Electron-acoustic solitary potential in nonextensive streaming plasma

The linear/nonlinear propagation characteristics of electron-acoustic (EA) solitons are examined in an electron-ion (EI) plasma that contains negative superthermal (dynamical) electrons as well as positively charged ions. By employing the magnetic hydrodynamic (MHD) equations and with the aid of the reductive perturbation technique, a Korteweg-de-Vries (KdV) equation is deduced. The latter admits soliton solution suffering from the superthermal electrons and the streaming flow. The utility of the modified double Laplace decomposition method (MDLDM) leads to approximate wave solutions associated with higher-order perturbation. By imposing finite perturbation on the stationary solution, and with the aid of MDLDM, we have deduced series solution for the electron-acoustic excitations. The latter admits instability and subsequent deformation of the wave profile and can’t be noticed in the KdV theory. Numerical analysis reveals that thermal correction due to superthermal electrons reduces the dimensionless phase speed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\bar{U}_{ph})$$\end{document}(U¯ph) for EA wave. Moreover, a random motion spread out the dynamical electron fluid and therefore, gives rise to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{U}_{ph}$$\end{document}U¯ph. A degree enhancement in temperature of superthermal (dynamical) electrons tappers of (increase) the wave steeping and the wave dispersion, enhancing (reducing) the pulse amplitude and the spatial extension of the EA solitons. Interestingly, the approximate wave solution suffers oscillation that grows in time. Our results are important for understanding the coherent EA excitation, associated with the streaming effect of electrons in the EI plasma being relevant to the earth’s magnetosphere, the ionosphere, the laboratory facilities, etc.

The reduction in binary interactions of the plasma components reduces the particles correlations that restore the ionized matter to non-extensive state. The latter has relevance to ionosphere 1 , magnetosphere 2 , solar winds 3 , laboratory plasma 4 , etc. Intriguingly, Maxwell's statistics fail to describe the dynamics of particles in non-extensive plasmas. Vasyliunas 5 introduced the distribution function that extended Lurentzian/kappa accounts correctly for the superthermal plasmas compositions. Importantly, a long tail associated with the Lorentzian/kappa particle distribution function shows deviation from the non-thermal thermodynamic equilibrium. Plasmas with low density and/or high-temperature 6 have fewer binary collisions and correlation effects among components, and they can become non-thermal. In such plasmas, the statistical distribution of particles changes dramatically, rendering the traditional Maxwell-Gibbs statistics useless. The kappa or extended Lorentzian distribution function was initially developed by Vasyliunas 7 to characterize the superthermal composition of the collisionless plasma in the magnetosphere. The extended Lorentzian function has a long-tailed particles distribution function, which deviates considerably from the thermodynamic equilibrium. Furthermore, when holds, the superthermal index ( κ e → ∞ ) associated with non-thermal constituents restores a Maxwellian plasma state. It's worth noting that superthermal particle states have been seen both in space and in laboratories. The reported thermal and superthermal velocity spectra for space plasmas 8-10 match well with the Lorentzian distribution function. The electron fluid in laser-induced plasma 11 achieves a nonequilibrium condition within the typical period, thus the kappa distribution function is suitable. The dispersion and damping rates measured for electron-acoustic waves (EAWs) in laboratory plasma 12 precisely match the calculated superthermal index κ e range of 3−4, validating the Lorentzian distribution function for hot electrons. Sultana et al. 13 studied the nonlinear development of ions acoustic (IA) excitations in plasma with kappa distributed electrons and discovered that superthermal electrons permit smaller shocks with greater amplitude. The kappa dispersed ions in magneto-dusty electron depletion plasma ignite the negative polarity oblique dust-acoustic isolated potentials studied by Shahmansouri and Alinejad in 14 . The Lorentzian plasma approximation may also be used to wave dynamics and related instabilities in the interstellar medium 15 , solar wind 16 , ionosphere 17 , auroral zone 18 , and other areas.
Fried and Gould 19 first put forward an idea for excitation of the electron-acoustic (EA) mode. They have pointed out that the EA potentials suffer a Landau's damping effect that decreases with the increase of wave

Governing equations and model
Here, we study the propagation characteristics for the electron-acoustic (EA)solitons in a nonnegative-ion (EI) plasma that comprises dynamical electrons, superthermal hot electrons, and stationary ions. The EI plasma is assumed to be immersed in a uniform magnetic field ( B (0)Ẑ ) in Z-direction. It is assumed that the electrons supper a constant shear flow (U (0) = aẐ) in the Z-axis, where a stands for the magnitude of the speed. Importantly, at electron dynamical scale the phase speed for EA wave is much larger as compared to the thermal speed of electrons, i.e ω/K << U th condition holds. Here U Th (= √ K B T h /m e ) represent the thermal speed, with K B the Boltzmann constant and T h is the temperature of the hot electrons. The nonlinear evaluation of EA mode is governed by the following fluids equations www.nature.com/scientificreports/ and where N c , U c and P c designated the number densities, speed and pressure respectively for the cold electrons. Moreover, is the electrostatic potential, e(m e ) is the electronic charge (mass) and N h (N i ) represents the number densities for hot electrons (ions). The hot electrons can be taken as inertialess for much large energy and therefore described by the following kappa distribution function (Baluku and Helberg 38 ) where the index ( κ e ) accounts for the superthermal electrons. In the presence of magnetic field, the electrons may experiences the drift motion as Tc ∇ ⊥ · N c ×Ẑ)/� c N c } are diamagnetic drift of the electron. The velocity component associated with the dynamical electrons turns out to be By using Eqs. (4-6) and after some algebraic manipulation, one can reduce Eq. (1) as designate the electron gyro frequency and the acoustic speed respectively with C is speed of light. The components of Eq. (2) in X, Y and Z-direction are Equation (3) can be expressed in the form as Equations (7)-(11) describe the evolution of electron-acoustic excitations in non-extensive plasma that comprises of dynamical electrons as well as kappa distributed electrons and stationary ions.

Linear stability/instability analysis
In order to examine stability/Instability conditions of a linear mode, we expand the relevant parameters in Eqs. (8)- (11) in the form exp[i(K x X + K y Y + K z Z − ωT)] up to first order. Thus after some algebraic manipulation, we obtain the following quartic equation Notice that in the limit when U (0) = 0 , the coefficients of ω 3 and ω vanish and therefor Eq. (12) reduces into a biquadratic equation as already obtained in 29 . Our interest is in the case when U (0) = 0 , the numerical solution of Eq. (12) reveals real as well as imaginary roots. The dimensionless imaginary root ( Imω ) in Eq. (12), corresponds to instability growth rate, depicted versus dimensionless wavenumber ( K ) in Fig. 1a with variations in streaming speed U (0) = 10 3 cm s −1 (solid curve), 2 × 10 3 cm s −1 (dashed blue curve) and 3 × 10 3 cm s −1 (dotted black curve). See the streaming effect of magnetoplasma rises the instability growth rate. The same is given in Fig. 1b versus K when the magnetic field B (0) = 10 5 G (solid curve), 2 × 10 5 G (dashed curve) and 3 × 10 5 G(dotted curve). It reveals that the intensification in the B (0) favors instability of the linear EA waves.

Non-linear wave analysis
For the nonlinear evaluation of electron-acoustic excitations in the nonextensive EI plasma, we use the reductive perturbation technique given by Washimi and Tanuili 39 . In this context, we chose the following stretching and the spatial-temporal variables as where U ph is phase velocity of the waves, K x , K y and K z are the direction cosines of the wave vector along X, Y and Z-axis respectively. The relevant plasma parameters are represented in the form as Figure 1. The dimensionless imaginary root Imω (whereω = ω/ω 0 with ω 0 = 10 12 rad/s) in Eq. (12), is plotted against the dimensionless wavenumber K (= K/K 0 with K 0 = 10 6 cm −1 ) with variation in (a) streaming speed U (0) = 10 3 cm s −1 (solid curve), 2 × 10 3 cm s −1 (dashed curve) and 3 × 10 3 cm s −1 (dotted curve). The same is depicted versing K when (b) magnetic field B (0) =10 5 G (solid curve), 2 × 10 5 G (dashed curve) and 3 × 10 5 G (dotted curve) when T c = 10 3 K(T h = 10 4 K) and κ e = 3 . www.nature.com/scientificreports/ similarly, the transverse components of electrons speed can be expressed as it should be noted that ǫ is a trivially very small dimensionless factor that calculates the energy of the dispersion and non-linearity. The occurrence of the magnetic field B (0) in the system leads to anisotropy, because of which the perpendicular components ( U cX and U cY ) can be stated in an upper order of the parameter ǫ than the corresponding factors of velocity U cZ . Therefore, the gyro-motion effects in the higher-order influence in the model. The transverse velocity components are extended to jump with ǫ 3/2 , while the corresponding component of velocity starts with ǫ . The components having ǫ 3/2 represent weak velocity perturbation as compared to the component having order ǫ. By using Eqs. (13)(14)(15) in Eqs. (7)(8)(9)(10)(11) we get lowest orders in ǫ where C 1 = (κ e − 1/2)/(κ e − 3/2) . By solving Eqs. (15)(16)(17)(18)(19) we can acquire phase speed of the EAWs as

Scientific Reports
h /K B T h stands for an expansion parameter. The expansion of the perturbation series beyond the first order of ǫ leads to the following perturbations ) . The next higher orders in ǫ for momentum and continuity equations respectively are (1) ∂ζ , (1) ∂ζ , (2) ∂ζ , For real values of all the parameters we have A < 0 and B > 0 , give rise to the negative potential of the EAWs pulses. To obtain a localized stationary solitary waves solution moving to the right, we transform the independent variables ζ and τ to new moving coordinates ( ξ = ζ − µ 0 τ ), where µ 0 is the speed of solitary waves in the new coordinate system. Also applying the vanishing conditions as → 0 , d�/dξ → 0 , d 2 �/dξ 2 → 0 at ξ → ±∞ , the localized solution of the equation (27) can be obtained as where δ 0 = 3µ 0 A and � = 2 B µ 0 represent the amplitude and spatial width of the EA solitons. The product δ 0 � 2 (= 12B/A) giving the constant values independent of µ 0 will suggest that the taller the amplitudes of the solitary waves will result in the faster and narrower the pulse shape accordance with the KdV theory. � = 2 B µ 0 suggests that increasing the solitary wave speed µ 0 will increase the amplitudes but a decrease will occur in its width. The expression for the electric field can be calculated as where X , Ŷ , and Ẑ are the unit vectors along X, Y and Z-axis respectively.

Modified double Laplace decomposition method (MDLDM)
This method is used here for the first to study the nonlinear evolution of solitary potential in nonextensive plasma. The modified double Laplace decomposition method has also extensively used to reduce the spatiotemporal solutions corresponds for the linear as well as nonlinear deferential equation [40][41][42][43][44] .
To discuss this method. consider the following non-linear problem of the form where L is highest order linear operator, ( L = D n (X, T) = ∂ n φ(X, T)/∂X n ), N is non-linear operator, and R is the operator contains the linear terms while f(X, T) is some external function. Consider a function φ(X, T) for X, T > 0 in XT − plane , the double Laplace transform of the function φ(X, T) is defined by 45 where S 1 and S 2 are complex numbers. The double Laplace transform for the partial derivatives of the function φ(X, T) can be represented as
The important results of our study are presented in the following discussion. Figure 2a depicts the dimensionless phase speed Ū ph (= U ph /U ph0 , verses the superthermal index (κ e ) , for the electron-acoustic (EA) solitary pulse at T h = 10 3 K(solid curve), 1.1 × 10 3 K(dashed curve),1.2 × 10 3 K(dotted curve). It reveals that the thermal correction of superthermal electrons decreases Ū ph . We have displayed Ū ph against κ e with variations in Fig. 2b temperature due to the dynamical electrons T c = 10 3 K(solid curve), 1.01 × 10 3 K(dashed curve),1.02 × 10 3 K(dotted curve). See the degree enhancement in T c gives rise to Ū ph . In a similar fashion, Fig. 2c,d, illustrate Ū ph verses κ e with variation in the electron streaming speed (U (0) ) , and www.nature.com/scientificreports/ obliquity parameter (K z ) . It infer that both the streaming electrons and the obliqueness (K z ) enhances the phase speed Ū ph . The lower and upper panels in Fig. 3a,b illustrate the dimensionless non linearity Ā (= A/A (0) ) and dispersion B (= B/B 0 ) coefficients, respectively with variations in thermal effects of hot (cold) electrons, i.e., T h (T c ) . Importantly note, thermal effect spread out the super-thermal electrons that in turn rises Ā and B . Moreover, opposite trend notices for Ā and B with enhancement in T c as shown in Fig. 3c,d. For the impact of relevant plasma parameters on the nonlinear steepening and dispersions effects, we have plotted in Fig. 4a-d coefficients Ā and B at different values of K z and U (0) respectively. Obviously, K z and U (0) reduces coefficient Ā and B . Thus it reveals that oblique propagation of EA excitations suffer reduction in the nonlinear pulse steepening and dispersion.
To show the impact of electronic temperature, we have given the wave solution (27) for pulse-shaped soliton against the spatial variable ξ (see Fig. 5a,b). Recall that T h (T c ) decreases(increases) coefficients Ā and B , and therefore rises (reduces) the pulse amplitude and spatial extension for solitary potentials. Likewise, the streaming speed (U (0) ) and the superthermality index ( κ e ) also impact the wave profiles as illustrated in Fig. 5c,d.
In Fig. 6a we compare our localized solution (27) with the series solution (44) obtained by the MDLDM method. Obviously, at τ > 0 the series MDLDM solution admits spatial deviation with amplification in pulse amplitude. The MDLDM solution in Fig. 6b at different values of κ e shows that variation in superthermality index significantly modifies the EA soliton. The 3D surface plots for both the localized and approximate solution have been shown in Fig. 6c,d. The MDLDM is an excellent tool to calculate analytical solutions of non-integrable system more accurately. The obtained results indicates that it is an effective tool for solving the KdV equation. One can see that the MDLDM solution satisfies the precise solution in Table 1 at τ = 0.1 and Table 2 at τ = 0.01 , as well as in Fig. 7a,b. From tables, it is observed that the absolute error between the localized and approximate solution is reducing and approaching to zero by taking larger values for the spatial variable ( −3 ≤ξ ≥ 3 ) with the temporal variable τ = 0.1 and 0.01 respectively. As we know that, the KdV equation permits solitary wave solutions in plasma,

Data availability
The data regarding this work is available within the manuscript.