Relativistic effect on time-fractional KdV-soliton and its propagation in electron beam plasmas having two kappa-distributed electrons

Abstract

In this work, we examine the nonlinear propagation of low-frequency ion-acoustic solitary waves in a collisionless, electropositive, relativistic electron-beam plasma composed of warm relativistic ions, superthermal cool and hot electron populations modeled by kappa distributions, and a finite-temperature relativistic electron beam. Starting from a relativistic multi-fluid description coupled with Poisson’s equation, we employ the reductive perturbation method to derive a planar Korteweg-de Vries (KdV) evolution equation that captures the interplay between quadratic nonlinearity and dispersion in this multi-component environment and then construct a time-fractional generalization of the KdV model via a variational formulation. The resulting time-fractional KdV equation is solved using the variational iteration technique, yielding first- and second-order analytical approximations that clearly display the impact of fractional temporal dynamics. Our parametric analysis shows that increases in the relativistic streaming factor, the hot-electron-to-ion density ratio, and the cool-electron spectral index reduce the linear phase velocity. In contrast, stronger cool-electron superthermality enhances the effective nonlinearity of the ion-acoustic mode. We further find that relativistic corrections substantially modify both the amplitude and width of classical solitons and their time-fractional counterparts: while standard KdV solitons maintain their form during propagation, fractional solitons undergo pronounced amplitude and structural changes reflecting the underlying memory effects. Overall, these findings outline a specific set of conditions in which relativistic movement, superthermal electron tails, and changes over time together influence how ion-acoustic solitary structures form, stay strong, and change in relativistic electron-beam plasmas.

1. Introduction

Low-frequency ion-acoustic (IA) solitary waves (IASWs) are a significant soliton-type nonlinear phenomenon that propagate through plasma systems, retaining their shape due to a balance between nonlinear and dispersive effects. They arise from the nonlinear interaction of ions and electrons and exhibit a variety of features depending on the plasma’s characteristics. Although electrons are most likely to follow a Maxwellian distribution, many astrophysical and space plasma observations have shown that the presence of high-energy-tail superthermal species can cause the particle distribution to deviate from it.^1–3 These energetic particles follow the κ-distribution^4–7:

$f_{\kappa }\left(u_p\right)=\left[\frac{N_0\Gamma \left(\kappa \right)}{\Gamma \left(\frac{\kappa -1}{2}\right)\sqrt{\pi \kappa u_t^2}}\right]{\left(1+\frac{u_p^2}{\kappa u_t^2}\right)}^{-\kappa },$

where $\mathrm{\Gamma }$ indicates the well-known gamma function, $\kappa$ (satisfying the condition $\kappa >3/2$ ) represents the spectral index, $N_0$ indicates the equilibrium density, $u_t$ represents the particle’s modified thermal speed having mass ( $m$ ) and temperature ( $\theta$ ) related via $u_t^2=2\frac{{\kappa }_B\theta }{m}\left(1-\frac{3}{2\kappa }\right)$ . Here, ${\kappa }_B$ stands for the Boltzmann constant. When $\kappa$ tends to $\infty$ , then the $\kappa$ -distribution tends to a Maxwellian distribution.

Suppose that $\mathrm{\Psi }$ represents the electrostatic potential and ${\theta }_e$ and ${N_0}_e$ represent, respectively, the temperature of electrons and equilibrium number density. Then, integrating the kappa-distribution throughout velocity space yields the following electron density:

$N_e\left(\Psi \right)={N_0}_e{\left[1-\frac{e\Psi }{\left(\kappa -3/2\right){\kappa }_B{\theta }_e}\right]}^{-\left(\kappa -1/2\right)}.$

Several satellite observations have verified the presence of plasmas with two-temperature electrons (hot and cool) in space and astrophysical settings,^8–11 and Schippers et al. ¹¹ observed that $\kappa$ -distributions (with lower values of $\kappa$ ) best fit the hot and cool electron velocity distributions. For the last few decades, various research has been carried out on plasmas with $\kappa$ -distributions. Baluku et al. have examined high-frequency electron-acoustic (EA) solitary waves (EASWs) ¹² and IASWs ⁵ in a plasma containing two-temperature kappa-distributed electrons. The propagation of IASWs in a magnetized plasma comprising fluid ions with a finite temperature and electrons with a $\kappa$ -distribution has been examined by Singh et al. ¹³ Alinejad et al. ¹⁴ have investigated the modulational instability of IAWs in an unmagnetized plasma containing two-temperature $\kappa$ -distributed electrons. Saberian et al. ¹⁵ have studied the formation and propagation of high-amplitude dust-acoustic (DA) solitary waves (DASWs) in a plasma comprising negatively charged dust grains (DGs) and e-p pairs with a kappa distribution. Also, Bhuyan et al. ⁷ have studied the effects of heavy ions and relativistic effects on the structure of IAWs in a plasma containing $\kappa$ -distributed electrons.

Due to the presence of electron beams in space and astrophysical environments,^8–11,16 numerous researchers have examined solitary waves (SWs) in electron-beam plasmas. For instance, Saini and Kourakis ¹⁷ have investigated the presence of arbitrary-amplitude IASWs in an unmagnetized plasma containing ions and κ-distributed electrons with a cold electron beam. Nejoh and Sanuki ¹⁸ have researched the presence of high-amplitude ion-acoustic waves in a plasma composed of warm ions and hot isothermal electrons under the influence of a warm electron beam. Esfandyari et al. ¹⁹ studied small-amplitude IASWs in a collisionless plasma composed of hot ions and isothermal electrons, with a cold relativistic electron beam. Employing the Sagdeev pseudopotential approach, Lakhina et al. ²⁰ have studied the characteristics of ion-and EA solitons (ESSs) in an unmagnetized multi-component plasma with ion and electron beams. Moreover, Saberian et al. ²¹ have investigated the propagation of IA solitons (IASs) in a plasma including a cold electron beam, warm ions, and superthermal $\kappa$ -distributed electrons.

Plasmas carrying particles moving with a speed very close to that of light produce substantial relativistic effects. Motivated by evidence of plasma particles’ relativistic behavior in astrophysical and space environments ²² and in laser–plasma interactions,^23,24 many researchers have focused on relativistic plasmas in recent decades. For example, Polyakov ²⁵ has assessed the possibility of IAWs arising in a relativistic plasma with a non-Maxwellian particle velocity distribution function. Gill et al. ²⁶ have theoretically investigated the propagation properties of IAWs in a magnetized, weakly relativistic e-p-i warm plasma and found that the width and amplitude of SWs are significantly affected by the relativistic factor. The nonlinear propagation of IASWs in an unmagnetized plasma comprising relativistic thermal ions, Boltzmann positrons, and nonextensive electrons has been studied theoretically and numerically by Hafez and Talukder. ²⁷ Also, the IASs have been examined by Kalita et al. ²⁸ in a high-relativistic e-p-i plasma. Very recently, Bhuyan et al.^7,29–31 have accounted for the relativistic streaming factor to analyze the structures of IAWs, dust-ion acoustic waves (DIAWs), and DA waves (DAWs) in various plasma environments. Earlier relativistic plasma studies have examined nonlinear wave evolution in specific configurations, such as electron plasma waves in quantum-degenerate media ³² and ion–dust systems incorporating quantum and relativistic corrections. ³³ These models, however, do not address ion-acoustic structures in a plasma simultaneously containing dual $\kappa$ -distributed electron populations and a relativistic inertial electron beam. The present work introduces two independent superthermal spectral indices, yielding distinct and competing nonlinear contributions that are absent in single-population or dust-based formulations. Moreover, the inclusion of a relativistic streaming beam adds inertial and energy-transfer effects that directly modify the nonlinear–dispersive balance governing solitary formation. These combined mechanisms generate coefficient structures and parametric sensitivities that cannot be reduced to earlier relativistic or quantum hydrodynamic treatments. Accordingly, the present formulation explores a distinct physical regime in which superthermality and relativistic beam dynamics jointly influence ion-acoustic solitary behavior.

From a mathematical modelling perspective, studies of relativistic ion-acoustic solitary waves (IASWs) have employed both perturbative and pseudopotential approaches to capture the nonlinear wave characteristics. In the weakly relativistic regime, the reductive perturbation method (RPM) has been used to derive Korteweg–de Vries (KdV) type evolution equations for IASWs, demonstrating how relativistic corrections influence amplitude and width in multi-component plasmas. ³⁴ More recently, Hosen et al. ³⁵ derived KdV and modified KdV (mKdV) equations for relativistic ion-acoustic solitary structures in magnetized dense plasmas, highlighting the effects of relativistic parameters on nonlinear characteristics using RPM techniques. Additionally, Sagdeev pseudopotential formulations have been applied to relativistic degenerate plasmas, providing insight into arbitrary-amplitude IASWs under different relativistic electron/positron distributions. ³⁶

Nowadays, it is impossible to find a discipline of mathematics, engineering, or physics that has not been influenced by fractional calculus, or more precisely, calculus that deals with derivatives of arbitrary orders. Recent studies^37,38 show that fractional-derivative formulations are more accurate for many physical systems. Again, different nonlinear phenomena in a variety of physical media, including various types of “acoustic waves, like solitons, cnoidal waves, shocks, etc.”, in plasma physics, are modeled and explained using the quadratic nonlinearity KdV equation (KdVE). So, numerous researchers have derived the time-fractional (TF) variant (TFV) of the KdVE from different perspectives and solved it using diverse techniques. Employing the Euler-Lagrange equation, El-Wakil et al. ³⁹ have derived a time-fractal form of the KdV equation by the variational approach^40–43 and solved it with the help of the variational-iteration technique.^44,45 Zhang ⁴⁶ used the Euler-Lagrange variational (ELV) method in the Riemann-Liouville (R-L) derivative (R-LD) sense to obtain the time-fractional version of the generalized KdV equation and demonstrated that the variational iteration method can effectively solve it. Moreover, utilizing the well-known Aboodh residual (AR) power series technique (ARPST) ⁴⁷ with the Aboodh transform (AT) iterative approach (ATIA), ⁴⁸ Alyousef et al. ⁴⁹ have studied the third- and fifth-order KdV-type equations in their respective time-fractional versions. Physically, the incorporation of fractional calculus into plasma wave modeling has been shown to capture memory and nonlocal temporal effects beyond classical KdV dynamics. For example, El-Wakil et al. ⁵⁰ derived a time-fractional KdV equation for IAWs in warm-ion plasmas and demonstrated that the fractional order significantly modifies soliton amplitude and width. This approach was further extended to electron-acoustic solitary structures in multi-component plasmas, confirming that fractional temporal evolution alters nonlinear propagation characteristics. ⁵¹

Although ion-acoustic solitary waves have been extensively studied in plasmas with $\kappa$ -distributed electrons, relativistic components, and electron beams, these effects have generally been treated separately and within classical (integer-order) evolution models. Time-fractional KdV formulations have also been developed, but primarily for non-relativistic plasma systems. To the best of our knowledge, no study has simultaneously incorporated relativistic streaming, warm relativistic ions, two-temperature $\kappa$ -distributed electrons, and a finite-temperature relativistic electron beam within a time-fractional KdV framework. In particular, the role of relativistic effects in modifying the structure and temporal evolution of fractional ion-acoustic solitons remains unexplored. The present work addresses this gap by providing a unified analysis of classical and time-fractional soliton dynamics in such relativistic electron-beam plasmas.

The manuscript is organized as follows: Section 1 provides the introduction, whereas Section 2 outlines the fluid plasma model with necessary assumptions. The derivation of the KdV equation with its solution is presented in Section 3. Section 4 derives the time-fractional KdV equation along with its solution. In Section 5, the results and discussions are incorporated. Finally, Section 6 provides concluding remarks and a summary of the significant results.

2. Assumption and model equations

The plasma system is supposed to be collisionless and electropositive. It comprises warm relativistic fluid ions, a pair of $\kappa$ -distributed electrons at different temperatures, and a relativistic inertial electron beam with non-zero temperature.

Assume that cool and hot electron species are denoted by $h$ and $c$ , respectively, and the subscript $j=h$ , $c$ . Then the charge neutrality condition of the system is stated by: $N_{0i}=N_{0c}+N_{0h}+N_{0b}$ , where $N_{0b}$ denotes the beam electrons’ unperturbed density, $N_{0i}$ denotes the ions’ density at the sheath edge, and $N_{0j}$ denotes the species $j$ ’s density at the sheath edge. Again, the $\kappa$ -distributed electrons’, both cool and hot, number density is given by:

$N_j=N_{0j}{\left[1-\frac{e\Psi }{Q_j{\kappa }_B{\theta }_{ej}}\right]}^{-P_j},$

with

$Q_j=\left({\kappa }_j-\frac{3}{2}\right)\hspace{0.17em}\&\hspace{0.17em}{P}_j=\left({\kappa }_j-\frac{1}{2}\right).$

Here, $e$ stands for the magnitude of electric charge, $\mathrm{\Psi }$ stands for the electrostatic potential, ${\theta }_{ej}$ denotes the temperature of the species $j\hspace{0.17em}\left(=h,c\right)$ at the sheath edge, ${\kappa }_j$ represents the superthermal index of the respective electron species $j\hspace{0.17em}\left(=h,c\right)$ , where ${\kappa }_j>\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ . All variables are normalized with respect to the equilibrium ion density, electron thermal speed, and Debye length. The following normalized equations govern the nonlinear dynamics of IASWs in the described plasma system ⁶ :

$\frac{{\partial n}_i}{\partial t}+\frac{\partial \left(n_i{u}_i\right)}{\partial x}=0,$

(1)

$\frac{\partial \left({\gamma }_i{u}_i\right)}{\partial t}+u_i\frac{\partial \left({\gamma }_i{u}_i\right)}{\partial x}+\frac{\partial \mathrm{\psi }}{\partial x}+\frac{{\sigma }_i}{n_i}\frac{\partial p_i}{\partial x}=0,$

(2)

$\frac{\partial p_i}{\partial t}+u_i\frac{\partial p_i}{\partial x}+3p_i\frac{\partial u_i}{\partial x}=0,$

(3)

$\frac{\partial n_b}{\partial t}+\frac{\partial \left(n_b{u}_b\right)}{\partial x}=0,$

(4)

$\frac{\partial \left({\gamma }_b{u}_b\right)}{\partial t}+u_b\frac{\partial \left({\gamma }_b{u}_b\right)}{\partial x}-\mu \frac{\partial \mathrm{\psi }}{\partial x}+\mu {\delta }_b\frac{{\sigma }_b}{n_b}\frac{\partial p_b}{\partial x}=0,$

(5)

$\frac{\partial p_b}{\partial t}+u_b\frac{\partial p_b}{\partial x}+3p_b\frac{\partial u_b}{\partial x}=0.$

(6)

Here, the normalized variables are defined as:

$n_i=N_i/N_{0i},n_b=N_b/N_{0i},u_i=U_i/c_s,u_b=U_b/c_s,c_s=\sqrt{{\kappa }_B{\theta }_{ec}/m_i},\mu =m_i/m_e,{\sigma }_i={\theta }_i/{\theta }_{ec},{\sigma }_b={\theta }_b/{\theta }_{ec},{\sigma }_h={\theta }_{eh}/{\theta }_{ec},{\delta }_b=N_{0b}/N_{0i},{\delta }_h=N_{0h}/N_{0i},p_i=\stackrel{\sim }{P_i}/N_{0i}{\kappa }_B{\theta }_i,p_b=\stackrel{\sim }{P_b}/N_{0b}{\kappa }_B{\theta }_b,x=X/{\lambda }_D,t=\tau /{\omega }_{pi}.$

In above equations, $N_i$ denotes the density of positive relativistic ions, $N_b$ denotes the density of relativistic beam electrons, $U_i$ denotes the relativistic positive ions’ velocity, $U_b$ denotes the velocity of the relativistic electron beam, $m_i$ denotes the relativistic ions’ mass, $m_e$ denotes the mass of the electron, ${\theta }_i$ denotes the relativistic ions’ temperature, ${\theta }_b$ denotes the relativistic electron beam’s temperature, ${\theta }_{eh}$ denotes the hot electron’s temperature, $\stackrel{\sim }{P_i}$ denotes the pressure of relativistic positive ions, $\stackrel{\sim }{P_b}$ denotes the pressure of relativistic electron beam, ${\lambda }_D$ denotes the Debye length with definition ${\lambda }_D=\sqrt{{ϵ}_0{\kappa }_B{\theta }_{ec}/e^2{N}_{0i}}$ , ${ϵ}_0$ is the free space’s electric permittivity, and ${\omega }_{pi}$ denotes the relativistic ion plasma frequency. Also, ${\gamma }_i=1/\sqrt{1-{\mathrm{u}}_{i0}^2/c_l^2}$ and ${\gamma }_b=1/\sqrt{{1-\mathrm{u}}_{b0}^2/c_l^2}$ are the relativistic Lorentz factors ⁷ corresponding to the warm-fluid ion and the inertial electron beam, respectively, where $u_{i0}$ is the unperturbed velocity of the relativistic ions, $u_{b0}$ is the unperturbed velocity of the relativistic electron beam, and $c_l$ is the speed of light. Moreover, Equations (1) to (6) are closed by the following Poisson’s equation:

$\frac{{\partial }^2\psi }{\partial x^2}+n_i-n_b-{\delta }_c{\left[1-\frac{\psi }{Q_c}\right]}^{-P_c}-{\delta }_h{\left[1-\frac{\psi }{{\sigma }_h{Q}_h}\right]}^{-P_h}=0,$

(7)

which can be simplified as:

$\frac{{\partial }^2\psi }{\partial x^2}=-n_i+n_b+{\delta }_c+{\delta }_h+C_1\psi +C_2{\psi }^2+\cdots ,$

(8)

with

$C_1={\delta }_c\left(\frac{P_c}{Q_c}\right)+\frac{{\delta }_h}{{\sigma }_h}\left(\frac{P_h}{Q_h}\right),C_2=\frac{{\delta }_c}{2}\left[\frac{{\kappa }_c^2-\frac{1}{4}}{{Q_c}^2}\right]+\frac{{\delta }_h}{2{\sigma }_h^2}\left[\frac{{\kappa }_h^2-\frac{1}{4}}{{Q_h}^2}\right].$

The normalized potential $\psi$ of Eqs. (7) and (8) is obtained as $\psi =e\mathrm{\Psi }/{\kappa }_B{\theta }_{ec}$ . The perturbed quantities are expanded in a power series of $\epsilon \ll 1$ as:

$n_i=1+{\displaystyle \sum }_{k=1}^{\infty }{\epsilon }^k{n}_{ik},$

(9)

$n_b=1+{\displaystyle \sum }_{k=1}^{\infty }{\epsilon }^k{n}_{bk},$

(10)

$u_i={\displaystyle \sum }_{k=0}^{\infty }{\epsilon }^k{u}_{ik},$

(11)

$u_b={\displaystyle \sum }_{k=0}^{\infty }{\epsilon }^k{u}_{bk},$

(12)

$p_i=1+{\displaystyle \sum }_{k=1}^{\infty }{\epsilon }^k{p}_{ik},$

(13)

$p_b=1+{\displaystyle \sum }_{k=1}^{\infty }{\epsilon }^k{p}_{bk},$

(14)

$\mathrm{\psi }={\displaystyle \sum }_{k=1}^{\infty }{\epsilon }^k{\psi }_k.$

(15)

For mathematical convenience, the equilibrium drift velocities of the ion fluid and the electron beam are assumed to be identical, that is, $u_{i0}=u_{b0}=V_0$ . We further assume that $V_0/c_l={\beta }_0$ . Under this assumption, the corresponding Lorentz factors associated with the ions and the beam electrons become identical, and may therefore be expressed by a single relativistic factor, denoted by $\gamma$ , which takes the following form:

${\gamma }_i={\gamma }_b=\gamma =1/\sqrt{1-V_0^2/c_l^2}\begin{array}{ll}& =\end{array}1+\frac{1}{2}{\left(\frac{V_0}{c_l}\right)}^2+\frac{3}{8}{\left(\frac{V_0}{c_l}\right)}^4+\cdots \approx 1+\frac{1}{2}{\left(\frac{V_0}{c_l}\right)}^2=1+\frac{1}{2}{\beta }_0^2.$

It is to be noted that ${\beta }_0$ is referred to as the relativistic streaming factor.^7,30,31

3. Nonlinear evolution equation with its solution

To study the relativistic effect on the properties of IASWs, it is convenient to introduce stretched coordinates that capture the slow evolution of weakly nonlinear ion-acoustic perturbations. Here, we consider stretched coordinates as^7,29–31:

$\zeta ={\epsilon }^{1/2}\left(x-\lambda t\right)\hspace{0.17em}\&\tau ={\epsilon }^{3/2}t,$

(16)

where $\epsilon \ll 1$ represents the small but finite wave amplitude, and $\lambda$ is the linear phase speed. This scaling ensures that the spatial variations occur over long wavelengths $\left(\frac{\partial }{\partial x}\sim {\epsilon }^{1/2}\right)$ and temporal evolution is slow $\left(\frac{\partial }{\partial t}\sim {\epsilon }^{3/2}\right)$ , so that nonlinear, dispersive, and temporal terms appear at the same perturbative order $O\left({\epsilon }^{3/2}\right)$ . Such a balanced ordering is necessary for the consistent derivation of the Korteweg–de Vries (KdV) evolution equation from the governing fluid equations. Similar multiscale asymptotic procedures have been successfully employed in recent studies of relativistic plasma systems to derive evolution equations for solitary waves.^52–54

Now, the stretched coordinates in (16) lead to

$\{\begin{array}{l}\frac{\partial }{\partial t}={\epsilon }^{3/2}\frac{\partial }{\partial \tau }-{\epsilon }^{1/2}\Omega \frac{\partial }{\partial \zeta },\\ \frac{\partial }{\partial x}={\epsilon }^{1/2}\frac{\partial }{\partial \zeta }.\end{array}$

(17)

Substituting the expansions of the perturbed quantities (Eqs. 9–15) into the normalized fluid equations (Eqs. 1–6 and 8) and applying the stretched-coordinate derivative operators defined in Eq. (17), the lowest-order terms in $\epsilon$ yield the first-order perturbation equations for the ion and beam dynamics:

$\left(u_{i0}-\lambda \right)\frac{\partial n_{i1}}{\partial \zeta }+\frac{\partial u_{i1}}{\partial \zeta }=0,$

(18)

${\gamma }_1\left(u_{i0}-\lambda \right)\frac{\partial u_{i1}}{\partial \zeta }+{\sigma }_i\frac{\partial p_{i1}}{\partial \zeta }+\frac{\partial {\psi }_1}{\partial \zeta }=0,$

(19)

$\left(u_{i0}-\lambda \right)\frac{\partial p_{i1}}{\partial \zeta }=0,$

(20)

$\left(u_{b0}-\lambda \right)\frac{\partial n_{b1}}{\partial \zeta }+\frac{\partial u_{b1}}{\partial \zeta }=0,$

(21)

${\gamma }_1\left(u_{b0}-\lambda \right)\frac{\partial u_{b1}}{\partial \zeta }+{\mu {\delta }_b\sigma }_b\frac{\partial p_{b1}}{\partial \zeta }-\mu \frac{\partial {\psi }_1}{\partial \zeta }=0,$

(22)

$\left(u_{b0}-\lambda \right)\frac{\partial p_{b1}}{\partial \zeta }=0,$

(23)

$C_1{\psi }_1-n_{i1}+n_{b1}=0,$

(24)

where ${\gamma }_1=1+\frac{3}{2}{\beta }_0^2$ . Clearly, Eqs.(20) and (23), respectively, show that $p_{i1}$ and $p_{b1}$ are constants. It is assumed that $p_{i1}=C^{*}$ and $p_{b1}=C_1^{*}$ .

The next order of $\epsilon$ yields the following reduced equations

$\frac{\partial n_{i1}}{\partial \tau }+\frac{\partial \left(n_{i1}{u}_{i1}\right)}{\partial \zeta }+\left(u_{i0}-\lambda \right)\frac{\partial n_{i2}}{\partial \zeta }+\frac{\partial u_{i2}}{\partial \zeta }=0,$

(25)

${\gamma }_1\frac{\partial u_{i1}}{\partial \tau }+\frac{1}{2}\left({\gamma }_2-\frac{3{\beta }_0\lambda }{c_l}\right)\frac{\partial u_{i1}^2}{\partial \zeta }-{\sigma }_i{n}_{i1}\frac{\partial p_{i1}}{\partial \zeta }$

$+{\gamma }_1\left(u_{i0}-\lambda \right)\frac{\partial u_{i2}}{\partial \zeta }+{\sigma }_i\frac{\partial p_{i2}}{\partial \zeta }+\frac{\partial {\psi }_2}{\partial \zeta }=0,$

(26)

$\frac{\partial p_{i1}}{\partial \tau }+3p_{i1}\frac{\partial u_{i1}}{\partial \zeta }+u_{i1}\frac{\partial p_{i1}}{\partial \zeta }+\left(u_{i0}-\lambda \right)\frac{\partial p_{i2}}{\partial \zeta }=0,$

(27)

$\frac{\partial n_{b1}}{\partial \tau }+\frac{\partial \left(n_{b1}{u}_{b1}\right)}{\partial \zeta }+\left(u_{b0}-\lambda \right)\frac{\partial n_{b2}}{\partial \zeta }+\frac{\partial u_{b2}}{\partial \zeta }=0,$

(28)

${\gamma }_1\frac{\partial u_{b1}}{\partial \tau }+\frac{1}{2}\left({\gamma }_2-\frac{3{\beta }_0\lambda }{c_l}\right)\frac{\partial u_{b1}^2}{\partial \zeta }-\mu {\delta }_b{\sigma }_b{n}_{b1}\frac{\partial p_{b1}}{\partial \zeta }$

$+{\gamma }_1\left(u_{b0}-\lambda \right)\frac{\partial u_{b2}}{\partial \zeta }+{\mu {\delta }_b\sigma }_b\frac{\partial p_{b2}}{\partial \zeta }-\mu \frac{\partial {\psi }_2}{\partial \zeta }=0,$

(29)

$\frac{\partial p_{b1}}{\partial \tau }+3p_{b1}\frac{\partial u_{b1}}{\partial \zeta }+u_{b1}\frac{\partial p_{b1}}{\partial \zeta }+\left(u_{b0}-\lambda \right)\frac{\partial p_{b2}}{\partial \zeta }=0,$

(30)

$\frac{{\partial }^2{\psi }_1}{\partial {\zeta }^2}-C_2{\psi }_1^2-C_1{\psi }_2+n_{i2}-n_{b2}=0,$

(31)

where ${\gamma }_2=1+\frac{9}{2}{\beta }_0^2$ .

Equations (18) to (20) give

$\{\begin{array}{l}{n}_{i1}=\frac{1}{{\gamma }_1{\left(u_{i0}-\lambda \right)}^2}{\psi }_1,\\ u_{i1}=-\frac{1}{{\gamma }_1\left(u_{i0}-\lambda \right)}{\psi }_1,\end{array}$

(32)

and Eqs. (21) to (23) give

$\{\begin{array}{l}{n}_{b1}=-\frac{\mu }{{\gamma }_1{\left(u_{b0}-\lambda \right)}^2}{\psi }_1,\\ u_{b1}=\frac{\mu }{{\gamma }_1\left(u_{b0}-\lambda \right)}{\psi }_1.\end{array}$

(33)

Equations (24), (32), and (33) together give the equation for the phase velocity as:

$\lambda =V_0\pm {\left(\frac{1+\mu }{C_1{\gamma }_1}\right)}^{1/2}.$

(34)

Using Eqs. (32) to (33), the set of PDEs comprising Eqs. (25) to (31) yields the nonlinear KdV equation as:

$\frac{\partial {\psi }_1\left(\zeta ,\tau \right)}{\partial \tau }+R{\psi }_1\left(\zeta ,\tau \right)\frac{\partial {\psi }_1\left(\zeta ,\tau \right)}{\partial \zeta }+S\frac{{\partial }^3{\psi }_1\left(\zeta ,\tau \right)}{\partial {\zeta }^3}=0,$

(35)

with the nonlinearity and dispersion coefficients, respectively, as

$R=\frac{\mu -1}{4{\gamma }_1^2\left(V_0-\lambda \right)} \left({\gamma }_2-3\lambda \frac{{\beta }_0}{c_l}\right)+\frac{1}{2}\left(\frac{\mu }{V_0-\lambda }-\frac{1}{{\gamma }_1}\right)+\frac{C_2}{C_1}\left(V_0-\lambda \right),S=\frac{\lambda -V_0}{2C_1}+\frac{3}{4{\gamma }_1\left(\lambda -V_0\right)}\left(C^{*}{\sigma }_i+C_1^{*}\mu {\delta }_b{\sigma }_b\right).$

For simplicity, Eq. (35) can be rewritten in the following form ( ${\psi }_1$ = $\psi )$ ,

${\psi }_{\tau }\left(\zeta ,\tau \right)+R\psi \left(\zeta ,\tau \right){\psi }_{\zeta }\left(\zeta ,\tau \right)+S{\psi }_{\zeta \zeta \zeta }\left(\zeta ,\tau \right)=0,$

(36)

where the subscripts (that is, $\tau \hspace{0.17em}\text{and}\hspace{0.17em}\zeta$ ) of $\psi$ are used for the partial derivative of $\psi$ with respect to the respective independent variables ( $\tau \hspace{0.17em}\text{and}\hspace{0.17em}\zeta$ ).

Equation (36) supports N-soliton solutions, where the one-soliton solution reads ⁵⁵ :

$\psi \left(\zeta ,\tau \right)={\psi }_{am}\hspace{0.17em}\sec \hspace{0.17em}{\mathrm{h}}^2\left(\frac{\eta }{W_0}\right).$

(37)

Here, ${\psi }_{am}=3{\lambda }_v/R$ and $W_0=\sqrt{4S/{\lambda }_v}$ are the amplitude and width of the IASW solitons, respectively, where ${\lambda }_v$ stands for the frame velocity.

Remember that the polarity of nonlinear structures described by the KdV Eq. (36) depends on the sign of the nonlinear coefficient $R$ , i.e., for $R>0$ , the KdV-soliton becomes compressive, while for $R<0$ , the KdV-soliton becomes rarefactive.

4. Time-fractional KdV equation with solution

4.1 Time-fractional KdV equation

Now, to examine the dynamical behavior of time fractional KdV (TFKdV) solitons in the current plasma model, the integer KdV Eq. (36) must be converted into its fractional version. Following the same methodology discussed in detail in Refs. ³⁹ , we ultimately derive the following FKdV equation:

$D_{\tau }^{\beta }\psi \left(\zeta ,\tau \right)+R\psi \left(\zeta ,\tau \right){\psi }_{\zeta }\left(\zeta ,\tau \right)+S{\psi }_{\zeta \zeta \zeta }\left(\zeta ,\tau \right)=0,$

(38)

where $0<\beta <1$ , indicates the fractional-order parameter (fractionality) and $\tau \in \left[a_0,b_0\right]$ . Here, ${D_{\tau }^{\beta }\equiv {}_0{}^{R}D}_{\tau }^{\beta }$ represents the right Riemann-Liouville (RL) fractional derivative operator. ⁵⁶

4.2. Solution of the time-fractional KdV equation

Several effective methods have been developed to analyze Eq. (38), providing highly accurate approximations. Among the most well-known are the VIM^44,45,57,58 and the Tantawy technique.^59–65 Since these methods have examined this equation in its general form, i.e., they include all relevant coefficients in their approximations, we can use one of these approximations to study the dynamical behavior of fractional KdV-solitons in the current plasma model and explore how fractionality influences these waves. According to the VIM, El-Wakil et al. ³⁹ used the following initial condition (IC) to derive an approximation Eq. (38)

${\psi }_{\left(0\right)}\left(\zeta ,0\right)\equiv {\psi }_{\left(0\right)}\equiv R_0{Y}^2=R_0\hspace{0.17em}\sec \hspace{0.17em}{\mathrm{h}}^2\left(c_0\zeta \right),$

(39)

where $R_0=3{\lambda }_0/R$ and $c_0=\sqrt{{\lambda }_0/S}$ .

Here, ${\psi }^{\left(0\right)}\left(\zeta ,\tau \right)$ is referred to as the zero-order approximation of the solution. In this solution, $R_0$ is the amplitude, $c_0$ is the width of the IASWs, and ${\lambda }_0$ is the speed of the traveling wave. According to the VIM, the approximation is assumed in the following form

${\psi }^{\left(i\right)}={\displaystyle \sum }_{i=0}^{\infty }{\psi }_{\left(i\right)}={\psi }_{\left(0\right)}+{\psi }_{\left(1\right)}+{\psi }_{\left(2\right)}+\cdots .$

(40)

Using Eqs. (39) and (40) and following the VIM, the solution up to first-order approximation is determined ³⁹

${\psi }^{\left(1\right)}={\psi }_{\left(0\right)}+{\psi }_{\left(1\right)}+\cdots ,$

(41)

with

${\psi }_{\left(0\right)}=R_0{Y}^2,{\psi }_{\left(1\right)}=c_0{R}_0{Y}^3\hspace{0.17em}\mathit{sin}\hspace{0.17em}h\left(c_0\zeta \right)\left[4c_0^{2}S+Y^2\left(R_{0}R-12c_0^{2}S\right)\right]\frac{{\tau }^{\beta }}{\Gamma \left(\beta +1\right)}.$

Also, the solution up to the second-order approximation is obtained ³⁹ as follows

${\psi }^{\left(2\right)}={\psi }_{\left(0\right)}+{\psi }_{\left(1\right)}+{\psi }_{\left(2\right)}+\cdots ,$

(42)

with

${\psi }_{\left(0\right)}=R_0{Y}^2,$

${\psi }_{\left(1\right)}=c_0{R}_0{Y}^3\hspace{0.17em}\mathit{sin}\hspace{0.17em}h\left(c_0\zeta \right)\left[4c_0^{2}S+Y^2\left(R_{0}R-12c_0^{2}S\right)\right]\frac{{\tau }^{\beta }}{\Gamma \left(\beta +1\right)},{\psi }_{\left(2\right)}=c_0^2{R}_0{Y}^2{I}_1\frac{{\tau }^{2\beta }}{\Gamma \left(2\beta +1\right)}+c_0^3{R}_0^{2}R\hspace{0.17em}\mathit{sin}\hspace{0.17em}h\left(c_0\zeta \right)Y^5{I}_2\frac{\Gamma \left(2\beta +1\right)}{{\left[\Gamma \left(\beta +1\right)\right]}^2}\frac{{\tau }^{3\beta }}{\Gamma \left(3\beta +1\right)},$

where

$I_1=16c_0^4{S}^2+8c_0^{2}S\left(5R_{0}R-63c_0^{2}S\right)Y^2+\left(3R_0^2{R}^2-176c_0^2{R}_{0}RS+1680c_0^4{S}^2\right)Y^4$

$\begin{array}{ll}& -\frac{7}{2}\left(R_0^2{R}^2-42c_0^2{R}_{0}RS+360c_0^2{S}^2\right)Y^6\end{array},$

$I_2=16c_0^4{S}^2+12c_0^{2}S\left(R_{0}R-14c_0^{2}S\right)Y^2+2\left(R_0^2{R}^2-32c_0^2{R}_{0}RS+240c_0^4{S}^2\right)Y^4\begin{array}{ll}& -\frac{5}{2}\left(R_0^2{R}^2-24c_0^2{R}_{0}RS+144c_0^4{S}^2\right)Y^6\end{array}.$

It is important to emphasize that the second-order solution ${\psi }^{\left(2\right)}\left(\zeta ,\tau \right)$ may exhibit a transition from rarefactive to compressive polarity as the fractional-time parameter $\beta$ and the evolution time $\tau$ vary. This behavior originates from the intrinsic memory effect introduced by the Caputo fractional derivative $D_{\tau }^{\beta }$ , which can be expressed as

$D_{\tau }^{\beta }f\left(\tau \right)=\frac{1}{\mathrm{\Gamma }\left(1-\beta \right)}{\int }_0^{\tau }\frac{f^{\prime }\left(s\right)}{{\left(\tau -s\right)}^{\beta }}\hspace{0.17em}ds,0<\beta <1.$

The integral kernel ${\left(\tau -s\right)}^{-\beta }$ implies the entire temporal history of the wave evolution, making the system fundamentally nonlocal in time. For smaller values of $\beta$ , the kernel decays more slowly, thereby enhancing the cumulative contribution of higher-order terms in the variational iteration approximation. In particular, terms proportional to ${\tau }^{2\beta }$ and ${\tau }^{3\beta }$ become increasingly significant as $\tau$ increases, effectively modifying the balance between nonlinear steepening and dispersive spreading. Consequently, beyond a certain evolution time, the soliton polarity can invert from rarefactive (negative potential) to compressive (positive potential). This polarity transition is therefore an intrinsic consequence of fractional-order plasma dynamics rather than a numerical artifact.

4.3. Physical interpretation of the fractional parameter $\mathit{\beta }$

The time-fractional derivative in the TFKdV equation, characterized by the parameter $\beta$ ( $0\hspace{0.17em}<\hspace{0.17em}\beta \le 1$ ), introduces a memory-dependent evolution in the plasma system. Physically, $\beta$ quantifies the degree of temporal nonlocality, capturing the effects of long-range correlations, anomalous transport, and non-Markovian responses in superthermal relativistic plasmas. When $\beta =1$ , the model reduces to the classical Markovian KdV dynamics, whereas $\beta \hspace{0.17em}<\hspace{0.17em}1$ corresponds to sub-diffusive scaling, reflecting slower soliton evolution and persistent structural features. These fractional effects naturally arise in plasmas containing dual κ-distributed electron populations and relativistic inertial electron beams, where velocity-space superthermality and delayed energy redistribution contribute to deviations from classical soliton behavior. The present interpretation aligns with previous studies of nonlinear evolutionary stages in dispersive $\kappa$ -distributed magnetized plasmas, ⁶⁶ providing an effective macroscopic description that bridges standard KdV-type evolution and more complex kinetic treatments. For the detailed numerical impact of $\beta$ on soliton amplitude and width, we refer to Figure 6(a) in the Results and Discussion section

5. Results and discussion

Here, we investigate the effects of the salient parameters on the structure and propagation of the KdV and TFKdV solitons in a relativistic, collisionless, electropositive plasma. It comprises warm relativistic fluid ions, two-temperature (cool and hot) $\kappa$ -distributed electrons, as well as a non-zero temperature relativistic inertial electron-beam. For numerical analysis, the values of the concerned parameters are considered as ⁶ : $1.8\le {\kappa }_c\le 3$ , $3\le {\kappa }_h\le 7$ , $\mu =1836$ , $0\le {\sigma }_i\le 0.3$ , $1\le {\sigma }_b\le 5$ , $100\le {\sigma }_h\le 140$ , $0.0015\le {\delta }_b\le 0.0033$ , $0.63\le {\delta }_h\le 0.65$ , ${\delta }_c=1$ , and the values of other parameters are chosen within physically meaningful ranges.

5.1. Astrophysical relevance of model parameters

The parameter ranges considered above are consistent with conditions encountered in both laboratory beam–plasma systems and astrophysical plasmas. The dual $\kappa$ -distributed electron populations with spectral indices $1.8\le {\kappa }_c\le 3$ and $3\le {\kappa }_h\le 7$ represent core–halo structures commonly reported in space plasma observations, where superthermal electron distributions are frequently modeled using finite $\kappa$ values. The smaller ${\kappa }_c$ values describe stronger superthermal tails of the cool electron component, while moderate ${\kappa }_h$ values correspond to less pronounced halo populations. The beam density ratio $0.0015\le {\delta }_b\le 0.0033$ represents a weak but finite inertial electron-beam component, consistent with laboratory beam–plasma experiments and space environments in which beam populations constitute a small fraction of the background plasma density. The temperature ratios $1\le {\sigma }_b\le 5$ and $100\le {\sigma }_h\le 140$ reflect realistic multi-temperature plasmas where halo electrons possess temperatures significantly higher than those of the core electrons. The ion temperature ratio $0\le {\sigma }_i\le 0.3$ remains within the regime of warm but subdominant ion thermal effects commonly adopted in ion-acoustic wave studies. The relativistic streaming factor $\gamma >1$ corresponds to mildly relativistic inertial electron beams, relevant to high-energy plasma environments such as pulsar magnetospheres and active galactic nuclei. Previous studies^6,67,68 have demonstrated that nonlinear ion-acoustic structures in relativistic and $\kappa$ -distributed plasmas are strongly influenced by such parameter regimes. Therefore, the adopted $\kappa$ indices, temperature ratios, beam fraction, and relativistic streaming parameters lie within physically meaningful laboratory and astrophysical ranges rather than being arbitrarily selected for mathematical illustration.

5.2. Analytical justification and robustness of solutions

The present study focuses on the analytical derivation and parameter-dependent behavior of time-fractional KdV soliton solutions in a relativistic, two-temperature $\kappa$ -electron plasma. Since the results are obtained via established analytical methods (reductive perturbation method, variational approach, and variational iteration technique), the soliton structures are exact within the adopted theoretical framework. Therefore, conventional uncertainty analysis, numerical convergence testing, or robustness evaluation is not performed in this context. The dependence of soliton properties on key plasma parameters has been thoroughly investigated through systematic parametric studies, providing insight into the behavior of the solutions under variations of physical parameters.

5.3. Parametric influence on nonlinear ion-acoustic structures

Figure 1(a) demonstrates that when the relativistic streaming factor ( ${\beta }_0$ ) increases, the phase velocity ( $\lambda$ ) drops. If the relativistic factor grows, so does the medium’s energy and wave number. Consequently, the phase velocity falls. This indicates that increasing the relativistic streaming factor enhances the effective inertial contribution of the beam, thereby modifying the dispersion relation and reducing the phase velocity. Figure 1(b) depicts the influence of the hot electron’s spectral index ( ${\kappa }_h$ ) on $\lambda$ and shows that $\lambda$ rises with increasing ${\kappa }_h$ . The increasing rate of $\lambda$ is more noticeable for the smaller ${\kappa }_h$ and larger values of the spectral index of cool electron ( ${\kappa }_c$ ). The IASWs’ phase velocity decreases with increasing hot-electron-to-ion density ratio, as shown in Figure 1(c). The increasing rate of $\lambda$ is remarkable for the smaller values of ${\kappa }_c\hspace{0.17em}\left(0\preccurlyeq {\kappa }_c\preccurlyeq 2.5\right)$ . As the hot-to-cool electron temperature ratio ( ${\sigma }_h$ ) increases, $\lambda$ correspondingly increases, as shown in Figure 1(d) for large values of ${\kappa }_c$ (cool) and small values of ${\kappa }_h$ (hot).

Figure 1.

Variation of the phase velocity ( $\lambda$ ) for the effects of (a) ${\beta }_0$ , when the values of the remaining parameters are taken as $V_0=0.03$ , $\mu =1836$ , ${\delta }_c=1$ , ${\delta }_h=0.65$ , ${\sigma }_h=100$ , and ${\kappa }_h=3$ ; (b) ${\kappa }_h$ , when the values of the remaining parameters are taken as in (a), except ${\beta }_0=0.3$ ; (c) ${\delta }_h$ , when the values of the remaining parameters are taken as in (b), except ${\kappa }_h=3.0$ ; (d) ${\sigma }_h$ , when the values of the remaining parameters are taken as in (c), except ${\delta }_h=0.65$ .

The nonlinearity of IASWs decreases as β₀ increases. At small values of κc, the superthermal impact of the cool electrons becomes high. Consequently, the nonlinearity coefficient ( $R$ ) rapidly increases over the range $1.6\preccurlyeq {\kappa }_c\preccurlyeq 1.8$ , as shown in Figure 2(a). Figure 2(b) illustrates the change in the $R$ under the influence of ${\kappa }_h$ and observes that ${\kappa }_h$ has a negligible impact on the shift in $R$ . Also, small values of ${\kappa }_c$ have a noticeable effect on the variation in $R$ . Figures 2(c) and (d), respectively, reveal the negligible impact of ${\delta }_h$ and ${\sigma }_h$ on the disparity of $R$ . In both cases, enhanced superthermality of the cold electrons leads to a rapid increase in the nonlinearity. That is, for small values of ${\kappa }_c$ , the nonlinearity rapidly flourishes.

Figure 2.

Variation of the nonlinearity coefficient ( $R$ ) for the effects of (a) ${\beta }_0$ , when the values of the remaining parameters taken as $V_0=0.03$ , $\mu =1836$ , ${\delta }_c=1$ , ${\delta }_h=0.65$ , ${\sigma }_h=100$ , ${\kappa }_h=3$ , and $c_l=3\times {10}^8$ ; (b) ${\kappa }_h$ , when the values of the remaining parameters are taken as in (a), except ${\beta }_0=0.3$ ; (c) ${\delta }_h$ , when the values of the remaining parameters are taken as in (b), except ${\kappa }_h=1.6$ ; (d) ${\sigma }_h$ , when the values of the remaining parameters are taken as in (c), except ${\delta }_h=0.63$ .

Figure 3(a) demonstrates that increasing the relativistic streaming factor ( ${\beta }_0$ ) leads to a decrease in the amplitude ( ${\psi }_{am}$ ) of the IASWs, which is more noticeable when the value of ${\kappa }_c$ is higher. As ${\beta }_{\mathit{0}}$ increases toward relativistic values, the effective relativistic inertia of the beam particles increases, which reduces the amplitude of the ion-acoustic solitary waves due to stronger inertial opposition to nonlinear steepening. Again, when ${\beta }_0$ increases, Lorentz’s force becomes more prominent, and for this force, the amplitude of the IASWs declines. The amplitude grows (in the range $1.8\preccurlyeq {\kappa }_c$ ) with ${\kappa }_h$ , as seen in Figure 3(b). As the superthermality ${\kappa }_h$ increases, more energy is concentrated in the medium, leading to additional energy gain for the waves. So, the amplitude of IASWs grows. It is to be noted that the Maxwellian cold electron effect is more dominant on the change of amplitude while ${\kappa }_h$ is small. Both parameters, ${\delta }_h$ and ${\sigma }_h$ , have a minor impact on the change in the IASW amplitude, as shown in Figures 3(c) and (d). It is worth mentioning that when ${\delta }_h$ ( ${\delta }_h$ ) is fixed, then the amplitude increases with increasing ${\kappa }_c$ , as shown in Figure 3(c) and (d).

Figure 3.

The amplitude ( ${\psi }_{am}$ ) of the KdV soliton for the effects of (a) ${\beta }_0$ , when the values of the remaining parameters taken as $V_0=0.03$ , $\mu =1836$ , ${\delta }_c=1$ , ${\delta }_h=0.63$ , ${\sigma }_h=100$ , ${\kappa }_h=1.6$ , $c_l=3\times {10}^8$ , and ${\lambda }_v=0.04$ ; (b) ${\kappa }_h$ , when the values of the remaining parameters are taken as in (a), except ${\beta }_0=0.3$ ; (c) ${\delta }_h$ , when the values of the remaining parameters are taken as in (b), except ${\kappa }_h=1.6$ ; (d) ${\sigma }_h$ , when the values of the remaining parameters are taken as in (c), except ${\delta }_h=0.63$ .

Figure 4(a) shows the soliton structure for the growing influence of ${\beta }_0$ . It is observed that the amplitude and width of the IASWs decrease as ${\beta }_0$ increases. The medium’s dispersion reduces as the relativistic streaming factor increases. Thus, the width of the IASWs decreases, since width is directly proportional to the dispersion coefficient of the KdV equation. Figure 4(b) highlights that in the range $1.51\preccurlyeq {\kappa }_c\preccurlyeq 1.59$ , both the amplitude and width grow with a rise in the spectral index of cool electron ( ${\kappa }_c$ ). An increase in ${\kappa }_c$ enhances the effective energy and dispersion of the medium; thus, waves receive additional energy from it. Consequently, the amplitude of IASWs increases. Furthermore, the medium’s dispersion improves, and so the width of the IASWs increases. An increase in either the temperature ratio ( ${\sigma }_b$ ) or the density ratio ( ${\delta }_b$ ) leads to a broader IASW width, as depicted in Figures 4(c) and (d).

Figure 4.

KdV soliton structure ( $\psi$ ) for the effects of (a) ${\beta }_0$ , when the values of the remaining parameters taken as $V_0=0.03$ , $c_l=3\times {10}^8$ , $\mu =1836$ , ${\delta }_b=0.0015$ , ${\delta }_c=1$ , ${\delta }_h=0.63$ , ${\sigma }_b=1.5$ , ${\sigma }_h=100$ , ${\sigma }_i=0.1$ , ${\kappa }_h=1.6$ , ${\kappa }_c=1.6$ , ${\lambda }_v=0.04$ , $p_{i1}=1$ , $p_{b1}$ =1, and $\tau =1$ ; (b) ${\kappa }_c$ , when the values of the remaining parameters are taken as in (a), except ${\beta }_0=0.3$ ; (c) ${\sigma }_b$ , when the values of the remaining parameters are taken as in (b), except ${\kappa }_c=1.6$ ; (d) ${\delta }_b$ , when the values of the remaining parameters are taken as in (c), except ${\sigma }_b=1.5$ .

Figure 5 explores the impacts of plasma parameters (like time fractional parameter ( $\beta$ ), relativistic streaming factor ( ${\beta }_0$ ), cool electron’s spectral index ( ${\kappa }_c$ ), and temperature ratio of electron beam to cool electron ( ${\sigma }_b$ ) on the approximation up to the first-order TFKdV soliton. From Figure 5(a), it is observed that both compressive and rarefactive TFKdV solitons (obtained from the approximation up to the first-order provided in Eq. (41) are produced. Figure 5(b) highlights the first-order approximation to the TFKdV soliton under the influence of ${\beta }_0$ and finds that in this situation, only a dip-shaped rarefactive soliton is produced. Figures 5(c) and (d) demonstrate the production of the first-order TFKdV soliton with increasing effects of ${\kappa }_c$ and ${\sigma }_b$ , respectively. In both cases, only dip-shaped rarefactive solitons are produced. In addition, due to the growing influence of ${\kappa }_c$ , both the amplitude and width of the IASWs are noticeably increased, whereas under the influence of ${\sigma }_b$ , the increase in width and amplitude is slight.

Figure 5.

Approximation up to the first-order TFKdV soliton ( ${\psi }^{\left(1\right)}$ ) under the effects of (a) $\beta$ , when the values of the remaining parameters taken as $V_0=0.03$ , $c_l=3\times {10}^8$ , ${\beta }_0=0.3$ , $\mu =1836$ , ${\delta }_b=0.0015$ , ${\delta }_c=1$ , ${\delta }_h=0.63$ , ${\sigma }_b=1.5$ , ${\sigma }_h=100$ , ${\sigma }_i=0.1$ , ${\kappa }_h=1.6$ , ${\kappa }_c=1.6$ , ${\lambda }_v=0.04$ , ${\lambda }_0=0.1$ , $p_{i1}=1$ , $p_{b1}$ =1, and $\tau =1$ ; (b) ${\beta }_0$ , when the values of the remaining parameters are taken as in (a), except $\beta =0.01$ ; (c) ${\kappa }_c$ , when the values of the remaining parameters are taken as in (b), except ${\beta }_0=0.3$ ; (d) ${\sigma }_b$ , when the values of the remaining parameters are taken as in (c), except ${\kappa }_c=1.51$ .

Figure 6 exposes the effects of related parameters fractionality $\beta$ , ${\beta }_0$ , ${\kappa }_c$ , and ${\sigma }_b$ on the approximation up to the second-order TFKdV soliton ( ${\psi }^{\left(2\right)}$ ). Figure 6(a) elucidates that for $\beta =0.01$ and $\beta =0.10$ , the width and amplitude increase, whereas those decrease for $\beta =0.80$ and $\beta =0.90$ . An increase in ${\beta }_0$ results in a decrease in both the amplitude and width of the second-order TFKdV soliton, depicted in Figure 6(b). Both amplitude and width of ${\psi }^{\left(2\right)}$ increase with rising ${\hspace{0.17em}\kappa }_c$ , while increasing ${\sigma }_b$ increases the width, as observed in Figures 6(c) and (d), respectively.

Figure 6.

Approximation up to the second-order TFKdV soliton ( ${\psi }^{\left(2\right)}$ ) for the effects of (a) $\beta$ , when the values of the remaining parameters taken as $V_0=0.03$ , $c_l=3\times {10}^8$ , ${\beta }_0=0.3$ , $\mu =1836$ , ${\delta }_b=0.0015$ , ${\delta }_c=1$ , ${\delta }_h=0.63$ , ${\sigma }_b=1.5$ , ${\sigma }_h=100$ , ${\sigma }_i=0.1$ , ${\kappa }_h=1.6$ , ${\kappa }_c=1.6$ , ${\lambda }_v=0.04$ , ${\lambda }_0=0.1$ , $p_{i1}=1$ , $p_{b1}$ =1, and $\tau =1$ ; (b) ${\beta }_0$ , when the values of the remaining parameters are taken as in (a), except $\beta =0.01$ ; (c) ${\kappa }_c$ , when the values of the remaining parameters are taken as in (b), except ${\beta }_0=0.3$ ; (d) ${\sigma }_b$ , when the values of the remaining parameters are taken as in (c), except ${\kappa }_c=1.51$ .

The propagation of the KdV soliton over time is shown in Figure 7(a). The soliton propagates forward in time, preserving its amplitude and width. Figure 7(b) explicates the first-order TFKdV soliton at different times and finds that the soliton propagates in the negative $\zeta$ -direction. The second-order TFKdV solitons for other times are displayed in Figure 7(c). It demonstrates that rarefactive solitons form at low time-parameter values, whereas compressive solitons form at high time-parameter values. The polarity transition observed in Figure 7(c) is a direct manifestation of the memory effect associated with the fractional-time derivative. Because the Caputo operator incorporates the entire temporal evolution of the wave through a slowly decaying kernel, higher-order contributions accumulate as the evolution time increases. As a result, the effective nonlinear–dispersive balance governing the solitary structure is modified, potentially leading to a gradual inversion of soliton polarity from rarefactive to compressive. This confirms that the observed behavior is physically linked to the nonlocal nature of fractional dynamics.

Figure 7.

Effect of time variation on the (a) KdV soliton ( $\psi$ ), when the values of the parameters taken as $V_0=0.03$ , $c_l=3\times {10}^8$ , ${\beta }_0=0.3$ , $\mu =1836$ , ${\delta }_b=0.0015$ , ${\delta }_c=1$ , ${\delta }_h=0.63$ , ${\sigma }_b=1.5$ , ${\sigma }_h=100$ , ${\sigma }_i=0.1$ , ${\kappa }_h=1.6$ , ${\kappa }_c=1.6$ , ${\lambda }_v=0.04$ , , $p_{i1}=1$ , and $p_{b1}$ =1; (b) first order TFKdV soliton ( ${\psi }^{\left(1\right)}$ ) for ${\lambda }_0=0.1$ and $\beta =0.5,$ when the values of the remaining parameters taken as in (a); and (c) second order TFKdV ( ${\psi }^{\left(2\right)}$ ) when the values of the remaining parameters are taken as in (b), except $\beta =0.9$ .

5.4. Modulational dynamics and envelope formation

Previous studies have shown that weakly nonlinear structures governed by KdV-type dynamics can evolve into amplitude-modulated wave packets through higher-order multiscale reductions, resulting in nonlinear Schrödinger descriptions of envelope states, as demonstrated by Majumdar et al. ⁶⁹ and Sarkar et al. ⁷⁰ In contrast, the present analysis examines a collisionless relativistic plasma composed of dual-κ electrons and a streaming electron beam, where evolution is governed by a time-fractional KdV equation derived through a variational approach. Introducing fractional temporal order alters the effective propagation scale and shifts the balance between nonlinear steepening and dispersive spreading at the solitary-wave level. Since envelope formation requires an additional slow modulation analysis beyond the perturbative order considered here, such a treatment is outside the scope of this study. Nonetheless, the modified dispersive structure resulting from the fractional relativistic framework indicates that modulational behavior could be significantly affected under a more comprehensive analysis, motivating future research.

5.5. Possible experimental relevance and validation

Although the present study is theoretical, the predicted relativistic and time-fractional effects could, in principle, be explored in laboratory beam–plasma or laser-produced plasma experiments. Relativistic electron beams can be generated in laser wakefield setups or linear accelerators interacting with pre-formed ion plasmas. The resulting ion-acoustic solitary waves could be probed using Langmuir probes, Thomson scattering, or fast optical diagnostics to measure amplitude, width, and phase velocity. Fractional-time effects, which model intrinsic memory in the plasma, may be effectively emulated in experiments by controlling parameters such as beam current, plasma density, and electron temperature. Adjusting these parameters could allow a qualitative reproduction of the nonlocal temporal dynamics predicted in our model. Observing the evolution of soliton polarity and structural changes over time would provide experimental insight into the roles of relativistic streaming and fractional-order effects. While a full experimental protocol is beyond the scope of the present work, these considerations outline a conceptual framework for validation. Such studies could bridge the gap between theoretical predictions and feasible laboratory observations. In this context, our results may guide future beam–plasma experiments aimed at exploring nonlinear soliton dynamics under relativistic and nonlocal temporal effects. Overall, the proposed approach emphasizes a qualitative, rather than quantitative, experimental confirmation of our findings.

5.6. Model limitations

The current model relies on several simplifying assumptions that limit its direct application. First, it assumes the plasma is collisionless, which may not fully account for collisional damping or thermalization effects found in real laboratory or space plasmas. Second, it uses a one-dimensional approximation, ignoring transverse variations and effects from multiple dimensions that could influence soliton behavior. Third, magnetic fields are neglected, so behavior in magnetized plasmas isn’t represented. Additionally, the time-fractional formulation is an idealized approach that introduces memory effects but doesn’t correspond to a directly measurable fractional parameter in lab systems. These assumptions mean that our results are mainly useful for qualitative insights rather than precise predictions. Future research could expand the model to incorporate collisions, higher-dimensional geometries, and magnetic-field effects to improve accuracy. Despite these limitations, the model offers valuable insights into how relativistic and superthermal factors affect soliton propagation.

6. Conclusion

A study has been conducted to analyze the effects of the relativistic streaming factor, superthermal index of cool and hot electrons, ratio of hot-to-cold electron temperatures, as well as hot electron-to-positive ions density ratio on the fundamental features like phase velocity, nonlinearity, amplitude, KdV and TFKdV soliton structures, as well as the propagation phenomena of IASWs in a collisionless, electropositive non-zero temperature relativistic electron-beam plasma environment, including relativistic fluid ions and $\kappa$ -distributed cool and hot electrons. For doing this, a KdV equation has been deduced by implementing the reductive perturbation scheme, and a time-fractional version of it has been obtained utilizing El-Wakil et al.’s procedure. ³⁹

It has been observed that an increase in the relativistic streaming factor or hot electron-to-ion density ratio or in the cool electron’s spectral index causes the phase velocity of IASWs to fall, whereas it rises with an increase in the hot electron’s superthermal index or ratio of hot-to-cold electron temperatures. The nonlinearity of IASWs changes inversely as the relativistic streaming factor increases. The high super-thermal impact of the cool electron causes a rapid increase in the nonlinearity coefficient in the range $1.6\preccurlyeq {\kappa }_c\preccurlyeq 1.8$ . The amplitude and width of the IASWs decrease when the relativistic factor grows, as then the medium’s dispersion is reduced. Within $1.51\preccurlyeq {\kappa }_c\preccurlyeq 1.59$ , the KdV soliton becomes taller and broader as the spectral index of cool electrons rises. The breadth of the KdV soliton increases with a boost in the beam-to-cool electron temperature ratio or the beam-to-ion density ratio, respectively.

Both compressive, hump-shaped, and rarefactive, dip-shaped first-order TFKdV solitons form as the time-fractional parameter varies. Under the influence of the relativistic streaming parameter, cool electron spectral index, and the electron beam-to-cool electron temperature ratio, only rarefactive first-order TFKdV solitons are observed. The amplitude and width of second-order TFKdV solitons decrease with increasing relativistic factor, while increasing the spectral index of cool electrons enhances both amplitude and width. The KdV solitons maintain their amplitude and width while moving forward in time, whereas the first-order TFKdV solitons shift left at different times. Furthermore, rarefactive second-order approximations to the TFKdV solitons occur at low values of the time parameter, whereas compressive approximations occur at higher values.

It is important to emphasize that these conclusions are confined to the physically admissible parameter ranges explored in this study. No extrapolation beyond the investigated relativistic streaming factors, $\kappa$ -indices, temperature ratios, density ratios, or fractional-order values has been made. All reported trends are directly supported by the analytical derivations and numerical illustrations provided. Furthermore, although the study is theoretical, the predicted relativistic and time-fractional effects could, in principle, be examined in laboratory beam-plasma or laser-produced plasma experiments, providing a conceptual pathway for qualitative experimental validation.

Future works: The current work includes several future research directions that could be pursued as extensions for the current investigation.

• Extension of the present plasma configuration to a fractional framework by introducing Caputo-type time-fractional derivatives into the evolution equations and then employing the Tantawy technique to construct rapidly convergent analytical approximations for the resulting fractional KdV/mKdV/Gardner-type models that capture memory-dependent nonlinear wave propagation in the current plasma model or any other complex plasma models. Application of the Tantawy technique, as a unifying perturbative-decomposition scheme, to systematically analyze a broader class of fractional evolutionary wave equations relevant to plasmas, such as fractional Burgers, KdV-Burgers, forced KdV, and coupled KdV-mKdV systems, to benchmark its accuracy and convergence against other modern fractional solvers and to map out its domain of optimal applicability.^71–73

• Formulation of nonpse lanar (cylindrical and spherical) variants of the reduced evolution equations arising from the present plasma model, followed by their fractional generalization, to study how geometric curvature and long-range memory jointly reshape the amplitude, width, and stability of electrostatic solitary and shock-like structures in laboratory, space, and astrophysical plasmas. ⁷⁴ Investigation of the interplay between geometric curvature and memory effects by deriving cylindrical and spherical time-fractional KdV/mKdV/Gardner equations and solving them via the Tantawy technique,^63,64 with an emphasis on how curvature-induced geometric dispersion competes with fractional-order–induced anomalous dissipation in determining the localization, steepening, and eventual decay of nonlinear wave packets.

• Incorporation of collisional damping, whether through effective friction terms modeling ion-neutral collisions, collisions between charged particles, or fractional dissipative operators, into the current plasma model will naturally lead to damped fractional KdV/modified KdV/Gardner equations. Subsequent analysis (via the Tantawy technique) of how combined memory and damping forces influence the long-term evolution and potential extinction of solitary excitations will clarify the competition between collisionless and collisional dissipation in the presence of memory. ⁶⁵

• Generalization of the present analysis to magnetized multi-component plasmas (including superthermal electrons, positrons, and dust grains) described by space-time fractional evolution equations, and use of the Tantawy technique in combination with Hirota-type or bilinear approaches to explore richer families of nonlinear structures (solitons, kinks, breathers, and cnoidal waves) and their sensitivity to both the fractional order and the magnetic field geometry. Exploration of higher-dimensional effects ⁷⁵ by deriving fractional Zakharov-Kuznetsov-type equations from generalized versions of the current model, followed by applying the Tantawy technique to track how memory and weak transverse modulations influence the formation, obliqueness, and stability of line and lump solitons in dusty and pair plasmas.

• Application of the Tantawy technique to inhomogeneous and forced fractional evolution equations emerging from plasmas with weak gradients in density, temperature, or external fields, to quantify how memory and inhomogeneity jointly modulate energy localization, the onset of shock formation, and the robustness of coherent structures against slow background variations.