Some electrostatic structures of the mKP equation for a nonthermal plasma system by a unified solver technique

Abstract

The reductive perturbation technique is used to obtain the Modified KP equation at critical densities distinguished by the MKPE in plasma ion pair with fast electron positron. The new structures reveal that super-solitary and period waveforms are derived via the mathematical analysis of Jacobi-elliptic function expansions (MJEFE) for MKPE. The electrostatic new structures such as super-soliton, cnoidals, shock-like, super-dispersive and superperiodic plasma waves which existed at critical densities have been introduced. The positrons (electron) nonthermality supports on nonlinear dispersive new structures have been discussed. Also, many of the obtained electrostatic solutions are important and may be applicable in ionosphere plasma observations.

Keywords:

• MKPE
• super-solutions
• Jacobian elliptic function
• applications in fluids

Mathematics Subject Classification (2010):

• 35C08
• 35A08
• 35Q51
• 35Q53

1. Introduction

The dynamical nonlinear studies for new nonlinear waves such as super nonlinear and shock waves play effective roles in applied physics of fluids and plasmas [1–5]. Supersolitons are distorted solitary waves that may be identified by the electric field's three local minima and three local maxima. Also, there have been very few reports of real satellite findings of supersoliton structures [6–8]. It was reported that the mKdV equation that accounted for the effects of a cubic nonlinearity has the periodical solutions of a super-nonlinear kind [6]. The NLS equation is used to examine electron-acoustic supernonlinear forms in plasmas via phase plane analysis. Also, the NLS equation waves are demonstrated by including an external periodical force in the dynamical behaviour [9]. The effects of generalized $(r,q)$ distributions on supernonlinear DA periodic waves have been investigated in dusty plasma using bifurcation of waves [10].

The physical importance of plasma containing pair-ions (PI) and their thermodynamically characteristics has a serious implementations in astronomy, terrestrial laboratories, solar wind studies [11–14], active galactic-nuclei [15,16], pulsar environments [17,18], nanotechnology [19] and plasma laboratories [20]. Non-thermal particle distributions have not been described by the thermality equilibrium states of particles [21]. These distributions observed by satellites and spacecrafts explain the non-Maxwellian states in Mars, Saturn, Earth bow-shock, Moon vicinity [22–28].

In fluid dynamics and plasma studies, accurate physical search shows the propagation of distinct nonlinear forms represents new profiles as bright soliton, dark shocklike, rational, supernonlinear periodic, explosive and huge structures [29–31]. The dark cylindrical soliton characteristics in nonthermality mesospheric fluid model have been inspected [29]. The time damped superthermal cylindrical structure has been discussed in space plasma [31]. Saha et al. studied the superperiodic progression in auroral zone [32]. Also, the localized, rational and huge propagations have been taken into in consideration for pairs plasma [24,33]. Finally, periodic chaotic and supernonlinearity profiles were investigated using dynamics stability of positrons acoustics mode in e-p-i plasmas [31].

On the other hand, in positron electron plasmas, new wave classes that do not present in the typical electron ion plasma were discovered. It was demonstrated that so-called periodical supernonlinear waveforms of the acoustic waves type, whose phase trajectories enclose the separatrix on the phase plane, can occur in the epid plasma [34–37]. Furthermore, the supernonlinear acoustic propagation in quantum electron positron plasma for the KdV and mKdV equations has been studied. The superperiodic waves are examined by some various system parameters [38]. The acoustic wave's properties and their fractal structures have been examined in auroral Earth's plasma. Also, the dynamics and existence of chaotic properties were studied by Lyapunov exponents [39]. Also, it was reported that the chaotic Alfven structures behaviour have been propagated in ionosphere plasma [40].

Abdelrahman and AlKhidhr in [41] introduced the math-solver to resolve the so famous duffing equation, which emerges from so many models of the NPDEs describing physical cases in applied science, using appropriate wave transformation. The duffing equation takes place in so many vital applications, see for example [42–54]. Our main aim is to apply the solver method to resolve the MKP equation in space of ionosphere fluid.

2. Unified solver

In view of unified solver technique [41], some family solutions of the following duffing model: (1) ${\mathrm{\Lambda }}_1{\mathrm{\Theta }}^{\mathrm{\prime \prime }}+{\mathrm{\Lambda }}_2{\mathrm{\Theta }}^3+{\mathrm{\Lambda }}_3\mathrm{\Theta }=0,$(1) are

Family I: (2) ${\mathrm{\Theta }}_1(x,t)=\pm \sqrt{\frac{-2{\mathrm{\Lambda }}_1}{{\mathrm{\Lambda }}_2}}m\mathrm{sn}(\zeta ).$(2) When $m\to 1$, Equation (2(2) ${\mathrm{\Theta }}_1(x,t)=\pm \sqrt{\frac{-2{\mathrm{\Lambda }}_1}{{\mathrm{\Lambda }}_2}}m\mathrm{sn}(\zeta ).$(2) ) becomes (3) ${\mathrm{\Theta }}_1(x,t)=\pm \sqrt{\frac{-2{\mathrm{\Lambda }}_1}{{\mathrm{\Lambda }}_2}}\tanh (\zeta ).$(3) Family II: (4) ${\mathrm{\Theta }}_2(x,t)=\pm \sqrt{\frac{-{\mathrm{\Lambda }}_1}{2{\mathrm{\Lambda }}_2}}m\mathrm{sn}(\zeta )-\sqrt{\frac{{\mathrm{\Lambda }}_1}{2{\mathrm{\Lambda }}_2}}m\mathrm{cn}(\zeta ).$(4) When $m\to 1$, Equation (4(4) ${\mathrm{\Theta }}_2(x,t)=\pm \sqrt{\frac{-{\mathrm{\Lambda }}_1}{2{\mathrm{\Lambda }}_2}}m\mathrm{sn}(\zeta )-\sqrt{\frac{{\mathrm{\Lambda }}_1}{2{\mathrm{\Lambda }}_2}}m\mathrm{cn}(\zeta ).$(4) ) becomes (5) ${\mathrm{\Theta }}_2(x,t)=\pm \sqrt{\frac{-{\mathrm{\Lambda }}_1}{2{\mathrm{\Lambda }}_2}}\tanh (\zeta )-\sqrt{\frac{{\mathrm{\Lambda }}_1}{2{\mathrm{\Lambda }}_2}}\mathrm{sech}(\zeta ).$(5) Family III: (6) ${\mathrm{\Theta }}_3(x,t)=\pm \sqrt{\frac{-{\mathrm{\Lambda }}_1}{2{\mathrm{\Lambda }}_2}}m\mathrm{sn}(\zeta )+\sqrt{\frac{{\mathrm{\Lambda }}_1}{2{\mathrm{\Lambda }}_2}}m\mathrm{cn}(\zeta ).$(6) When $m\to 1$, Equation (6(6) ${\mathrm{\Theta }}_3(x,t)=\pm \sqrt{\frac{-{\mathrm{\Lambda }}_1}{2{\mathrm{\Lambda }}_2}}m\mathrm{sn}(\zeta )+\sqrt{\frac{{\mathrm{\Lambda }}_1}{2{\mathrm{\Lambda }}_2}}m\mathrm{cn}(\zeta ).$(6) ) becomes (7) ${\mathrm{\Theta }}_3(x,t)=\pm \sqrt{\frac{-{\mathrm{\Lambda }}_1}{2{\mathrm{\Lambda }}_2}}\tanh (\zeta )+\sqrt{\frac{{\mathrm{\Lambda }}_1}{2{\mathrm{\Lambda }}_2}}\mathrm{sech}(\zeta ).$(7) Family IV: (8) ${\mathrm{\Theta }}_4(x,t)=\pm \sqrt{\frac{2{\mathrm{\Lambda }}_1}{{\mathrm{\Lambda }}_2}}m\mathrm{cn}(\zeta ).$(8) As long as $m\to 1$, Equation (8(8) ${\mathrm{\Theta }}_4(x,t)=\pm \sqrt{\frac{2{\mathrm{\Lambda }}_1}{{\mathrm{\Lambda }}_2}}m\mathrm{cn}(\zeta ).$(8) ) becomes (9) ${\mathrm{\Theta }}_4(x,t)=\pm \sqrt{\frac{2{\mathrm{\Lambda }}_1}{{\mathrm{\Lambda }}_2}}\mathrm{sech}(\zeta ).$(9)

3. Mathematical model

In plasma model containing pairs ion, the nonthermal electrons effect on the soliton shape has been examined [25]. In this study, a two-dimensional fluid system in [25] with extra fast positrons is considered. The model equation for this fluid reads (10) $\begin{array}{rl}& \frac{\mathrm{\partial }{n}_{+}}{\mathrm{\partial }t}+\frac{\mathrm{\partial }(n_{+}{u}_{+})}{\mathrm{\partial }x}+\frac{\mathrm{\partial }(n_{+}{V}_{+})}{\mathrm{\partial }y}=0,\end{array}$(10) (11) $\begin{array}{rl}& \frac{\mathrm{\partial }{n}_{-}}{\mathrm{\partial }t}+\frac{\mathrm{\partial }}{\mathrm{\partial }x}(n_{-}{u}_{-})+\frac{\mathrm{\partial }}{\mathrm{\partial }y}(n_{-}{V}_{-})=0,\end{array}$(11) (12) $\begin{array}{rl}& \frac{\mathrm{\partial }{u}_{+}}{\mathrm{\partial }t}+u_{+}\frac{\mathrm{\partial }{u}_{+}}{\mathrm{\partial }x}+V_{+}\frac{\mathrm{\partial }{u}_{+}}{\mathrm{\partial }y}+\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }x}=0,\end{array}$(12) (13) $\begin{array}{rl}& \frac{\mathrm{\partial }{u}_{-}}{\mathrm{\partial }t}+u_{-}\frac{\mathrm{\partial }{u}_{-}}{\mathrm{\partial }x}+V_{-}\frac{\mathrm{\partial }{u}_{-}}{\mathrm{\partial }y}-\mathrm{\alpha Q}\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }x}=0,\end{array}$(13) (14) $\begin{array}{rl}& \frac{\mathrm{\partial }{V}_{+}}{\mathrm{\partial }t}+u_{+}\frac{\mathrm{\partial }{V}_{+}}{\mathrm{\partial }x}+V_{+}\frac{\mathrm{\partial }{V}_{+}}{\mathrm{\partial }y}+\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }y}=0,\end{array}$(14) (15) $\begin{array}{rl}& \frac{\mathrm{\partial }{V}_{-}}{\mathrm{\partial }t}+u_{-}\frac{\mathrm{\partial }{V}_{-}}{\mathrm{\partial }x}+V_{-}\frac{\mathrm{\partial }{V}_{-}}{\mathrm{\partial }y}-\mathrm{\alpha Q}\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }y}=0.\end{array}$(15) (16) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{y}^2}={\delta }_n{n}_{-}-n_{+}+{\delta }_e(1-\mathrm{\beta \varphi }+\beta {\varphi }^2){\mathrm{e}}^{\varphi }\\ & -{\delta }_p(1+\mathrm{\sigma \beta \varphi }+{\sigma }^2\beta {\varphi }^2){\mathrm{e}}^{(-\mathrm{\sigma \varphi })}.\end{array}$(16) The equilibrium condition implies (17) ${\delta }_n+{\delta }_e=1+{\delta }_p,$(17) $n_{-,+}$ is density of ions. $\sigma =T_e/T_p$ (electrons to positrons temperatures ratio), $\alpha =(\frac{Z_{-}}{Z_{+}})$, where $Z_{\pm }$ are charges, ${\delta }_n=\frac{n_{-0}{Z}_{-}}{n_{+0}{Z}_{+}}$, ${\delta }_e=n_{e0}/n_{+0}{Z}_{+},$ and ${\delta }_p=n_{p0}/n_{+0}{Z}_{+}$, $\beta =\frac{4\rho }{1+3\rho }$, where $\rho$ is the fast electrons (positrons) parameter. To obtain KP for electrostatic IA waves, the stretched time (space) coordinates given in the form: (18) $T={ϵ}^{\frac{3}{2}}t,X={ϵ}^{\frac{1}{2}}(x-\mathrm{\lambda t}),\mathrm{and}Y=ϵy,$(18) where ϵ is a dimensionless small expansion parameter and λ is the velocity of IAWs. All dependent variables in the system are expanded in power series of ϵ as (19) $\begin{array}{rl}{n}_{+,-}& =1+{ϵ}^2{n}_{+,-}^{(1)}+{ϵ}^4{n}_{+,-}^{(2)}+{ϵ}^6{n}_{+,-}^{(3)}+..,\\ u_{+,-}& ={ϵ}^2(u_{+,-}^{(1)}+{ϵ}^2{u}_{+,-}^{(2)}+{ϵ}^4{u}_{+,-}^{(3)}+..),\\ V_{+,-}& ={ϵ}^3(V_{+,-}^{(1)}+{ϵ}^2{V}_{+,-}^{(2)}+{ϵ}^4{V}_{+,-}^{(3)}+..),\\ \varphi & ={ϵ}^2({\varphi }^{(1)}+{ϵ}^2{\varphi }^{(2)}+{ϵ}^4{\varphi }^{(3)}+..).\end{array}$(19) The boundary conditions are $|\xi |\to \mathrm{\infty },n_{+,-}=1,u_{+,-}=V_{+,-}=0,\varphi =0.$ By using the lowest order and next higher order in ϵ, one can obtain the MKPE in the form: (20) $\frac{\mathrm{\partial }}{\mathrm{\partial }X}(\frac{\mathrm{\partial }{\varphi }^{(1)}}{\mathrm{\partial }T}+A{\varphi }^{(1)}\frac{\mathrm{\partial }{\varphi }^{(1)}}{\mathrm{\partial }X}+R\frac{{\mathrm{\partial }}^3{\varphi }^{(1)}}{\mathrm{\partial }{X}^3})+P\frac{{\mathrm{\partial }}^2{\varphi }^{(1)}}{\mathrm{\partial }{Y}^2}=0.$(20) where $\begin{array}{rl}A& =\frac{3}{{\lambda }^4}(1-{\delta }_n{\alpha }^2{Q}^2-\frac{{\lambda }^4}{3}({\delta }_e-{\sigma }^2{\delta }_p))\\ & \times \frac{\lambda }{2(1-\beta )({\delta }_e+\sigma {\delta }_p)},\\ R& =-\frac{(3\rho +1)\lambda }{2(\rho -1)({\delta }_e+\sigma {\delta }_p)},P=\frac{\lambda }{2},\\ \lambda & =\sqrt{\frac{1+\mathrm{\alpha Q}{\mu }_{-}}{(1-\beta )({\delta }_e+{\delta }_p\sigma )}}.\end{array}$For small amplitude limits, a critical density point is foreseeable to reproduction at the nonlinearity vanishing point. So, the transformations $\tau ={ϵ}^{3}t$ for times and $\xi =ϵ(x-\mathrm{\lambda t}),\eta ={ϵ}^{2}y$ for space, ϵ ($\lambda )$ is small number (IAs speed). Modified KP equation at $Q=Q_c$ is given by (21) $\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }(\frac{\mathrm{\partial }}{\mathrm{\partial }\tau }\varphi +G{\varphi }^2\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }\xi }+R\frac{{\mathrm{\partial }}^3\varphi }{\mathrm{\partial }{\xi }^3})+P\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{\eta }^2}=0,$(21) with (22) $\begin{array}{rl}{Q}_c& =\pm \sqrt{\frac{3+({\sigma }^2{\delta }_p-{\delta }_e){\lambda }^4}{3{\alpha }^2{\delta }_n}},\end{array}$(22) (23) $\begin{array}{rl}G& =\frac{\begin{array}{c}(15\rho +1)({\delta }_e+{\sigma }^3{\delta }_p){\lambda }^6\\ -15(3\rho +1)({\alpha }^3{Q}^3{\delta }_n+1)\end{array}}{4(\rho -1)({\delta }_e+\sigma {\delta }_p){\lambda }^5},\\ R& =-\frac{(3\rho +1)\lambda }{2(\rho -1)({\delta }_e+\sigma {\delta }_p)},P=\frac{\lambda }{2}.\end{array}$(23) By using transformations: (24) $\begin{array}{rl}\chi & =\mathrm{L\xi }+\mathrm{M\eta }-\tau ({\upsilon }_1+{\upsilon }_2),\end{array}$(24) (25) $\begin{array}{rl}t& =\tau ,\end{array}$(25) (26) $\begin{array}{rl}\varphi (\chi )& =\varphi (x,y,t),\end{array}$(26) where L is the x cosine direction and M is the y cosine direction.

The MKP equation is reduced to (27) $3\theta \frac{{\mathrm{d}}^2\varphi }{\mathrm{d}{\chi }^2}+h{\varphi }^3-3(v-S)\varphi =0.$(27)

4. Application

Here, we applied the unified solver method to solve Equation (27(27) $3\theta \frac{{\mathrm{d}}^2\varphi }{\mathrm{d}{\chi }^2}+h{\varphi }^3-3(v-S)\varphi =0.$(27) ). Comparing this equation with Equation (1(1) ${\mathrm{\Lambda }}_1{\mathrm{\Theta }}^{\mathrm{\prime \prime }}+{\mathrm{\Lambda }}_2{\mathrm{\Theta }}^3+{\mathrm{\Lambda }}_3\mathrm{\Theta }=0,$(1) ) gives ${\mathrm{\Lambda }}_1=3\theta$, ${\mathrm{\Lambda }}_2=h$ and ${\mathrm{\Lambda }}_3=-3(v-S)$. Thus the solutions of Equation (21(21) $\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }(\frac{\mathrm{\partial }}{\mathrm{\partial }\tau }\varphi +G{\varphi }^2\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }\xi }+R\frac{{\mathrm{\partial }}^3\varphi }{\mathrm{\partial }{\xi }^3})+P\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{\eta }^2}=0,$(21) ) and its related electrostatic field and the related electrostatic field $E_f(x,t)$ are:

Family I: (28) $\begin{array}{rl}{u}_1(x,t)& =\pm \sqrt{\frac{-6\theta }{h}}m\mathrm{sn}(x-(S-\theta (1+m^2))t),\end{array}$(28) (29) $\begin{array}{rl}{E}_{f1}(x,t)& =-\sqrt{6}m\sqrt{-\frac{\theta }{h}}\mathrm{cn}(\mathrm{St}-(m^2+1)\mathrm{\theta t}-x)\mathrm{dn}\\ & \times (\mathrm{St}-(m^2+1)\mathrm{\theta t}-x)\end{array}$(29) At $m\to 1$, the solution of Equation (28(28) $\begin{array}{rl}{u}_1(x,t)& =\pm \sqrt{\frac{-6\theta }{h}}m\mathrm{sn}(x-(S-\theta (1+m^2))t),\end{array}$(28) ) goes to (30) $\begin{array}{rl}{u}_1(x,t)& =\pm \sqrt{\frac{-6\theta }{h}}\tanh (x-(S-2\theta )t),\end{array}$(30) (31) $\begin{array}{rl}{E}_{f1}(x,t)& =-\sqrt{6}\sqrt{-\frac{\theta }{h}}{\mathrm{sech}}^2(t(S-2\theta )-x).\end{array}$(31) Family II: (32) $\begin{array}{rl}{u}_2(x,t)& =\pm \sqrt{\frac{-3\theta }{2h}}m\mathrm{sn}(x-(S-\frac{1}{2}\theta (2-m^2))t)\\ & -\sqrt{\frac{3\theta }{2h}}m\mathrm{cn}(x-(S-\frac{1}{2}\theta (2-m^2))t),\end{array}$(32) (33) $\begin{array}{rl}{E}_{f2}(x,t)& =\sqrt{\frac{3}{2}}m\mathrm{dn}(\mathrm{St}+(0.5m^2-1.)\mathrm{\theta t}-x)\\ & \times (\sqrt{\frac{\theta }{h}}\mathrm{sn}(\mathrm{St}+(0.5m^2-1.)\mathrm{\theta t}-x)\\ & -\sqrt{-\frac{\theta }{h}}\mathrm{cn}(\mathrm{St}+(0.5m^2-1.)\mathrm{\theta t}-x).\end{array}$(33) At $m\to 1$, the solution of Equation (32(32) $\begin{array}{rl}{u}_2(x,t)& =\pm \sqrt{\frac{-3\theta }{2h}}m\mathrm{sn}(x-(S-\frac{1}{2}\theta (2-m^2))t)\\ & -\sqrt{\frac{3\theta }{2h}}m\mathrm{cn}(x-(S-\frac{1}{2}\theta (2-m^2))t),\end{array}$(32) ) goes to (34) $\begin{array}{rl}{u}_2(x,t)& =\pm \sqrt{\frac{-3\theta }{2h}}\tanh (x-(S-\frac{1}{2}\theta )t)\\ & -\sqrt{\frac{3\theta }{2h}}\mathrm{sech}(x-(S-\frac{1}{2}\theta )t).\end{array}$(34) (35) $\begin{array}{rl}{E}_{f2}(x,t)& =1.22474\sqrt{\frac{\theta }{h}}\tanh (t(S-0.5\theta )-x)\\ & \times \mathrm{sech}(t(S-0.5\theta )-x)\\ & -1.22474\sqrt{-\frac{\theta }{h}}{\mathrm{sech}}^2(t(S-0.5\theta )-x)\end{array}$(35) Family III: (36) $\begin{array}{rl}{u}_3(x,t)& =\pm \sqrt{\frac{-3\theta }{2h}}m\mathrm{sn}(x-(S-\frac{1}{2}\theta (2-m^2))t)\\ & +\sqrt{\frac{3\theta }{2h}}m\mathrm{cn}(x-(S-\frac{1}{2}\theta (2-m^2))t).\end{array}$(36) (37) $\begin{array}{rl}{E}_{f3}(x,t)& =-\sqrt{\frac{3}{2}}\mathrm{mdn}(\mathrm{St}+(0.5m^2-1.)\mathrm{\theta t}-x)\\ & (\sqrt{-\frac{\theta }{h}}\mathrm{cn}(\mathrm{St}+(0.5m^2-1.)\mathrm{\theta t}-x)\\ & +\sqrt{\frac{\theta }{h}}\mathrm{sn}(\mathrm{St}+(0.5m^2-1.)\mathrm{\theta t}-x)).\end{array}$(37)

At $m\to 1$, the solution of Equation (36(36) $\begin{array}{rl}{u}_3(x,t)& =\pm \sqrt{\frac{-3\theta }{2h}}m\mathrm{sn}(x-(S-\frac{1}{2}\theta (2-m^2))t)\\ & +\sqrt{\frac{3\theta }{2h}}m\mathrm{cn}(x-(S-\frac{1}{2}\theta (2-m^2))t).\end{array}$(36) ) goes to (38) $\begin{array}{rl}{u}_3(x,t)& =\pm \sqrt{\frac{-3\theta }{2h}}\tanh (x-(S-\frac{1}{2}\theta )t)\\ & +\sqrt{\frac{3\theta }{2h}}\mathrm{sech}(x-(S-\frac{1}{2}\theta )t),\end{array}$(38) (39) $\begin{array}{rl}{E}_{f3}(x,t)& =-1.22474\sqrt{-\frac{\theta }{h}}{\mathrm{sech}}^2(t(S-0.5\theta )-x)\\ & -1.22474\sqrt{\frac{\theta }{h}}\tanh (t(S-0.5\theta )-x)\\ & \times \mathrm{sech}(t(S-0.5\theta )-x).\end{array}$(39) Family IV: (40) $\begin{array}{rl}{u}_4(x,t)& =\pm \sqrt{\frac{6\theta }{h}}m\mathrm{cn}(x-(S-\theta (1-2m^2))t),\end{array}$(40) (41) $\begin{array}{rl}{E}_{f4}(x,t)& =-\sqrt{6}m\sqrt{\frac{\theta }{h}}\mathrm{dn}(\mathrm{St}+(1.m^2-0.5)\mathrm{\theta t}-x)\\ & \times \mathrm{sn}(\mathrm{St}+(1.m^2-0.5)\mathrm{\theta t}-x).\end{array}$(41) At $m\to 1$, the solution of Equation (40(40) $\begin{array}{rl}{u}_4(x,t)& =\pm \sqrt{\frac{6\theta }{h}}m\mathrm{cn}(x-(S-\theta (1-2m^2))t),\end{array}$(40) ) (42) $\begin{array}{rl}{u}_4(x,t)& =\pm \sqrt{\frac{6\theta }{h}}\mathrm{sech}(x-(S+\theta )t),\end{array}$(42) (43) $\begin{array}{rl}{E}_{f4}(x,t)& =-2.44949\sqrt{\frac{\theta }{h}}\tanh (t(0.5\theta +S)-x)\\ & \times \mathrm{sech}(t(0.5\theta +S)-x).\end{array}$(43)

5. Results and discussion

Nonthermal plasma system contains electron-positron-ions (e-p-i), and it should be emphasized that the KP equation cannot represent the solitary propagations in the system at a particular value known as the critical point. As a result, the MKP equation was developed to describe the system at this important point. A computational technique used to find different solutions to the MKP problem. The propagation of nonlinear solitonic like and other excitations of two direction IAs and its related electrostatic fields $E_{\mathrm{f}}$ are depicted by MKP equation (27(27) $3\theta \frac{{\mathrm{d}}^2\varphi }{\mathrm{d}{\chi }^2}+h{\varphi }^3-3(v-S)\varphi =0.$(27) ). The new solitons and electric excitations obtained in this study have been examined using plasma parameters linked space plasma data (${\delta }_p=0.7,{\delta }_e=0.8,u=0.5,\sigma =1.0,\alpha =1.0,L=0.7,\upsilon =0.01)$ [31,55]. At critical value $Q=Q_c$, the KP dynamical behaviour cannot describe the wave mode. So, new derived MKP solutions can characterize this investigation at those critical nonlinear points by EMJEF. The EMJEF accords a lot of different solution profiles. Equations (28(28) $\begin{array}{rl}{u}_1(x,t)& =\pm \sqrt{\frac{-6\theta }{h}}m\mathrm{sn}(x-(S-\theta (1+m^2))t),\end{array}$(28) ) and (29(29) $\begin{array}{rl}{E}_{f1}(x,t)& =-\sqrt{6}m\sqrt{-\frac{\theta }{h}}\mathrm{cn}(\mathrm{St}-(m^2+1)\mathrm{\theta t}-x)\mathrm{dn}\\ & \times (\mathrm{St}-(m^2+1)\mathrm{\theta t}-x)\end{array}$(29) ) are introduced solutions ${\varphi }_c=u_1(x;t)$ and $E_{f1}(x,t)$ that give solutions in three forms which respect on model densities and temperatures as plotted in Figures 1–6. The cnoidal form and its super electrostatic form are the first announced forms as drawn in Figures 1 and 2. Another forms insert a new super periodic potential propagation and related super field soliton as in Figures 3 and 4. The third shock potential and related solitonic structures are given in Figures 5 and 6. From the aforementioned relationships, it appears that the nonthermal fast parameters reduced all profile amplitudes in three type waves in addition to phase change that appears clearly in electrostatic field's structures. Equations (32(32) $\begin{array}{rl}{u}_2(x,t)& =\pm \sqrt{\frac{-3\theta }{2h}}m\mathrm{sn}(x-(S-\frac{1}{2}\theta (2-m^2))t)\\ & -\sqrt{\frac{3\theta }{2h}}m\mathrm{cn}(x-(S-\frac{1}{2}\theta (2-m^2))t),\end{array}$(32) ) and (36(36) $\begin{array}{rl}{u}_3(x,t)& =\pm \sqrt{\frac{-3\theta }{2h}}m\mathrm{sn}(x-(S-\frac{1}{2}\theta (2-m^2))t)\\ & +\sqrt{\frac{3\theta }{2h}}m\mathrm{cn}(x-(S-\frac{1}{2}\theta (2-m^2))t).\end{array}$(36) ) defined both forms ${\varphi }_c=u_2(x;t)$, ${\varphi }_c=u_3(x;t)$ that identify some solitonic types as in Figures 7–11. It was plotted for variable values of $\chi$ and $\rho$. Figures 7 and 8 perform a periodical and super-nonlinear pictures. Figure 9 denotes influential supersolitary shape. In another result, the shock (soliton) form is set in Figures 10 and 11. Also, for Equations (32(32) $\begin{array}{rl}{u}_2(x,t)& =\pm \sqrt{\frac{-3\theta }{2h}}m\mathrm{sn}(x-(S-\frac{1}{2}\theta (2-m^2))t)\\ & -\sqrt{\frac{3\theta }{2h}}m\mathrm{cn}(x-(S-\frac{1}{2}\theta (2-m^2))t),\end{array}$(32) ), (36(36) $\begin{array}{rl}{u}_3(x,t)& =\pm \sqrt{\frac{-3\theta }{2h}}m\mathrm{sn}(x-(S-\frac{1}{2}\theta (2-m^2))t)\\ & +\sqrt{\frac{3\theta }{2h}}m\mathrm{cn}(x-(S-\frac{1}{2}\theta (2-m^2))t).\end{array}$(36) ), it was reported that $\rho$ rebates amplitudes without no variation in phases as described in Figures 7–11.

On the other hand, the related field equations (33(33) $\begin{array}{rl}{E}_{f2}(x,t)& =\sqrt{\frac{3}{2}}m\mathrm{dn}(\mathrm{St}+(0.5m^2-1.)\mathrm{\theta t}-x)\\ & \times (\sqrt{\frac{\theta }{h}}\mathrm{sn}(\mathrm{St}+(0.5m^2-1.)\mathrm{\theta t}-x)\\ & -\sqrt{-\frac{\theta }{h}}\mathrm{cn}(\mathrm{St}+(0.5m^2-1.)\mathrm{\theta t}-x).\end{array}$(33) ), (37(37) $\begin{array}{rl}{E}_{f3}(x,t)& =-\sqrt{\frac{3}{2}}\mathrm{mdn}(\mathrm{St}+(0.5m^2-1.)\mathrm{\theta t}-x)\\ & (\sqrt{-\frac{\theta }{h}}\mathrm{cn}(\mathrm{St}+(0.5m^2-1.)\mathrm{\theta t}-x)\\ & +\sqrt{\frac{\theta }{h}}\mathrm{sn}(\mathrm{St}+(0.5m^2-1.)\mathrm{\theta t}-x)).\end{array}$(37) ) describing some new electrostatic structures as super, periodic and train soliton fields are shown in Figures 12, 13 and 14. It was noted that fast positron changes the electrostatic wave phase shift. Furthermore, Equations (40(40) $\begin{array}{rl}{u}_4(x,t)& =\pm \sqrt{\frac{6\theta }{h}}m\mathrm{cn}(x-(S-\theta (1-2m^2))t),\end{array}$(40) ) and (41(41) $\begin{array}{rl}{E}_{f4}(x,t)& =-\sqrt{6}m\sqrt{\frac{\theta }{h}}\mathrm{dn}(\mathrm{St}+(1.m^2-0.5)\mathrm{\theta t}-x)\\ & \times \mathrm{sn}(\mathrm{St}+(1.m^2-0.5)\mathrm{\theta t}-x).\end{array}$(41) ) are different solution forms ${\varphi }_c=u_4(x;t)$ and $E_{f4}(x,t)$ which have four solutions in which its differences with $\chi$ and $\rho$ are represented in Figures 15–18. Interestingly, Figures 15 and 16 expound the wave periodic and soliton trains. Moreover, super-nonlinearity and solitonic waves are plotted in Figures 17 and 18. Finally, for any solution described by (40(40) $\begin{array}{rl}{u}_4(x,t)& =\pm \sqrt{\frac{6\theta }{h}}m\mathrm{cn}(x-(S-\theta (1-2m^2))t),\end{array}$(40) ), as established in Figures 15–18, it was reported that the fast factor $\rho$ shows a noticeable modification in solitary phases and decreases its amplitudes.

Figure 1. Graph of cnoidal wave ${\varphi }_c$ for $\chi$ & $\rho$.

Figure 2. Graph of $E_{f1}(x,t)$ with $\chi$ and $\rho$.

Figure 3. Graph of periodic wave ${\varphi }_c$ with $\chi$ & $\rho$.

Figure 4. Graph of $E_{f1}(x,t)$ with $\chi$ and $\rho$.

Figure 5. Graph of shock wave ${\varphi }_c$ with $\chi$ & $\rho$.

Figure 6. Graph of $E_{f1}(x,t)$ with $\chi$ and $\rho$.

Figure 7. Graph of cnoidal wave $R_e{\varphi }_c$ with $\chi$ & $\rho$.

Figure 8. Graph of periodic wave $R_e{\varphi }_c$ with $\chi$ & $\rho$.

Figure 9. Graph of super-periodic wave $I_m{\varphi }_c$ with $\chi$ & $\rho$.

Figure 10. Graph of shock wave $R_e{\varphi }_c$ with $\chi$ & $\rho$.

Figure 11. Graph of soliton wave $I_m{\varphi }_c$ with $\chi$ v $\rho$.

Figure 12. Graph of $\mathrm{Re}{E}_{f3}(x,t)$ with $\chi$ and $\rho$.

Figure 13. Graph of $\mathrm{Im}{E}_{f3}(x,t)$ with $\chi$ and $\rho$.

Figure 14. Graph of $|E_{f3}(x,t)|$ with $\chi$ and $\rho$

Figure 15. Graph of periodic wave $|{\varphi }_c|$ with $\chi$ & $\rho$.

Figure 16. Graph of soliton-train wave $|{\varphi }_c|$ with $\chi$ & $\rho$.

Figure 17. Graph of super periodic soliton $I_m{\varphi }_c$ with $\chi$ & $\rho$.

Figure 18. Graph of soliton wave $|{\varphi }_c|$ with $\chi$ & $\rho$.

6. Conclusion

It is noted that at certain value called critical point, the KP equation cannot depict the solitonic propagations in the system. So, the MKP equation acquired for describing the system at this critical point. A symbolic computational method employed for obtaining various MKP equation solutions. Most potential and field excitations obtained here as shocks, solitonlike, cnoidals, supershocks, trains and super-nonlinearity are presented from the study of MKPE which characterized wave solutions in nonthermal fluid plasmas via EMJEF. This investigation announced that the parameters $\rho$ can control the solitary characteristics in plasma. Accordingly, these fast electron (positron) parameters may reduce the profiles amplitude and sometimes modify its wave phases. Finally, the results of new structures obtained from MKPE might be reused to establish the acoustic electrostatical plasma wave in Earth' s ionosphere.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Additional information

Funding

The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number (IF2/PSAU/2022/01/20300).