{1}+126\,{k}^{10}{{\gamma }_{1}}^{2}+270\,{k}^{8}{{\gamma }_{1}}^{3}+300\,{k}^{6}{{\gamma }_{1}}^{4}+180\,{k}^{4}{{\gamma }_{1}}^{5}+54\,{k}^{2}{{\gamma }_{1}}^{6}+6\,{{\gamma }_{1}}^{7},\\ B & = & 72\,{k}^{12}{\gamma }_{1}+306\,{k}^{10}{{\gamma }_{1}}^{2}+504\,{k}^{8}{{\gamma }_{1}}^{3}-16\,{k}^{8}{\gamma }_{1}{\gamma }_{2}+396\,{k}^{6}{{\gamma }_{1}}^{4}-48\,{k}^{6}{{\gamma }_{1}}^{2}{\gamma }_{2}\\ & & +144\,{k}^{4}{{\gamma }_{1}}^{5}-36\,{k}^{6}{\gamma }_{1}{\gamma }_{3}+48\,{k}^{6}{{\gamma }_{2}}^{2}-48\,{k}^{4}{{\gamma }_{1}}^{3}{\gamma }_{2}+18\,{k}^{2}{{\gamma }_{1}}^{6}-45\,{k}^{4}{{\gamma }_{1}}^{2}{\gamma }_{3}+66\,{k}^{4}{\gamma }_{1}{{\gamma }_{2}}^{2}\\ & & -16\,{k}^{2}{{\gamma }_{1}}^{4}{\gamma }_{2}-9\,{k}^{2}{{\gamma }_{1}}^{3}{\gamma }_{3}+18\,{k}^{2}{{\gamma }_{1}}^{2}{{\gamma }_{2}}^{2},\\ C & = & 48\,{k}^{12}{\gamma }_{1}+162\,{k}^{10}{{\gamma }_{1}}^{2}+198\,{k}^{8}{{\gamma }_{1}}^{3}-24\,{k}^{8}{\gamma }_{1}{\gamma }_{2}+102\,{k}^{6}{{\gamma }_{1}}^{4}-48\,{k}^{6}{{\gamma }_{1}}^{2}{\gamma }_{2}+18\,{k}^{4}{{\gamma }_{1}}^{5}\\ & = & -36\,{k}^{6}{\gamma }_{1}{\gamma }_{3}+48\,{k}^{6}{{\gamma }_{2}}^{2}-24\,{k}^{4}{{\gamma }_{1}}^{3}{\gamma }_{2}+8\,{k}^{4}{\gamma }_{1}{{\gamma }_{2}}^{2}.\end{array}\end{eqnarray}$