Nonlinear dynamics of the dissipative ion-acoustic solitary waves in anisotropic rotating magnetoplasmas

Wedad Albalawi; Carlos A. Fotsing; Camus G. L. Tiofack; Alim; Alidou Mohamadou; Rania A. Alharbey; Samir A. El-Tantawy

doi:10.1515/phys-2025-0209

Article Open Access

Nonlinear dynamics of the dissipative ion-acoustic solitary waves in anisotropic rotating magnetoplasmas

Wedad Albalawi , Carlos A. Fotsing , Camus G. L. Tiofack , Alim , Alidou Mohamadou , Rania A. Alharbey and Samir A. El-Tantawy

Published/Copyright: October 8, 2025

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From the journal Open Physics Volume 23 Issue 1

Abstract

This study explores the nonlinear dynamics of ion-acoustic waves (IAWs) in a magnetized, collisional, anisotropic rotating plasma that includes hot ions, superthermal electrons, and positrons. Anisotropic ion pressure is defined using the Chew–Goldberger–Low theory. Our linear analysis shows that pressure anisotropy notably impacts wave frequency, particularly for shorter wavelengths, and identifies a threshold wavenumber beyond which wave propagation is impossible. We derive a nonlinear damped Zakharov–Kuznetsov equation by applying the reductive perturbation technique. This equation describes the phase velocity and profile of ion-acoustic solitary waves, which are significantly influenced by superthermal, electron–positron temperature ratio, pressure anisotropy, the Coriolis force, and ion collisions. Our numerical analysis reveals that IAWs propagate in the plasma in a direction parallel to the magnetic field with a phase velocity that is unaffected by the plasma rotation frequency Ω 0 , the magnetic field through ω c i , or the perpendicular pressure component P ⊥ . The phase velocity increases with the κ index and parallel pressure P ‖ and decreases with the positron temperature ratio σ . Moreover, it is found that the wave amplitude decreases with increasing ion pressure ( P ‖ ) and the electron–positron temperature ratio ( σ ) . On the contrary, the amplitude increases with rising superthermality κ , while collisions cause the wave amplitude to spread. The Coriolis force exclusively affects the width of electrostatic waves. The results of this study are particularly relevant for understanding wave behavior in astrophysical and space environments, especially within Earth’s magnetosphere, where nonthermal electrons and positrons coexist with anisotropic pressure ions.

Keywords: anisotropic pressure; damped Zakharov–Kuznetsov equation; ion-acoustic waves; dissipative solitary waves; collisional plasma; kappa distribution; magnetized rotating plasma

1 Introduction

In recent years, the study of ion-acoustic waves (IAWs) in electron–positron–ion (e–p–i) plasma has garnered significant attention, enhancing our understanding of nonlinear phenomena in both space and laboratory settings [1–9]. IAWs are a pivotal feature of plasma systems, distinguished by their low-frequency oscillations primarily driven by ion motion and often accompanied by electron and positron density fluctuations. Washimi and Taniuti pioneered the investigation of ion-acoustic solitary waves (IASWs) in plasma [10], along with Sagdeev’s theoretical framework [11]. This established a core nonlinear theory of IAWs, which Ikezi et al. [12] later validated through experiments. Since then, advancements in understanding IASWs have been noteworthy by many authors [13,14]. For instance, research on the dynamics of these waves in magnetized plasma has revealed both subsonic and supersonic compressive solitons, using the pseudopotential method to explore how various plasma parameters influence wave profiles [15]. Plasma waves have also been investigated in semiconductor materials during photothermal excitation [16,17]. Studies on large-amplitude solitary waves in warm magnetoplasma have utilized the Sagdeev pseudopotential approach to examine wave structures across different amplitudes. Researchers have analyzed the impact of magnetic fields and plasma temperatures on the formation and stability of these waves, showing how factors like ion temperature and Mach number influence soliton behavior [18]. Additionally, investigations into small-amplitude solitary waves in magnetized ion-beam plasma have highlighted the coexistence of slow and fast modes under varying conditions [19]. Furthermore, recent studies have delved into the complex behaviors of nonlinear IAW structures such as periodic waves, solitary waves, and breathers in auroral magnetoplasmas [20].

The presence of a strong magnetic field in a collisionless medium results in plasma anisotropy, where the plasma exhibits distinct behaviors in parallel and perpendicular directions relative to the magnetic field [21]. The Chew–Goldberger–Low (CGL) theory [22], formulated in 1956, aptly describes this anisotropy, provided there is no coupling between the parallel and perpendicular degrees of freedom [23]. To analyze such plasmas, separate equations of state are required for the perpendicular and parallel ion pressures, denoted as P i ⊥ and P i ‖ , where P i ⊥ ( P i ‖ ) are the perpendicular (parallel, respectively) components of the ion pressure relative to the ambient magnetic field. Wave–particle interactions may reduce this anisotropy by establishing correlations between these directions [24,25]. In space plasmas, phenomena such as plasma convection cause magnetic compression and expansion along field lines, leading to variations in the perpendicular and parallel temperatures [24]. These anisotropic conditions are commonly observed in regions like the magnetosphere and the near-Earth magnetosheath [21,26,27]. Adnan et al. investigated small amplitude ion-acoustic solitons in weakly magnetized plasma, focusing on anisotropic ion pressure [28]. Similarly, Chatterjee et al. examined obliquely propagating IASWs and double layers in magnetized dusty plasma with anisotropic ion pressure [29]. Additionally, the effects of pressure anisotropy on nonlinear electrostatic excitations in magnetized e–p–i plasmas were explored, revealing that anisotropic ion pressure significantly impacts the width and amplitude of solitary waves [30,31]. Furthermore, it was found that the increase in wave frequency induced by anisotropic ion pressure alters wave dispersion and enhances wave crest accumulation, potentially leading to the development of modified instability modes [32]. A decrease in pressure anisotropy has also been shown to enhance both the amplitude and width of dust-acoustic rogue waves [33,34]. Alternatively, the effect of a magnetic field on the motion of two rigid objects traveling linearly through an incompressible fluid with a couple stress characteristics has been explored [35].

The velocity distribution of plasma particles influences the behavior of wave motions in plasma. The Maxwellian distribution has been the standard choice for representing these particles for many decades. However, due to advancements in the study of space [36–39] and laboratory plasmas [40,41], non-Maxwellian particle distributions have become more prevalent. These distributions more accurately depict energetic particles with high-energy (superthermal) velocity distributions, often exhibiting superthermal tails. The Kappa velocity distribution is particularly effective in describing these tails. In 1968, Vasyliunas [36] introduced the Kappa distribution to match observational data. Today, this distribution is widely used to explain various astrophysical and space plasma phenomena, including those in the auroral zone [42], Earth’s magnetosphere [43], in the interstellar medium [44], and in the solar wind [45]. Recent research has delved into the impact of superthermal electron distributions on solitary waves [46–49]. Saini et al. [50] examined the dynamics of electrostatic solitary excitations with superthermal electrons using a pseudopotential approach, concluding that for a fixed Mach number, the solitary wave profile becomes steeper and wider compared to typical plasma structures. Further studies on superthermal particles’ role in electrostatic wave packets in electron–ion [51] and e–p–i [52,53] plasmas have shown, using the nonlinear Schrödinger equation, that superthermality increases the modulational instability of these wave packets. Singh et al. [54] recently studied the thermal effects of ions on IAWs in magnetized superthermal plasma via the Sagdeev potential approach, achieving results consistent with Viking satellite observations in the auroral region.

Wave propagation in plasmas with anisotropic pressure can experience significant changes. When introduced, magnetic fields induce the Lorentz force, which affects the motion of charged particles and modifies wave dispersion, stability, and energy redistribution [55]. Additionally, plasma rotation brings in the Coriolis force [56], which alters wave characteristics, especially in rotating plasma environments like fusion reactors or planetary magnetospheres [57]. These combined effects create a complex environment for understanding wave behavior. Theoretical and experimental studies have shown that even a weak Coriolis force can lead to interesting phenomena in astrophysical environments [57]. Given the Coriolis force’s important role in rotating space plasmas, many researchers have attempted to analyze wave dynamics in the presence of the Coriolis force [58–61].

In addition to the effects of anisotropy and non-Maxwellian distributions, ion-neutral collisions play a crucial role in many plasmas. These collisions introduce dissipation and damping mechanisms that influence wave stability and soliton formation, acting as a source of energy loss and further complicating plasma dynamics. The behavior of IASWs in such complex plasmas can be described using extended nonlinear equations, such as the Zakharov–Kuznetsov (ZK) equation with a damping term. This equation incorporates the effects of anisotropy, magnetic fields, collisions, rotation, and non-Maxwellian velocity distributions. The ZK equation generalizes the Korteweg–de Vries equation to higher dimensions and can describe the evolution of ion-acoustic solitons under various physical conditions. Several researchers have recently focused on studying dissipation phenomena in plasmas [62–64]. Their findings indicate that dissipation significantly impacts the profile of IASWs in plasma. While there has been extensive research on the effects of anisotropic ion pressure, studies on collisional magnetized nonthermal e–p–i rotating plasmas with superthermal electrons and positrons remain limited. This project aims to explore the nonlinear dynamics of ion-acoustic solitons in such a plasma, characterized by anisotropic, magnetized, and rotating conditions with a Kappa velocity distribution. The focus will be on understanding the influences of Coriolis forces, magnetic fields, ion-neutral collisions, and anisotropic pressure on soliton formation, stability, and propagation. By deriving and solving the ZK equation under these complex conditions, this study will deepen our comprehension of IAW dynamics in plasma systems with more realistic physical characteristics. These insights are essential for advancing our knowledge of phenomena in space physics, fusion research, and astrophysical systems.

This article is organized as follows: The basic governing equations for describing our anisotropy e–p–i plasma system are shown in Section 2. In Section 3 linear structure is carried out. Section 4 contains the derivation of the damped Zakharov–Kuznetsov (dZK)-type equation using the reductive perturbation method (RPM). The analytical solutions representing dissipated solitary pulses are given in Section 4.1 and numerical simulations are shown in Section 4.2. Finally, the summary of our research work is concluded in Section 5.

2 Basic governing equations

We consider a collisional, magnetized, three-component e–p–i plasma that rotates under the influence of the Coriolis force. The electrons and positrons in this plasma are superthermal, while the ions experience weak collisions with neutrals. These ions are considered inertial ( m i ≫ m e , p ) , having mass m i , velocity u i , charge Z i e , collision frequency ν n , and an anisotropic ion pressure tensor P ¯ ¯ i modeled by the CGL theory [22]. The magnetic field is uniform and oriented along the z → -axis ( B → = B 0 e → z ) . The plasma rotates with a low frequency Ω 0 around the z → -axis due to the Coriolis effect, with negligible centrifugal force. We assume that the electrostatic wave propagates in all three spatial directions, i.e., ∇ → = ( ∂ x , ∂ y , ∂ z ) . At thermal equilibrium, the plasma composition is such that μ e = 1 + μ p , where μ s = n s 0 ⁄ n i 0 , with n i 0 and n s 0 ( s = e , p ) denoting the equilibrium number densities of ions, electrons, and positrons, respectively. The following basic equations govern the dynamics of IAWs in the current plasma model [28,30,62]:

(1) ∂ t n i + ∇ ⋅ ( n i u i ) = 0 ,

(2) ∂ t u i + ( u i ⋅ ∇ ) u i + Z i e m i ∇ ϕ − Z i e m i c B 0 u i ∧ e z − 2 u i ∧ Ω + 1 m i n i ∇ ⋅ P ¯ ¯ i + ν n u i = 0 ,

(3) ∇ 2 ϕ = − 4 π e ( n p + Z i n i − n e ) ,

where e denotes the elementary charge, n s denotes the number density of particle s ( s = e , p , i ) , and ϕ denotes the electrostatic potential. P ¯ ¯ i represents the pressure tensor. The number densities of electrons and positrons are defined by [28]

(4) n s = n s 0 1 + q s ϕ k B T s ( κ s − 3 ⁄ 2 ) − ( κ s − 1 ⁄ 2 ) ,

with q s = e if s = p and q s = − e if s = e .

In this work, the pressure differs between the perpendicular and parallel directions, leading to plasma anisotropy under a strong magnetic field. Consequently, the pressure tensor takes the form [65]

(5) P ¯ ¯ i = P i ⊥ 0 0 0 P i ⊥ 0 0 0 P i ‖ = P i ⊥ I ¯ ¯ + ( P i ‖ − P i ⊥ ) e z e z ,

where I ¯ ¯ is the unit tensor and e z is the unit vector along the magnetic field direction. According to the CGL theory [22], the scalar pressures P i ⊥ and P i ‖ are defined as follows:

(6) P i ⊥ = P i 0 ⊥ n i n i 0 ; P i ‖ = P i 0 ‖ n i n i 0 3 ,

where P i 0 ⊥ ( Z i n i 0 k B T i ⊥ ) and P i 0 ‖ ( Z i n i 0 k B T i ‖ ) define the perpendicular and parallel pressures, respectively, at equilibrium. In isotropic plasmas, P i ⊥ = P i ‖ . In the three spatial directions ( x , y , z ) through which the wave propagates, the plasma dynamics equations are as follows:

(7) ∂ t n i + ∂ x ( n i u i x ) + ∂ y ( n i u i y ) + ∂ z ( n i u i z ) = 0 ,

(8) n i ∂ t u x + n i ( u x ∂ x + u y ∂ y + u z ∂ z ) u x + n i ∂ x Φ − ( ω c i + 2 Ω 0 ) n i u y + P i ⊥ ∂ x n i + ν n n i u x = 0 ,

(9) n i ∂ t u y + n i ( u x ∂ x + u y ∂ y + u z ∂ z ) u y + n i ∂ y Φ + ( ω c i + 2 Ω 0 ) n i u x + P i ⊥ ∂ y n i + ν n n i u y = 0 ,

(10) ∂ t u z + ( u x ∂ x + u y ∂ y + u z ∂ z ) u z + ∂ z Φ + P i ‖ n i ∂ z n i + ν n u z = 0 ,

(11) ( ∂ x 2 + ∂ y 2 + ∂ z 2 ) Φ + n i − 1 − α 1 Φ − α 2 Φ 2 − α 3 Φ 3 + ⋯ = 0 ,

where the coefficients α 1 , α 2 , and α 3 are functions of the κ parameter given by

(12) α 1 = ( 1 + ( 1 + σ ) μ p ) ( κ − 1 ⁄ 2 ) ( κ − 3 ⁄ 2 ) , α 2 = ( 1 + ( 1 − σ 2 ) μ p ) ( κ 2 − 1 ⁄ 4 ) 2 ( κ − 3 ⁄ 2 ) 2 , α 3 = ( 1 + ( 1 + σ 3 ) μ p ) ( κ 2 − 1 ⁄ 4 ) ( κ + 3 ⁄ 2 ) 6 ( κ − 3 ⁄ 2 ) 3 ,

with κ e = κ p = κ for simplification of calculations.

Here, n i is the ion number density normalized by its equilibrium value n i 0 , u is the ion velocity considered fluid and normalized by the ion sound speed C s i ( = ( Z i k B T e ⁄ m i ) 1 ⁄ 2 ) , and Φ is the electrostatic potential normalized by k B T e ⁄ e . The time variable is normalized by the plasma frequency ω p i ( = ( 4 π Z i n i 0 e 2 ⁄ m i ) 1 ⁄ 2 ) . The spatial variables x , y , and z are normalized by the Debye length λ D i ( = ( k B T e ⁄ 4 π Z i n i 0 e 2 ) 1 ⁄ 2 ) . Here, k B is the Boltzmann constant, e is the elementary charge, and σ ( = T e ⁄ T p ) is the temperature ratio of electrons to positrons.

3 Linear structure

The analysis involves performing a Fourier analysis of Eqs. (7)–(11), assuming small perturbations of the form ∼ exp [ j ( k → ⋅ r → − ω t ) ] around the equilibrium value, where j 2 = − 1 , k 2 = k ⊥ 2 + k ‖ 2 , with k ⊥ 2 = k x 2 + k y 2 and k ‖ 2 = k z 2 , and ω represents the perturbation frequency. In the limit of large wavelength values ( k ≪ 1 ) , the dispersion relation of waves propagating in a uniform magnetic field with anisotropic ion pressure is obtained and defined as

(13) − ω + 1 k 2 + α 1 + P ⊥ ( ω + j ν ) k ⊥ 2 ( ω + j ν ) 2 − ( ω c i + 2 Ω 0 ) 2 + 1 k 2 + α 1 + P ‖ k ‖ 2 ( ω + j ν ) = 0 ,

where ν = ν n . In the case of a non-collisional plasma ( ν = 0 ) and low rotation ( ω c i ≫ Ω 0 ) , the dispersion relation reduces to

(14) 1 k 2 + α 1 + P ⊥ k ⊥ 2 ω 2 − ω c i 2 + 1 k 2 + α 1 + P ‖ k ‖ 2 ω 2 = 1 .

Eq. (14) defines the dispersion relation of waves in a non-collisional and non-rotational magnetized e–p–i plasma with anisotropic ion pressure. In this limiting case, Eq. (14) aligns with Eq. (21) in Adnan et al. [30] and Eq. (32) in Ullah Khan et al. [66]. Solving Eq. (14) reveals two ion oscillation modes within this model: the slow or acoustic mode, characterized by the frequency ω − , and the fast mode, characterized by the frequency ω + such that

(15) ω ± 2 = 1 2 F ± 1 2 F 2 − 4 C ,

where F = k 2 k 2 + α 1 + P ⊥ k ⊥ 2 + P ‖ k ‖ 2 + ω c i 2 and C = 1 k 2 + α 1 + P ‖ k ‖ 2 ω c i 2 . Next, we will examine the behavior of ion oscillations in both the perpendicular and parallel directions of the magnetic field.

3.1 Parallel component

To consider the propagation of the wave in the direction of the magnetic field, we set k ⊥ = 0 and k ‖ = k . Consequently, Eq. (13) becomes

(16) ω 2 + j ν ω − 1 k 2 + α 1 + P ‖ k 2 = 0 .

The solution to this equation is complex and expressed as ω = ω r + j ω i m , given by

(17) ω r = ± 1 k 2 + α 1 + P ‖ k 2 − 1 4 ν 2 1 ⁄ 2 , ω i m = − 1 4 ν 2 .

Ion damping occurs due to collisions between ions and neutrals. For large wavelengths ( k ≪ 1 ) , there is a threshold value k s for the wavenumber below which the wave is damped. Consequently, ion oscillations occur in the medium when k s < k ( ω r 2 > 0 ) . In this context, k s is defined as

(18) k s = ν 2 α 1 1 + α 1 P ‖ .

For k s > k , ion oscillations cease, and the medium becomes strongly dissipative.

We will now numerically analyze the behavior of the dispersion relation in Eq. (16) for various values of the spectral parameter ( κ ) , the positron concentration ( μ p ) , the pressure ( P ‖ ) , the temperature ratio ( σ ) , and the collision frequency ( ν ) .

Figure 1 depicts the linear properties of the dispersion concerning the wavenumber k . Figure 1(a) demonstrates that the frequency of ion oscillations increases with both the wavenumber k and the spectral parameter. The superthermality of the plasma induces ion instability at low wavenumbers. The spectral index κ does not affect the imaginary part of the wave frequency.

$Figure 1 Plot of the real part of wave frequency ω r {\omega }_{r} versus wavenumber k k (a) for different values of the spectral parameter κ \kappa with μ p = 0.4 , σ = 0.5 {\mu }_{p}=0.4,\sigma =0.5 , P ‖ = 0.3 {P}_{\Vert }=0.3 , and ν = 0.02 \nu =0.02 and (b) for different values of parallel component of anisotropy pressure P ‖ {P}_{\Vert } with μ p = 0.4 , κ = 3 , σ = 0.2 {\mu }_{p}=0.4,\kappa =3,\sigma =0.2 , and ν = 0.04 \nu =0.04 .$

Figure 1

Plot of the real part of wave frequency ω r versus wavenumber k (a) for different values of the spectral parameter κ with μ p = 0.4 , σ = 0.5 , P ‖ = 0.3 , and ν = 0.02 and (b) for different values of parallel component of anisotropy pressure P ‖ with μ p = 0.4 , κ = 3 , σ = 0.2 , and ν = 0.04 .

Figure 1(b) illustrated the variations of the real part of the oscillation frequency ω r as a function of the wavenumber k , for different values of the anisotropic pressure ( P ‖ ) . The oscillation frequency increases with both the wavenumber k and the anisotropic pressure ( P ‖ ) . The pressure anisotropy does not affect the oscillation frequency for large wavelengths, where the oscillation frequency remains nearly constant. For short wavelengths, the pressure anisotropy increases the oscillation frequency. The pressure anisotropy does not impact the imaginary part of the oscillations ω i m . Figure 2 illustrates the parameter σ and positron concentration μ p effects on the ion frequency ω r . It shows that the ion frequency decreases with increasing temperature ratio σ (panel (a)) and increasing positron concentration μ p (panel (b)), while it increases with the wavenumber k . Like the spectral parameter κ and anisotropic pressure ( P ‖ ) , σ and μ p have no effects on the imaginary frequency ω i m .

$Figure 2 Plot of the real part of wave frequency ω r {\omega }_{r} : (a) for different values of ratio temperature σ \sigma with μ p = 0.5 {\mu }_{p}=0.5 and (b) for different values of concentration of positron μ p {\mu }_{p} with σ = 0.5 \sigma =0.5 where κ = 4 , P ‖ = 0.06 \kappa =4,{P}_{\Vert }=0.06 , and ν = 0.02 \nu =0.02 .$

Figure 2

Plot of the real part of wave frequency ω r : (a) for different values of ratio temperature σ with μ p = 0.5 and (b) for different values of concentration of positron μ p with σ = 0.5 where κ = 4 , P ‖ = 0.06 , and ν = 0.02 .

Figure 3 demonstrates that the imaginary part of the oscillation frequency is negative and consists of two regimes: the transient regime ( 0 ≤ k ≤ 0.04 ) and the permanent regime ( k ≥ 0.04 ) . The transient regime corresponds to the decreasing variations of the imaginary frequency ( ω i m ) . This is because for some values of the wavenumber below the threshold wavenumber k s (Eq. (17)), the real part of the frequency becomes imaginary. This transient regime results from damping oscillations, i.e., the wavenumber is below the threshold wavenumber ( k < k s ) . The threshold wavenumber is k s = 0.04 . The permanent regime ( k ≥ 0.04 ) is constant and corresponds to ω i m = − ν ⁄ 2 . An increase in collision frequency leads to strong attenuation of oscillations in the plasma.

$Figure 3 Plot of the imaginary part of frequency versus wavenumber k k for different values of collision frequency ν \nu with parameters fixed: μ p = 0.4 , κ = 5 , σ = 0.5 {\mu }_{p}=0.4,\kappa =5,\sigma =0.5 , and P ‖ = 0.05 {P}_{\Vert }=0.05 .$

Figure 3

Plot of the imaginary part of frequency versus wavenumber k for different values of collision frequency ν with parameters fixed: μ p = 0.4 , κ = 5 , σ = 0.5 , and P ‖ = 0.05 .

We have demonstrated that the parallel component P ‖ of the anisotropic pressure increases the ion oscillation frequency without affecting the damping of these oscillations. Now, let us consider the influence of the perpendicular component P ⊥ on ion oscillations.

3.2 Perpendicular component

In the direction perpendicular to the uniform magnetic field, we assume the wavenumber in the parallel direction to be zero ( k ‖ = 0 ) and the wavenumber in the perpendicular direction to be k ⊥ = k . Thus, Eq. (13) reduces to:

(19) ω 3 + 2 j ν ω 2 − [ 1 k 2 + α 1 + P ⊥ k 2 + ( ω c i + 2 Ω 0 ) 2 + ν 2 ] ω − j 1 k 2 + α 1 + P ⊥ ν k 2 = 0 .

The complex solutions ( ω r ; ω i m ) of this Eq. (19) are obtained by the Cardan method and are given by

(20) ω r = 3 2 ( r + r ′ ) , ω i m = 1 2 ( r − r ′ ) − b 0 3 ,

where

(21) r = 1 2 q 0 + q 0 2 + 4 27 p 0 3 3 , r ′ = 1 2 − q 0 + q 0 2 + 4 27 p 0 3 3 , q 0 = 1 27 ( − 2 b 0 3 + 9 c 0 b 0 − 27 d 0 ) , p 0 = 1 3 ( 3 c 0 − b 0 2 ) , d 0 = 1 k 2 + α 1 + P ⊥ ν k 2 , c 0 = 1 k 2 + α 1 + P ⊥ k 2 + ( ω c i + 2 Ω 0 ) 2 + ν 2 , b 0 = 2 ν .

Assuming no collisions ( ν → 0 ) , low plasma rotation ( ω c i ≫ Ω 0 ) , and cold ions ( P ⊥ = 0 ) in a Maxwellian plasma with n e o = n p 0 and T e = T p , Eq. (19) becomes

(22) ω 2 − ω c i 2 + k 2 k 2 + ( 1 + σ ) μ p = 0 .

This Eq. (22) is similar to Eqs. ( 30 ) and ( 21 ) in previous studies [62,67], respectively. The difference lies in the normalization condition and the distribution function used.

The numerical study of the dispersion relation in the direction perpendicular to the magnetic field is presented in the following paragraph. As with the parallel component, we will investigate the impact of plasma parameters ( κ , μ p , σ , Ω 0 , P ⊥ , ν ) on ion oscillations. Figure 4 shows the variation of the frequency as a function of the wavenumber k . Figure 4(a) illustrates the variation of the real part of the frequency as a function of the wavenumber k for different values of the spectral parameter κ . The oscillation frequency increases with the wavenumber k . An increase in the spectral parameter leads to a rise in the oscillation frequency. For zero wavenumber, the frequency is non-zero ( ω = 0.63 ) , indicating the existence of electrostatic waves. Figure 4(b) indicates that an increase in positron concentration leads to a decrease in ion frequency, resulting in greater stability for these ions. Conversely, the frequency increases with the wavenumber k . Higher wavenumbers correspond to higher frequencies of ion oscillations. The positron concentration raises the frequency of electrostatic waves ( ω ≈ 0.84 ; k → 0 ) compared to the influence of the spectral parameter ( ω = 0.63 ; k → 0 ) . Figure 4(c) illustrates the decrease in the imaginary part of the ion frequency with the wavenumber k . We observe that an increase in positrons leads to a reduction in the damping of oscillations in the transient regime k < 2.3 , thereby diminishing the effect of energy dissipation in the plasma.

$Figure 4 The variation of wave frequency versus wavenumber k k . (a) Real part frequency ω r {\omega }_{r} for different values of the spectral parameter κ \kappa with μ p = 0.2 , σ = 0.8 , P ⊥ = 0.06 , Ω 0 = 0.05 , ω c i = 0.3 {\mu }_{p}=0.2,\sigma =0.8,{P}_{\perp }=0.06,{\Omega }_{0}=0.05,{\omega }_{ci}=0.3 and ν = 0.03 \nu =0.03 . (b) Real part frequency ω r {\omega }_{r} for different values of concentration positron μ p {\mu }_{p} and (c) imaginary part for different values of concentration positrons μ p {\mu }_{p} with σ = 0.5 , κ = 4 \sigma =0.5,\kappa =4 , P ⊥ = 0.4 , Ω 0 = 0.6 , ω c i = 0.3 {P}_{\perp }=0.4,{\Omega }_{0}=0.6,{\omega }_{ci}=0.3 , and ν = 0.08 \nu =0.08 .$

Figure 4

The variation of wave frequency versus wavenumber k . (a) Real part frequency ω r for different values of the spectral parameter κ with μ p = 0.2 , σ = 0.8 , P ⊥ = 0.06 , Ω 0 = 0.05 , ω c i = 0.3 and ν = 0.03 . (b) Real part frequency ω r for different values of concentration positron μ p and (c) imaginary part for different values of concentration positrons μ p with σ = 0.5 , κ = 4 , P ⊥ = 0.4 , Ω 0 = 0.6 , ω c i = 0.3 , and ν = 0.08 .

For k > 2.3 , the frequency ω i m becomes constant (permanent regime). The frequency ω i m remains negative, indicating the damping of ion oscillations in this plasma model. Just as with positron concentration, a decrease in positron temperature ( σ ) reduces the oscillation frequency ω r . Notably, the frequency ω r increases when k = 0 ( ω r ≈ 1.22 when k = 0 ) compared to the parameters μ p and κ (curve not shown).

In Figure 5, we investigate the effects of plasma rotation on the real and imaginary components of the dispersion relation. Figure 5(a) shows that the frequency of electrostatic wave oscillations and the wavenumber k increase with an increase in the plasma rotation frequency. For infinite wavenumbers, the oscillation frequency becomes linear and is no longer affected by plasma rotation. Therefore, the plasma rotation frequency does not influence wave oscillations at short wavelengths. Figure 5(b) shows that the imaginary part of the frequency is always negative due to the effects of oscillation damping. The frequency ω i m decreases as a function of the wavenumber k and passes through a critical value k c (which varies depending on the values of Ω 0 ), then changes concavity and becomes constant beyond a particular value of k . Additionally, as the plasma rotation frequency increases, the frequency ω i m decreases, indicating that the oscillations become less damped. The permanent regime is characterized by the wavenumber values where the frequency ω i m remains constant and negative.

$Figure 5 The variation of wave frequency versus wavenumber k k . (a) Real part frequency ω r {\omega }_{r} and (b) imaginary part for different values of plasma rotating frequency Ω 0 {\Omega }_{0} with parameters fixed σ = 0.5 , κ = 4 , \sigma =0.5,\kappa =4, P ⊥ = 0.4 , μ p = 0.6 , ω c i = 0.3 {P}_{\perp }=0.4,{\mu }_{p}=0.6,{\omega }_{ci}=0.3 and ν = 0.08 \nu =0.08 .$

Figure 5

The variation of wave frequency versus wavenumber k . (a) Real part frequency ω r and (b) imaginary part for different values of plasma rotating frequency Ω 0 with parameters fixed σ = 0.5 , κ = 4 , P ⊥ = 0.4 , μ p = 0.6 , ω c i = 0.3 and ν = 0.08 .

The collision frequency does not affect the real part of the frequency (curve not shown). The variation of the imaginary part of the wave frequency as a function of k is always negative. It presents two zones: a decreasing part (transient zone) and a part where the frequency remains constant (permanent regime), as shown in Figure 6. However, the collision frequency strongly affects the imaginary component ω i m . The higher the collision frequency, the more ω i m decreases, making the wave oscillations more damped.

$Figure 6 Plot of the imaginary part of the frequency as a function of k k for different values of ν \nu with σ = 0.8 , κ = 2 , P ⊥ = 0.006 , μ p = 0.2 , \sigma =0.8,\kappa =2,{P}_{\perp }=0.006,{\mu }_{p}=0.2, ω c i = 0.3 {\omega }_{ci}=0.3 , and Ω 0 = 0.05 {\Omega }_{0}=0.05 .$

Figure 6

Plot of the imaginary part of the frequency as a function of k for different values of ν with σ = 0.8 , κ = 2 , P ⊥ = 0.006 , μ p = 0.2 , ω c i = 0.3 , and Ω 0 = 0.05 .

In Figure 7, we have plotted the real and imaginary components of ω as a function of the wavenumber k for four different values of the perpendicular component of the anisotropy pressure. Figure 7(a) shows that for short wavelengths, the growth rate of the ion acoustic frequency ω r increases with the wavenumber k and with increasing values of the anisotropy pressure P ⊥ . The increase in ion pressure P ⊥ leads to instability of IAWs. Pressure anisotropy does not affect waves at low wave numbers, ensuring stability. It should be noted that in the absence of anisotropy pressure ( P ⊥ = 0 ) , the frequency of ion waves tends to a constant limit value. Figure 7(b) shows that ω i m decreases as a function of k up to a critical value of the wavenumber ( k < k c ) , then changes concavity ( k > k c ) and becomes constant at higher wavenumbers. For k < k c , the increase in ion pressure anisotropy decreases the imaginary component of the frequency, causing the oscillations to become more damped. Conversely, for k > k c , the anisotropy increases the frequency ω i m , reducing the damping effect.

$Figure 7 Variation of the real part ω r {\omega }_{r} (a) and the imaginary part ω i m {\omega }_{im} (b) of the wave frequency against the wave number k k for different values of P ⊥ {P}_{\perp } . Along with σ = 0.8 , κ = 2 , ν = 0.04 , μ p = 0.2 , ω c i = 0.3 \sigma =0.8,\kappa =2,\nu =0.04,{\mu }_{p}=0.2,{\omega }_{ci}=0.3 , and Ω 0 = 0.05 {\Omega }_{0}=0.05 .$

Figure 7

Variation of the real part ω r (a) and the imaginary part ω i m (b) of the wave frequency against the wave number k for different values of P ⊥ . Along with σ = 0.8 , κ = 2 , ν = 0.04 , μ p = 0.2 , ω c i = 0.3 , and Ω 0 = 0.05 .

4 Nonlinear structure

In this section, we transform Eqs (7)–(11) into a single nonlinear equation that encapsulates the dynamics of IASWs. This transformation will be accomplished through the RPM, which effectively examines the nonlinear properties of finite-amplitude waves. We can streamline the system’s complexity by utilizing the concept of stretched coordinates. The independent variables are defined as follows:

(23) X = ε 1 ⁄ 2 x , Y = ε 1 ⁄ 2 y , Z = ε 1 ⁄ 2 ( z − v p t ) , T = ε 3 ⁄ 2 t ,

with 0 < ε ≪ 1 as the parameter indicating the weakness of nonlinearity, and v p representing the phase velocity of the wave. We will expand the perturbed quantities in terms of a power series in ε around their equilibrium values as follows:

(24) n i = 1 + ε n i ( 1 ) + ε 2 n i ( 2 ) + ε 3 n i ( 3 ) + ⋯ , u x , y = ε 3 ⁄ 2 u x , y ( 1 ) + ε 3 u x , y ( 2 ) + ε 5 ⁄ 2 u x , y ( 3 ) + ⋯ , u z = ε u z ( 1 ) + ε 2 u z ( 2 ) + ε 3 u z ( 3 ) + ⋯ , Φ = ε Φ ( 1 ) + ε 2 Φ ( 2 ) + ε 3 Φ ( 3 ) + ⋯ , ν n = ε 3 ⁄ 2 ν 0 .

Considering Eqs (23) and (24) in Eqs (7)–(11), we derive at first order in ε

(25) u x ( 1 ) = − 1 + P ⊥ α 1 ω c i + 2 Ω 0 ∂ Y Φ ( 1 ) , u y ( 1 ) = 1 + P ⊥ α 1 ω c i + 2 Ω 0 ∂ X Φ ( 1 ) , u z ( 1 ) = v p n i ( 1 ) = 1 v p Φ ( 1 ) + P ‖ v p n i ( 1 ) , n i ( 1 ) = α 1 Φ ( 1 ) .

By combining the components u z ( 1 ) and n i ( 1 ) , we derive the phase velocity v p of IAW propagation

(26) v p 2 = 1 α 1 + P ‖ .

Notably, the phase velocity v p is independent of the perpendicular pressure component P ⊥ . Instead, this speed is influenced by the spectral parameter κ , the positron concentration μ p through α 1 , and the parallel pressure component P ‖ . The phase velocity v p increases with both the parallel pressure component and the spectral index κ , as illustrated in Figure 8(a). Conversely, this speed decreases with increasing ratio temperature σ , as shown in Figure 8(b). It is also noteworthy that for κ = 3 ⁄ 2 , the phase velocity is not defined. As κ → ∞ , the phase velocity reaches a steady state, rendering the waves stable. Therefore, IAWs achieve stability exclusively in Maxwellian plasmas. Proceeding with the development to second order in ε 1 ⁄ 2 , we obtain

(27) u x ( 2 ) = v p 1 + P ⊥ α 1 ( ω c i + 2 Ω 0 ) 2 ∂ Z X 2 Φ ( 1 ) , u y ( 2 ) = v p 1 + P ⊥ α 1 ( ω c i + 2 Ω 0 ) 2 ∂ Z Y 2 Φ ( 1 ) ,

as well as at second order in ε of the Poisson equation, we have

(28) ( ∂ X 2 + ∂ Y 2 + ∂ Z 2 ) Φ ( 1 ) + n i ( 2 ) − α 1 Φ ( 2 ) − α 2 Φ ( 1 ) 2 = 0 .

$Figure 8 Plot phase speed v p {v}_{p} versus the superthermality index κ \kappa for different values: (a) of anisotropy pressure parallel component P ‖ {P}_{\Vert } where σ = 0.8 \sigma =0.8 and (b) of concentration positrons σ \sigma where P ‖ = 0.5 {P}_{\Vert }=0.5 with parameter fixed μ p = 0.4 {\mu }_{p}=0.4 .$

Figure 8

Plot phase speed v p versus the superthermality index κ for different values: (a) of anisotropy pressure parallel component P ‖ where σ = 0.8 and (b) of concentration positrons σ where P ‖ = 0.5 with parameter fixed μ p = 0.4 .

At third order in the ε 1 ⁄ 2 expansion for the continuity equations and the z component of the momentum equation, incorporating Eqs (25)–(27) and combining them with the derivative with respect to Z of Eq. (28), we derive the following nonlinear differential equation:

(29) ∂ T Φ ( 1 ) + A Φ ( 1 ) ∂ Z Φ ( 1 ) + B ∂ Z ( ∂ X 2 + ∂ Y 2 ) Φ ( 1 ) + C ∂ Z 3 Φ ( 1 ) + ν Φ ( 1 ) = 0 ,

with

(30) A = C [ α 1 2 ( 3 + 4 α 1 P ‖ ) − 2 α 2 ] , B = C 1 + ( 1 + α 1 P ‖ ) ( 1 + α 1 P ⊥ ) ( ω c i + 2 Ω 0 ) 2 , C = 1 2 α 1 3 ( 1 + α 1 P ‖ ) , ν = ν 0 2 .

Eq. (29) is recognized as the dZK nonlinear equation, which incorporates the nonlinear coefficient A , dispersion coefficients B , C , and dissipation ν , all defined in Eq. (30). Our results align with previous works regarding the limiting cases discussed. For example, in the absence of collisions ( ν = 0 ) with neutrals, positrons μ p = 0 , and the Coriolis force Ω 0 = 0 , Eq. (30) matches Eq. ( 33 ) from the study of Adnan et al. [28]. In the case of a non-collisional e–p–i plasma with low rotation ( ω c i ≫ Ω 0 ) within a uniform magnetic field, where propagation takes place in the ( y ; z ) plane, Eqs (29) and (30) are equivalent to equations ( 39 ) and ( 40 ) found in the study of Adnan et al. [30].

4.1 Solitary pulse solution

To model the existence of electrostatic waves qualitatively, we typically consider a solitary wave solution of the dZK Eq. (29), analogous to the KdV equation. However, the dZK Eq. (29) is a non-integrable system due to collisions between ions and neutrals ( ∂ T E ≠ 0 ) . We will simplify this by adopting an approximate solution of Eq. (29), neglecting collisions, and introducing the independent variable:

(31) ζ = l x X + l y Y + l z Z − u 0 T ,

where l x , l y , and l z are the direction cosines of the wave vector k → such that l x 2 + l y 2 + l z 2 = 1 , u 0 the constant propagation velocity of the soliton. By neglecting collisions, we obtain the standard ZK equation given by [68]

(32) ∂ T Φ ( 1 ) + A Φ ( 1 ) ∂ Z Φ ( 1 ) + B ∂ Z ( ∂ X 2 + ∂ Y 2 ) Φ ( 1 ) + C ∂ Z 3 Φ ( 1 ) = 0 .

By introducing the variable defined in (31) into Eq. (32), we obtain the following energy density:

(33) 1 2 d Φ ( 1 ) d ζ 2 + V ( Φ ( 1 ) ) = 0 ,

where V ( Φ ( 1 ) ) is the pseudo-potential defined by

(34) V ( Φ ( 1 ) ) = − 1 2 u 0 l z [ l z 2 C + B ( l x 2 + l y 2 ) ] × Φ ( 1 ) 2 1 − A 3 u 0 Φ ( 1 ) .

Figure 9 shows the impact of P ⊥ , P ‖ , and Ω 0 on the pseudo-potential V . This potential exhibits a compressive solitary wave structure with a single well. It is shown that as P ‖ and P ⊥ grow, the potential depth decreases (Figure 9(a) and (b)), leading to a reduction in both the amplitude and width of the compressive solitary waves. In contrast, Figure 9(c) demonstrate the opposite behavior where it is observed that the amplitude and width of the corresponding compressive soliton become greater as Ω 0 enhances. The integration of Eq. (33) provides the soliton solution described by

(35) Φ ( 1 ) ( ζ ) = Φ 0 ( 1 ) sech 2 ( ζ ⁄ L ) ,

where Φ 0 ( 1 ) denotes the amplitude and L represents the width of the soliton, defined by

(36) Φ 0 ( 1 ) = 3 u 0 A l z , L = 4 l z [ C l z 2 + B ( l x 2 + l y 2 ) ] u 0 .

The approximate solution of Eq. (29) is derived by applying the conservation of momentum [62,69,70], resulting in

(37) E = ∫ − ∞ + ∞ Φ ˜ ( 1 ) 2 d ζ ˜ and ∂ T E = − 2 ν E .

From Eq. (35), we derive the approximate solution of the dZK equation in the following form

(38) Φ ˜ ( 1 ) ( ζ ˜ ) = Φ ˜ 0 ( 1 ) sech 2 ( ζ ˜ ⁄ L ˜ ) .

Here, Φ ˜ 0 ( 1 ) = Φ 0 ( 1 ) ( T ) and L ˜ = L ( T ) represent the amplitude and width of the soliton, respectively, both dependent on time. By substituting Eq. (38) into Eq. (37), and applying the condition ν = 0 , we determine the amplitude and width of the soliton as defined by

(39) Φ ˜ 0 ( 1 ) = 3 u ˜ l z A , L ˜ = 4 l z ( B ( l x 2 + l y 2 ) + C l z 2 ) u ˜ ,

with u ˜ = u 0 e − ν T and ζ ˜ = l x X + l y Y + l z Z − u ˜ T . Thus, the approximate solution of the dZK equation is:

(40) Φ ˜ ( 1 ) = 3 u 0 l z A e − ν T sech 2 u 0 4 l z ( B ( l x 2 + l y 2 ) + C l z 2 ) e − ν 2 T ζ ˜ ⋅

$Figure 9 Variation of pseudo-potential V V against Φ ( 1 ) {\Phi }^{\left(1)} for different values of parallel pressure component P ‖ {P}_{\Vert } (a) with Ω 0 = 0.1 {\Omega }_{0}=0.1 and P ⊥ = 0.1 {P}_{\perp }=0.1 , perpendicular pressure component P ⊥ {P}_{\perp } (b) with Ω 0 = 0.1 {\Omega }_{0}=0.1 and P ‖ = 0.06 {P}_{\Vert }=0.06 , Coriolis force parameter (c) with P ⊥ = 0.1 {P}_{\perp }=0.1 and P ‖ = 0.03 {P}_{\Vert }=0.03 . Parameters fixed: κ = 3 , μ p = 0.6 , σ = 0.2 \kappa =3,{\mu }_{p}=0.6,\sigma =0.2 , and ω c i = 0.4 {\omega }_{ci}=0.4 .$

Figure 9

Variation of pseudo-potential V against Φ ( 1 ) for different values of parallel pressure component P ‖ (a) with Ω 0 = 0.1 and P ⊥ = 0.1 , perpendicular pressure component P ⊥ (b) with Ω 0 = 0.1 and P ‖ = 0.06 , Coriolis force parameter (c) with P ⊥ = 0.1 and P ‖ = 0.03 . Parameters fixed: κ = 3 , μ p = 0.6 , σ = 0.2 , and ω c i = 0.4 .

4.2 Numerical analysis

In this section, we will numerically analyze the effects of plasma parameters on the occurrence and propagation of solitary waves in the plasma, specifically focusing on P ⊥ , P ‖ , Ω 0 , and μ p . Figure 10 presents the influence of superthermality on the coefficients A , B , and C for varying values of the parallel pressure component. We observe that for small values of the spectral parameter κ , A decreases while P ‖ increases, B rises with κ and P ‖ , and C increases with κ but decreases with P ‖ . Superthermality diminishes the nonlinear effects, whereas anisotropy in pressure enhances the nonlinear effect. The medium exhibits increased dispersion with higher levels of superthermality. Like the parallel pressure component, the positron concentration exerts the same influence on nonlinearity ( coefficient A ) as P ‖ (curve is not shown). The coefficients A and C are independent of the parameters P ⊥ and Ω 0 , and the magnetic field, as indicated in Eq. (30). Figure 11 depicts the effect of the parallel ( P ‖ ) and perpendicular ( P ⊥ ) pressure components on the width of solitary waves propagating in the plasma. Figure 11(a) and (b) reveals that increasing anisotropy pressure leads to an increase in soliton width. This width also rises with the spectral parameter. For high values of κ ( κ → + ∞ ) , the soliton width tends to stabilize at a constant value. Anisotropy pressure thus increases the width of IASWs in this plasma model.

$Figure 10 Variation of coefficient (a) A , (b) B A,(b)B , and (c) C C against κ \kappa for different values of parallel component pressure P ‖ {P}_{\Vert } . Parameters fixed: ω c i = 0.4 , Ω 0 = 0.3 , σ = 1 {\omega }_{ci}=0.4,{\Omega }_{0}=0.3,\sigma =1 , and μ p = 0.7 {\mu }_{p}=0.7 .$

Figure 10

Variation of coefficient (a) A , (b) B , and (c) C against κ for different values of parallel component pressure P ‖ . Parameters fixed: ω c i = 0.4 , Ω 0 = 0.3 , σ = 1 , and μ p = 0.7 .

$Figure 11 Variation of width L L versus κ \kappa for different values of P ‖ {P}_{\Vert } with P ⊥ = 0.8 {P}_{\perp }=0.8 (a) and P ⊥ {P}_{\perp } with P ‖ = 0.01 {P}_{\Vert }=0.01 (b). Parameters fixed: μ p = 0.2 {\mu }_{p}=0.2 , σ = 0.2 \sigma =0.2 , ω c i = 0.4 {\omega }_{ci}=0.4 , Ω 0 = 0.05 {\Omega }_{0}=0.05 , u 0 = 0.3 {u}_{0}=0.3 , and ν = 0 \nu =0 .$

Figure 11

Variation of width L versus κ for different values of P ‖ with P ⊥ = 0.8 (a) and P ⊥ with P ‖ = 0.01 (b). Parameters fixed: μ p = 0.2 , σ = 0.2 , ω c i = 0.4 , Ω 0 = 0.05 , u 0 = 0.3 , and ν = 0 .

We recall that, unless otherwise stated, we have taken l x = 0.3 , l y = 0.5 , and l z = 1 − l x 2 − l y 2 . Figure 12 illustrates that in the absence of collisions ( ν = 0 ) , the amplitude of solitary waves decreases as the parallel pressure component P ‖ and the electron–positron temperature ratio σ increase. Conversely, the amplitude increases with superthermality, eventually stabilizing as κ → + ∞ . Consequently, Maxwellian plasmas solitons exhibit higher amplitudes than those in superthermal plasmas. It is important to note that the coefficient A in Eq. (30) is unaffected by the perpendicular pressure component’s soliton amplitude.

$Figure 12 Variation of amplitude Φ 0 ( 1 ) {\Phi }_{0}^{\left(1)} versus κ \kappa for different values of P ‖ {P}_{\Vert } with σ = 0.2 \sigma =0.2 , μ p = 0.2 {\mu }_{p}=0.2 (a) and σ \sigma with P ‖ = 0.6 {P}_{\Vert }=0.6 , μ p = 0.2 {\mu }_{p}=0.2 (b). Parameters fixed: u 0 = 0.3 {u}_{0}=0.3 and ν = 0 \nu =0 .$

Figure 12

Variation of amplitude Φ 0 ( 1 ) versus κ for different values of P ‖ with σ = 0.2 , μ p = 0.2 (a) and σ with P ‖ = 0.6 , μ p = 0.2 (b). Parameters fixed: u 0 = 0.3 and ν = 0 .

In the presence of collisions, the amplitude of the waves diminishes exponentially over time, as illustrated in Figure 13. The wave energy dissipates, leading the wave to become evanescent. This amplitude decreases with an increase in the parallel pressure component and increases with superthermality κ . In the absence of collisions ( ν = 0 ) , we will now present the numerical study of the behavior of the parameters P ⊥ , P ‖ , κ , and Ω 0 on the electrostatic potential that characterizes the propagation of electrostatic waves in the plasma. Figure 14 illustrates the variations of the electrostatic potential Φ ( 1 ) as a function of the independent variable ζ . Figure 14(a) reveals that for various values of superthermality κ , the amplitude and width of the soliton increase with rising values of the spectral parameter κ .

$Figure 13 Variation of amplitude Φ ˜ 0 ( 1 ) {\tilde{\Phi }}_{0}^{\left(1)} versus T T for different values of superthermality κ \kappa with P ‖ = 0.6 {P}_{\Vert }=0.6 (a) and parallel component pressure P ‖ {P}_{\Vert } with κ = 4 \kappa =4 (b). Parameters fixed: μ p = 0.6 {\mu }_{p}=0.6 , σ = 0.2 \sigma =0.2 , u 0 = 0.3 {u}_{0}=0.3 and ν = 0.4 \nu =0.4 .$

Figure 13

Variation of amplitude Φ ˜ 0 ( 1 ) versus T for different values of superthermality κ with P ‖ = 0.6 (a) and parallel component pressure P ‖ with κ = 4 (b). Parameters fixed: μ p = 0.6 , σ = 0.2 , u 0 = 0.3 and ν = 0.4 .

$Figure 14 Plot electrostatic potential Φ ( 1 ) {\Phi }^{\left(1)} against space variable ζ \zeta for different values of superthermality κ \kappa with P ‖ = 0.3 , σ = 0.5 {P}_{\Vert }=0.3,\sigma =0.5 , μ p = 0.6 {\mu }_{p}=0.6 (a) and parallel component pressure with κ = 5 \kappa =5 , σ = 0.8 \sigma =0.8 , μ p = 0.4 {\mu }_{p}=0.4 (b) with other parameters fixed at: Ω 0 = 0.1 {\Omega }_{0}=0.1 , P ⊥ = 0.1 {P}_{\perp }=0.1 , u 0 = 0.3 {u}_{0}=0.3 , and ω c i = 0.4 {\omega }_{ci}=0.4 .$

Figure 14

Plot electrostatic potential Φ ( 1 ) against space variable ζ for different values of superthermality κ with P ‖ = 0.3 , σ = 0.5 , μ p = 0.6 (a) and parallel component pressure with κ = 5 , σ = 0.8 , μ p = 0.4 (b) with other parameters fixed at: Ω 0 = 0.1 , P ⊥ = 0.1 , u 0 = 0.3 , and ω c i = 0.4 .

Superthermality decreases nonlinearity by enhancing the dispersion effect of the plasma. As κ → ∞ , the amplitude tends to a constant value. Figure 14(b) illustrates that for various values of the parallel component P ‖ , the amplitude of IASWs decreases while the soliton width increases. Consequently, the parallel component of anisotropy pressure reduces wave amplitude and increases the width of these waves, making the plasma more nonlinear and dispersive. Therefore, increasing anisotropy pressure tends to lower the system’s energy and stabilize the waves. Additionally, anisotropy pressure elevates the amplitude of solitary waves compared to superthermal plasma. Figure 15 illustrates the effect of the perpendicular pressure component P ⊥ on the propagation of solitary waves. It shows that while the amplitude of electrostatic waves remains unchanged, the width of the waves increases. Thus, an increase in anisotropy pressure in the perpendicular component enhances the dispersiveness of the plasma medium while maintaining the nonlinear effect. Consequently, the waves propagate perpendicularly to the magnetic field with constant energy.

$Figure 15 Plot electrostatic potential Φ ( 1 ) {\Phi }^{\left(1)} against space variable ζ \zeta for different values of perpendicular component pressure P ⊥ {P}_{\perp } with other parameters fixed at: P ‖ = 0.6 {P}_{\Vert }=0.6 , κ = 5 \kappa =5 , σ = 0.8 \sigma =0.8 , μ p = 0.4 {\mu }_{p}=0.4 , Ω 0 = 0.1 {\Omega }_{0}=0.1 , u 0 = 0.3 {u}_{0}=0.3 , and ω c i = 0.4 {\omega }_{ci}=0.4 .$

Figure 15

Plot electrostatic potential Φ ( 1 ) against space variable ζ for different values of perpendicular component pressure P ⊥ with other parameters fixed at: P ‖ = 0.6 , κ = 5 , σ = 0.8 , μ p = 0.4 , Ω 0 = 0.1 , u 0 = 0.3 , and ω c i = 0.4 .

When considering the low rotation of the plasma with a frequency Ω 0 around the magnetic axis, known as the Coriolis effect, it is evident that as the frequency Ω 0 increases, the electrostatic waves maintain their amplitude while the soliton width decreases, resulting in a pointed, spiky soliton profile. This is depicted in Figure 16. This implies that the rotation of the plasma and its coupling effect with the magnetic field through ω c i impact the dispersive properties of the wave profile. This finding aligns with the results of Farooq and Mushtaq [62]. When the parallel component of the magnetic field aligns with the rotation, leading particles to follow the field lines (strong field), the magnetic effect is amplified.

$Figure 16 Plot electrostatic potential Φ ( 1 ) {\Phi }^{\left(1)} against space variable ζ \zeta for different values of low frequency Ω 0 {\Omega }_{0} with other parameters fixed at: P ‖ = 0.6 {P}_{\Vert }=0.6 , κ = 5 \kappa =5 , σ = 0.8 \sigma =0.8 , μ p = 0.4 {\mu }_{p}=0.4 , P ⊥ = 0.1 {P}_{\perp }=0.1 , u 0 = 0.3 {u}_{0}=0.3 , and ω c i = 0.4 {\omega }_{ci}=0.4 .$

Figure 16

Plot electrostatic potential Φ ( 1 ) against space variable ζ for different values of low frequency Ω 0 with other parameters fixed at: P ‖ = 0.6 , κ = 5 , σ = 0.8 , μ p = 0.4 , P ⊥ = 0.1 , u 0 = 0.3 , and ω c i = 0.4 .

In a non-conservative plasma, dissipative solitons can form due to the balance between loss and gain terms, alongside the balance between dispersion and nonlinear effects. Additionally, the interaction between dispersion and dissipation effects can give rise to shock waves. In Figure 17, we illustrate the evolution of IASWs in the plasma by numerically plotting the solution of the ZK equation given in Eq. (40). Figure 17(a) demonstrates that in a non-collisional magnetized plasma ( ν = 0 ) , the electrostatic waves maintain stability over time, with the amplitude and width of the soliton remaining constant as they propagate. Conversely, Figure 17(b) explores the solitary wave solution of Eq. (29) propagating in a collisional magnetized plasma ( ν = 0.04 ) in the absence of anisotropy pressure ( P ‖ = P ⊥ = 0 ) to highlight the effect of dissipation due to collisions. In this scenario, we observe that the amplitude of IASWs decreases while the width increases over time, all while conserving their fundamental properties. Figure 17(c) shows the combined effects of collisions and anisotropy pressure. We observe that coupling anisotropy pressure with collisions decreases the soliton amplitude by about twice as much as when considering the impact of collisions alone. The wave attenuates rapidly over time.

$Figure 17 3D plot of electrostatic potential Φ ˜ ( 1 ) {\tilde{\Phi }}^{\left(1)} versus space variable ζ ˜ \tilde{\zeta } and time variable T T with (a) ν = 0 \nu =0 , P ⊥ = 0.1 {P}_{\perp }=0.1 and P ‖ = 0.6 {P}_{\Vert }=0.6 , (b) ν = 0.04 \nu =0.04 and P ⊥ = P ‖ = 0 {P}_{\perp }={P}_{\Vert }=0 and (c) ν = 0.04 \nu =0.04 , P ⊥ = 0.1 {P}_{\perp }=0.1 and P ‖ = 0.6 {P}_{\Vert }=0.6 . Parameters fixed: κ = 4 \kappa =4 , μ p = 0.4 {\mu }_{p}=0.4 , σ = 0.2 \sigma =0.2 , Ω = 0.3 \Omega =0.3 , ω c i = 0.4 {\omega }_{ci}=0.4 , and u 0 = 0.3 {u}_{0}=0.3 .$

Figure 17

3D plot of electrostatic potential Φ ˜ ( 1 ) versus space variable ζ ˜ and time variable T with (a) ν = 0 , P ⊥ = 0.1 and P ‖ = 0.6 , (b) ν = 0.04 and P ⊥ = P ‖ = 0 and (c) ν = 0.04 , P ⊥ = 0.1 and P ‖ = 0.6 . Parameters fixed: κ = 4 , μ p = 0.4 , σ = 0.2 , Ω = 0.3 , ω c i = 0.4 , and u 0 = 0.3 .

An increase in collision frequency results in the formation of a shorter and wider dissipative soliton in the magnetized e–p–i plasma. The soliton amplitude significantly decreases with rising collision frequency while the soliton width sharply increases, as shown in Figure 18. Based on the numerical results in Figure 18, it is observed that at higher collision frequencies compared to dispersion, the formation of a shock wave in the magnetized e–p–i plasma can be expected.

$Figure 18 3D plot of electrostatic potential Φ ˜ ( 1 ) {\tilde{\Phi }}^{\left(1)} versus space variable ζ ˜ \tilde{\zeta } and collisional frequency ν \nu . Parameters fixed: κ = 4 \kappa =4 , μ p = 0.4 {\mu }_{p}=0.4 , σ = 0.2 \sigma =0.2 , Ω = 0.3 \Omega =0.3 , ω c i = 0.4 {\omega }_{ci}=0.4 , P ⊥ = 0.1 {P}_{\perp }=0.1 , P ‖ = 0.6 {P}_{\Vert }=0.6 , T = 2.2 T=2.2 , and u 0 = 0.5 {u}_{0}=0.5 .$

Figure 18

3D plot of electrostatic potential Φ ˜ ( 1 ) versus space variable ζ ˜ and collisional frequency ν . Parameters fixed: κ = 4 , μ p = 0.4 , σ = 0.2 , Ω = 0.3 , ω c i = 0.4 , P ⊥ = 0.1 , P ‖ = 0.6 , T = 2.2 , and u 0 = 0.5 .

5 Conclusion

In this study, we conducted analytical and numerical investigations into the linear and nonlinear propagation of electrostatic excitations within a collisional, magneto-rotating e–p–i plasma. This plasma is notable for its anisotropic ion pressure, ion collisions with neutrals, and the presence of superthermal electrons and positrons, characterized by a κ -distribution function. The anisotropic nature of the plasma is described by an asymmetric ion pressure tensor and is modeled using the CGL theory [22]. We have derived the dispersion equation, breaking it into components parallel and perpendicular to the magnetic field. The solution of this equation is expressed as a complex function, ( ω = ω r + j ω i m ) . The oscillation of IAWs characterizes the parallel component, while the oscillation of electrostatic cyclotron waves characterizes the perpendicular component. The linear regime can be summarized as follows:

In the parallel component of the magnetic field in the dispersion relation:
1. We have demonstrated that there is a threshold wavenumber k s , below which the real part of the frequency ω r becomes imaginary, ( ω r 2 < 0 ) . This condition implies k s > k and ω r = 0 when k = 0 , indicating the existence of IAWs.
2. The growth rate of the real frequency ω r for ions increases with both superthermality κ and the wavenumber k . Consequently, ions exhibit instability at shorter wavelengths.
3. The anisotropic pressure P ‖ affects ion oscillations in this plasma model by increasing their frequency. A rise in P ‖ leads to the instability of ion waves within the plasma.
4. The real part ω r of the ion frequency increases with the wavenumber k and decreases with both the electron–positron temperature ratio σ and the positron concentration μ p . Consequently, an increase in positron concentration μ p or a decrease in the positron temperature σ leads to a reduction in the frequency of ion oscillations, thereby stabilizing them.
5. The imaginary part ω i m of the frequency is negative and decreases with increasing wavenumber k , indicating damped ion oscillations and the propagation of evanescent waves. The damping of oscillations is unaffected by σ , μ p , and P ‖ . However, the damping increases with rising collision frequency ν , as illustrated in Figure 3.
6. We can predict that in the parallel component to the magnetic field, the waves propagate at the speed v p = lim k → 0 ω r k = 1 α 1 + P ‖ 1 ⁄ 2 .
7. The collision frequency ν does not affect the wave propagation speed in the plasma.
In the perpendicular component of the magnetic field of the dispersion relation
1. For low collision frequencies ν ≪ 1 , the frequency of ion oscillations is not zero when k = 0 . In this case, the frequency is approximately ω r ≃ ω c i + 2 Ω 0 , which characterizes the presence of electrostatic waves.
2. The growth rate of the frequency ω r for ion oscillations on the phonon branch increases with the superthermality κ of the plasma and with the wavenumber. At infinite wavenumbers, ions become highly agitated and unstable due to their high frequency.
3. The frequency ω r of oscillations increases with the wavenumber k and decreases with the positron concentration μ p and the electron–positron temperature ratio σ . An increase in μ p or σ stabilizes the ions. On the contrary, an increase in the plasma rotation frequency Ω 0 leads to a rise in the perturbation frequency of the ions. As shown in Figure 5(a), for short wavelengths ( k > 1 ) , the dispersion becomes linear, and the Coriolis effect diminishes due to the anisotropic nature of the plasma.
4. The imaginary part ω i m of the perturbation frequency decreases as a function of the wavenumber k and remains consistently negative. It exhibits a transient regime and a steady-state regime. Numerical analysis indicates that increasing the positron concentration μ p (or the electron–positron temperature ratio σ ), as well as the plasma rotation frequency Ω 0 , causes the growth rate ω i m of the frequency to increase and approach zero, as shown in Figures 4(b) and 5(b). Additionally, the increase in the effect of the Coriolis force leads to a reduction in damped oscillations, thereby promoting wave propagation.
5. Increasing the collision frequency ν > 0 between ions and neutrals leads to a reduction in the growth rate ω i m ( < 0 ) of the frequency, which causes the damping of waves in the plasma, preventing propagation.
6. The oscillation frequency ω r increases with increasing anisotropic pressure P ⊥ . This pressure anisotropy only affects waves at short wavelengths, where ions are highly excited and their frequency is proportional to the wavenumber ( ω r ∝ k ) . Conversely, in the absence of pressure anisotropy ( P ⊥ = 0 ) and as k → + ∞ , ω r = 1 , indicating that ions reach a permanent oscillation regime with a constant frequency. In the dimensional case ( ω r → ω ′ ⁄ ω p i ) , this constant frequency is the plasma frequency ( ω ′ = ω p i ( = 4 π Z i n i 0 e 2 ⁄ m i ) ) . Note that an increase in the perpendicular pressure component P ⊥ leads to the damping of waves in the medium.

The dZK equation, which models the evolution of IASWs, is derived using the RPM. The nonlinear and dispersion coefficients depend on various factors, including the κ index of superthermality, the parallel P ‖ and perpendicular P ⊥ pressure components, the plasma rotation frequency Ω 0 , the positron concentration μ p , and the electron positron temperature ratio σ . The damping coefficient, however, remains constant and is solely dependent on the collision frequency ν . The summary of the nonlinear regime is as follows:

IAWs propagate in the plasma in a direction parallel to the magnetic field with a phase velocity that is unaffected by the plasma rotation frequency Ω 0 , the magnetic field through ω c i , or the perpendicular pressure component P ⊥ . The phase velocity increases with the κ index and parallel pressure P ‖ and decreases with the temperature ratio σ .
The nonlinear effect increases with the parallel pressure component P ‖ and decreases with higher values of the κ index of superthermality of electrons and positrons. In the parallel direction to the magnetic field, the dispersion effect decreases with the parallel pressure component P ‖ and increases with the perpendicular pressure component P ⊥ in the perpendicular direction to the magnetic field.
Ion pressure anisotropy has a significant impact on the amplitude and width of IASWs. The wave amplitude decreases with increasing ion pressure ( P ‖ ) and the electron–positron temperature ratio ( σ ) . On the contrary, the amplitude increases with rising superthermality κ . As for soliton width, it increases with higher anisotropic pressure components ( P ⊥ and P ‖ ) . The higher the ion pressure, the more the solitary waves spread.
The Coriolis force, expressed through Ω 0 , exclusively affects the width of electrostatic waves. With increased plasma rotation, the width of the waves becomes narrower.
Collisions between ions and neutrals greatly affect the propagation properties of solitary waves in the plasma. Our analysis reveals that as the collision frequency ν increases, the medium becomes more strongly damped, resulting in an exponential decrease in the amplitude of solitary waves. Over time, the wave spreads while conserving its properties. When anisotropic pressure is combined with collisions, the soliton amplitude is reduced by approximately twice as much compared to the effect of collisions alone.
An increase in collision frequency results in a shorter and broader dissipative soliton in the magnetized e–p–i plasma.

In summary, our work offers a qualitative description of solitary wave observations in rotating space plasmas and astrophysical environments with high magnetic fields, such as the magnetosphere and the vicinity of Earth’s magnetosheath, where nonthermal electrons and positrons coexist with anisotropic pressure ions.

6 Future work

In this study, the small-amplitude ion-acoustic dissipative solitons have been investigated by analyzing the damped ZK equation. Some discrepancies may appear between theoretical and practical observations, which requires some processing of the theoretical model to achieve complete agreement between theoretical results and experimental observations. To this end, in future works, we can follow the same work in the literature [71–75] by including the fifth-order dispersion in the governing equations to study the properties of large-amplitude dissipative nonlinear structures that can arise and propagate in the current plasma model or in any other model can describe by these equations. Additionally, the fractional calculus has played a vital role in recent times due to its ability to analyze various types of fractional evolutionary wave equations (EWEs) that can be used to model many different phenomena in many scientific disciplines [76–79]. Furthermore, studying the fractional EWEs is of great benefit in providing explanations for some unknown phenomena while studying these phenomena using the same EWEs in their integer forms. Therefore, in our future works, we intend to study the same current model in its fractional form to investigate the dynamics of propagation of the dissipative ion-acoustic solitary and cnoidal waves under the influence of fractionality. To do this, we will use the most recent methods discovered in analyzing fractional differential equations, such as the Tantawy technique and the new iterative method, and the residual power series method which have proven effective in many applications [80–84].

tantawy@sci.psu.edu.eg

Acknowledgments

The authors express their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project Number (PNURSP2025R157), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Funding information: The authors express their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project Number (PNURSP2025R157), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. Also, this research work was funded by Institutional Fund Projects under grant no. (IFPIP: 981-247-1443). The authors gratefully acknowledge technical and financial support provided by the Ministry of Education and King Abdulaziz University, DSR, Jeddah, Saudi Arabia.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing does not apply to this article as no data sets were generated or analyzed during the current study.

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Received: 2025-03-28

Revised: 2025-06-29

Accepted: 2025-07-01

Published Online: 2025-10-08

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https://doi.org/10.1515/phys-2025-0209

Keywords for this article

anisotropic pressure; damped Zakharov–Kuznetsov equation; ion-acoustic waves; dissipative solitary waves; collisional plasma; kappa distribution; magnetized rotating plasma

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