Dissipative ion-acoustic solitary structures in magneto-rotating collisional non-Maxwellian plasmas in the framework of a forced damped Zakharov-Kuznetsov approach

Abstract

This work examines low-frequency ion-acoustic solitary structures in a weakly rotating, magnetized, collisional electron-positron-ion plasma. In this model, the electrons and positrons obey a q-nonextensive distribution, while ions are treated as a warm fluid subject to ion-neutral collisions and an externally applied periodic force. By employing the reductive perturbation technique, the governing fluid and Poisson equations are reduced to a forced damped Zakharov-Kuznetsov (FDZK) equation that consistently incorporates four key physical ingredients in a unified framework: nonextensive statistics, collisional dissipation, Coriolis effects due to plasma rotation, and external energy injection through a periodic source term. In the collisionless, force-free limit, an exact compressive solitary wave solution of the underlying Zakharov–Kuznetsov equation is obtained, and the corresponding pseudo-potential analysis clarifies the constraints on the existence region of ion-acoustic solitary waves and their polarity. For weak damping and finite periodic forcing, approximate time-dependent solitary solutions of the FDZK equation are constructed using energy-type conservation arguments, allowing the combined impact of dissipation and external driving on the soliton profile (amplitude, width, and speed) to be quantified. The impact of various related parameters on the dissipative soliton profiles is numerically investigated. These findings offer a physically transparent picture of how nonthermal statistics, rotation, and external excitations jointly shape nonlinear ion-acoustic dynamics in realistic space and laboratory plasmas, such as planetary and pulsar magnetospheres, the solar wind, and magnetized laboratory devices.

1. Introduction

Research conducted in both space and laboratory environments has demonstrated that plasmas can generate a diverse range of nonlinear structures, shaped by the system’s specific conditions and configurations, with commonly observed perturbations including ion-acoustic waves (IAWs), dust acoustic waves, dust-ion acoustic waves, and electron acoustic waves. The behavior of IAWs, a fundamental plasma wave phenomenon, has been extensively studied for decades through both theoretical and experimental approaches, beginning with the theoretical prediction of ion-acoustic solitary waves (IASWs) by Washimi and Taniuti, ¹ who modeled plasmas with cold ions and isothermal electrons, followed by the first experimental observation of these solitons by Ikezi et al. ² Subsequent refinements to the theory were made to address discrepancies between experimental and theoretical results, incorporating factors such as finite ion temperature,^3,4 trapped electron populations,^5,6 and higher-order nonlinear effects. Solitary waves (SWs) are nonlinear, localized structures that arise from a balance between nonlinearity and dispersion, remaining a focal point of research due to their significant practical applications.^7–9 IASWs, in particular, have been extensively investigated in plasma physics and complex plasmas,^10–14 with both theoretical and experimental studies exploring their formation across diverse physical conditions. These waves have been examined in contexts ranging from laboratory^15,16 to space plasmas,^17–19 with various studies^20–22 modeling them under different assumptions such as cold ions paired with non-adiabatic electrons described by vortex-like, ²⁰ nonthermal (Cairns distribution), ²² bi-Maxwellian, ²¹ or nonextensive distributions^23,24 to understand their properties and behavior better.

In recent years, significant interest has grown in nonextensive statistical mechanics, which generalizes traditional thermodynamics by accounting for deviations from the Boltzmann-Gibbs-Shannon entropy framework. This nonextensive extension, first introduced by Rényi ²⁵ and later formalized by Tsallis, ²⁶ provides a more versatile approach to describing statistical equilibrium in complex systems by incorporating an entropic index q. This parameter quantifies the level of nonextensivity where q = 1 recovers the classical Boltzmann-Gibbs extensive statistics, while values diverging from unity capture non-additive behaviors observed in models with memory effects, long-range interactions, or fractal structures. This formalism has proven particularly valuable in plasma physics, astrophysics, and other domains where traditional statistical mechanics fails to describe observed non-equilibrium phenomena. Tsallis’ non-additive entropy and its generalized statistics have proven effective in modeling nonextensive phenomena,^27–32 offering an alternative to the Maxwellian distribution of Boltzmann-Gibbs statistics, which, while universally valid for macroscopic ergodic equilibrium systems, often fails to describe long-range interacting systems like plasmas and gravitational systems in non-equilibrium states. The entropic index q with q < 1 corresponds to super-extensive systems^33,34 and q > 1 to sub-extensive systems. ³⁵ Recent work by Verheest ³⁶ further refines this framework, proposing that only the ranges 1/3 < q < 1 (super-extensive) and q > 1 (sub-extensive) can adequately explain nonextensive effects on nonlinear structures, providing critical constraints for applying Tsallis statistics to complex physical systems.

The swirling motions of magnetized conducting media like plasmas occur in geophysical systems, laboratory experiments, and cosmic environments, enabling studies of linear wave propagation that highlight the Coriolis force’s role^37,38 in ionospheric dynamics, a phenomenon crucial for astrophysical processes such as sunspot formation, stellar evolution, and magnetized celestial body dynamics. When stars collapse into neutron stars, their dramatically reduced moment of inertia results in extreme rotational acceleration due to the conservation of angular momentum. Meanwhile, the stability of both linear and nonlinear plasma waves is heavily influenced by strong magnetic fields in astrophysical and geophysical contexts. This plasma rotation, driven by magnetic fields, significantly impacts phenomena in space plasmas and laboratory devices, ³⁹ demonstrating its broad relevance across scales from experimental setups to cosmic structures.

The complex nonlinear dynamics of plasma particles necessitate specialized analytical approaches, such as perturbation techniques, with the reductive perturbation technique (RPT)^1,40 being particularly effective for deriving the governing equations of nonlinear solitary waves, as evidenced by extensive research. This methodology builds upon foundational models such as the Korteweg-de Vries (KdV) equation, ⁴¹ the first theoretical framework for SWs, which was later extended to three dimensions by the Zakharov-Kuznetsov (ZK) equation. ⁴² Studies show that collisional interactions between plasma components dramatically modify solitary wave and shock wave dynamics.^43–45 These waves may gradually dissipate due to the presence of dissipation in the plasma medium. Seminal numerical explorations of damped solitary wave dynamics in magnetized plasmas were conducted by Krapak and Morfill, ⁴⁶ who examined energy dissipation mechanisms. Beyond interspecies collisions, the ion-acoustic solitary waves can dissipate if the ions become significantly hotter than the electrons, highlighting the influence of ion temperature.^47,48 Since most practical systems are out of equilibrium, nearly all waves experience some degree of energy loss.

The influence of external periodic excitations on observable physical phenomena varies significantly across different contexts. Recent years have seen a growing interest in analyzing nonlinear solitary waves under such perturbations, although few studies have incorporated both damping and forcing parameters. Chatterjee et al. ⁴⁹ explored dust-ion-acoustic solitary waves using the damped forced KdV (DFKdV) equation in superthermal plasmas with κ-distributed electrons, while Ali et al. ⁵⁰ investigated electron acoustic solitary waves through the derivation of the forced KdV (FKdV) equation under similar plasma conditions, highlighting the significant role of external forces and damping in shaping wave dynamics. Chowdhury et al. ⁵¹ examined IASWs using a FKdV-like Schamel equation in superthermal plasmas with trapped electrons, while Paul et al. ⁵² analyzed the impact of dust-ion collisions on solutions of the damped Korteweg-de Vries-Burgers (KdVB) equation. Mandi et al. ⁵³ further explored damping and periodic forcing effects through a damped forced modified KdV equation, followed by Paul et al. ⁵⁴ who studied dust ion acoustic waves in nonextensive plasmas using a damped forced KdVB framework. Most recently, Ozah et al. ⁵⁵ investigated relativistic IASWs by deriving the forced Zakharov-Kuznetsov (FZK) equation for κ-distributed electrons in superthermal plasmas, collectively advancing understanding of wave dynamics under external forcing and dissipation. A FZK equation has also been derived by El-Awady et al. ⁵⁶ to investigate dust-ion-acoustic solitary waves in a magnetized dusty plasma subjected to an external periodic perturbation system. Parallel numerical studies by Adnan et al. ⁵⁷ and Rakib and Sultana ⁵⁸ have explored the characteristics of solitary structures in magnetized, non-rotating plasmas with non-Maxwellian distributions, providing a basis for understanding parametric influences on wave dispersion and nonlinearity. Moreover, El-Awady et al. ⁵⁹ studied nonlinear characteristics of IASWs in a plasma system and obtained the forced damped ZK (FDZK) equation. Our present contribution distinguishes itself from this prior body of work in several key, synthesizing aspects. While earlier studies have examined forced KdV models or rotating magnetized plasmas in isolation, this work is, to our knowledge, the first to derive and solve a FDZK equation for a magneto-rotating, collisional electron-positron-ion (e-p-i) plasma. Specifically, we unify four critical physical elements that have not been combined in a single ZK-type model: (i) collisional damping (via ion-neutral collisions), (ii) external periodic forcing, (iii) Coriolis force effects due to finite plasma rotation, and (iv) a nonthermal electron-positron component described by a q-nonextensive distribution. This unified approach allows us to investigate the concurrent and competing influences of energy injection (forcing), energy dissipation (collisions), rotational stabilization, and nonthermal statistics on solitary wave morphology and stability. Consequently, our analytical solution and numerical analysis reveal new, nuanced regimes of wave behavior, such as the transition from coherent solitary waves to forced, modulated, and aperiodic structures that cannot be captured by models lacking one or more of these physical ingredients, thereby offering a more comprehensive framework for interpreting nonlinear wave dynamics in complex, realistic plasma environments. It is important to note that the study of dissipative ion-acoustic solitary structures in a magneto-rotating, collisional, non-Maxwellian e-p-i plasma is applicable to several realistic environments where such multi-component, magnetized, and rotating plasmas are observed. For instance, planetary magnetospheres, ⁶⁰ pulsar magnetospheres, ⁶¹ solar wind and coronal plasmas, ⁶² as well as accretion disks around compact objects, are characterized by rapid rotation, strong magnetization, and collisional plasmas that support such nonlinear structures. ⁶³ On Earth, the ionosphere and magnetosphere also host collisional plasmas with nonthermal electrons. ⁶⁴ Moreover, laboratory devices such as tokamaks and stellarators ⁶⁵ can reproduce magnetized, rotating plasma conditions with external periodic forcing, making them viable platforms for observing dissipative solitary waves.

Building upon previous research,^49–55 we employed the RPT to derive the forced damped ZK (FDZK) equation for a magneto-rotating, collisional e-p-i plasma with isotropic ion pressure, while neglecting ionization and recombination effects from inelastic ion collisions. By incorporating a variable charge density source function into the Poisson equation, we obtained the FDZK equation, with the primary objective of determining its approximate analytical solitary wave solution under conditions where ion temperature, collision frequency, positron concentration, electron temperature, rotation frequency, and both the strength and frequency of the periodic force remain small. Additionally, we investigated how variations in the nonextensive parameter q influence the morphology and structural characteristics of these solitary waves, providing insights into their behavior under different plasma conditions.

This work is structured as follows: Section 2 presents the fluid model governing the plasma system, while Section 3 examines the nonlinear wave structure, with subsections 3.1 and 3.2 detailing pulse-soliton characteristics and numerical analyses of the effects of various physical parameters. Finally, a summary of the major findings along with the conclusion is provided in Section 4.

2. Fluid model

This manuscript investigates the behavior of IASWs in a weakly rotating, magnetized e-p-i plasma with q-nonextensive distribution subjected to an external periodic force. The ions are modeled as a fluid with charge z _i e, mass m _i , and velocity u _i , incorporating collisions with neutral atoms or molecules. The model is subjected to an external magnetic field B = B₀ e _z along the z-axis, while plasma rotation at a low frequency Ω = Ω₀ e _z is considered to account for Coriolis force effects aligned with the magnetic field direction. The nonthermal character of the plasma is modeled through a q-nonextensive distribution for both positrons and electrons. At the same time, ions are treated as a cold inertial component $\left(T_i\ll T_{e,p}\right)$ to examine the nonlinear properties of IASWs. The system is subjected to an external periodic force along the z-axis, which is incorporated via a source term S(z, t) in Poisson’s equation, enabling the examination of forced wave propagation in this configuration. Under thermal equilibrium conditions, the charge neutrality relation z _i n_i0 + n_p0 = n_e0 holds, where $n_{s0}\left(s=e,p,i\right)$ represents the unperturbed number density of each species. For wave propagation in three spatial dimensions $\left(x,y,z\right)$ under the specified conditions, the plasma is modeled as a rotating fluid with ion-neutral collisions, modeled by the following system of equations:

$\frac{\partial }{\partial t}{n}_i+u_{\mathit{ix}}\left(\frac{\partial }{\partial x}{n}_i\right)+n_i\left(\frac{\partial }{\partial x}{u}_{\mathit{ix}}\right)+u_{\mathit{iy}}\left(\frac{\partial }{\partial y}{n}_i\right)+n_i\left(\frac{\partial }{\partial y}{u}_{\mathit{iy}}\right)+u_{\mathit{iz}}\left(\frac{\partial }{\partial z}{n}_i\right)+n_i\left(\frac{\partial }{\partial z}{u}_{\mathit{iz}}\right)=0,$

(1)

$n_i\frac{\partial }{\partial t}{u}_{\mathit{ix}}+n_i\left(u_{\mathit{ix}}\frac{\partial }{\partial x}{u}_{\mathit{ix}}+u_{\mathit{iy}}\frac{\partial }{\partial y}{u}_{\mathit{ix}}+u_{\mathit{iz}}\frac{\partial }{\partial z}{u}_{\mathit{ix}}\right)+n_i\left(\frac{\partial \varphi }{\partial x}\right)-\left({\omega }_{\mathit{ci}}+2{\mathrm{\Omega }}_0\right)n_i{u}_{\mathit{iy}}+{\sigma }_i\left(\frac{\partial }{\partial x}{n}_i\right)+\nu n_i{u}_{\mathit{ix}}=0,$

(2)

$n_i\frac{\partial }{\partial t}{u}_{\mathit{iy}}+n_i\left(u_{\mathit{ix}}\frac{\partial }{\partial x}{u}_{\mathit{iy}}+u_{\mathit{iy}}\frac{\partial }{\partial y}{u}_{\mathit{iy}}+u_{\mathit{iz}}\frac{\partial }{\partial z}{u}_{\mathit{iy}}\right)+n_i\left(\frac{\partial \varphi }{\partial y}\right)+\left({\omega }_{\mathit{ci}}+2{\mathrm{\Omega }}_0\right)n_i{u}_{\mathit{ix}}+{\sigma }_i\left(\frac{\partial }{\partial y}{n}_i\right)+\nu n_i{u}_{\mathit{iy}}=0,$

(3)

$n_i\frac{\partial }{\partial t}{u}_{\mathit{iz}}+n_i\left(u_{\mathit{ix}}\frac{\partial }{\partial x}{u}_{\mathit{iz}}+u_{\mathit{iy}}\frac{\partial }{\partial y}{u}_{\mathit{iz}}+u_{\mathit{iz}}\frac{\partial }{\partial z}{u}_{\mathit{iz}}\right)+n_i\left(\frac{\partial \varphi }{\partial z}\right)+{\sigma }_i\left(\frac{\partial }{\partial z}{n}_i\right)+n_i\nu u_{\mathit{iz}}=0\cdot$

(4)

The plasma model is completed using Poisson’s equation with source term^55,66:

$\frac{{\partial }^2\varphi }{\partial x^2}+\frac{{\partial }^2\varphi }{\partial y^2}+\frac{{\partial }^2\varphi }{\partial z^2}+n_i+{\mu }_p{n}_p-{\mu }_e{n}_e-S(z,t)=0\cdot$

(5)

In the preceding equations (1)-(5), n _s is the density number of particle normalized by its equilibrium value $n_{s0}\left(s=e,p,i\right)$ , u _i is the ion speed normalized by ion sound velocity $C_{si}\left(={(z_i{k}_B{T}_e/m_i)}^{1/2}\right)$ and ϕ the electrostatic potential normalized by k _B T _e /e. Time variable is normalized by the inverse of the plasma frequency ${\omega }_{\mathit{pi}}\left(={(4\pi n_{i0}{e}^2/m_i)}^{1/2}\right)$ , ω _ci , ν and Ω ₀ are normalized by ω _pi while the spatial variables x, y and z are normalized by the Debye length ${\lambda }_{\mathit{Di}}\left(=k_B{T}_e/4z_i\pi n_{\mathit{i0}}{e}^2\right)$ , where k _B is the Boltzmann constant, e the elementary charge, μ _s = n_s0/n_i0 and ${\sigma }_i\left(=T_i/z_i{T}_e\right)$ the ratio of ion-electron temperature. S(z, t) is a source term of periodic force that arises from experimental conditions. ⁶⁶

The normalized q−nonextensive electron and positron are given by:

$\begin{array}{c}{n}_e={\left[1+\left(q-1\right)\varphi \right]}^{\frac{3q-1}{2q-2}}\simeq 1+\frac{1}{2}\left(3q-1\right)\varphi +\frac{1}{8}\left(q+1\right)\left(3q-1\right){\varphi }^2-\frac{1}{48}\left(q^2-2q-3\right)\left(3q-1\right){\varphi }^3\cdots ,\hfill \\ n_p={\left[1-\left(q-1\right)\sigma \varphi \right]}^{\frac{3q-1}{2q-2}}\simeq 1-\frac{1}{2}\left(3q-1\right)\sigma \varphi +\frac{1}{8}\left(q+1\right)\left(3q-1\right){\sigma }^2{\varphi }^2+\frac{1}{48}\left(q^2-2q-3\right)\left(3q-1\right){\sigma }^3{\varphi }^3\cdots ,\hfill \end{array}$

(6)

where σ = T _e /T _p is the ratio of electron-positron temperature. Taking into account equation (6) in equation (5), we obtain the following equation:

$\frac{{\partial }^2\varphi }{\partial x^2}+\frac{{\partial }^2\varphi }{\partial y^2}+\frac{{\partial }^2\varphi }{\partial z^2}+n_i+{\alpha }_3{\varphi }^3-{\alpha }_2{\varphi }^2-{\alpha }_1\varphi -\left(S+1\right)=0,$

(7)

with coefficients α₁, α₂ and α₃ containing information about plasma via q−index parameter, are express by:

$\begin{array}{c}{\alpha }_1=\frac{1}{2}\left(3q-1\right)\left(1+{\mu }_p\left(\sigma +1\right)\right),\hfill \\ {\alpha }_2=\frac{1}{8}\left(q+1\right)\left(3q-1\right)\left(1+{\mu }_p\left(1-{\sigma }^2\right)\right),\hfill \\ {\alpha }_3=\frac{1}{48}\left(3q-1\right)\left(q-3\right)\left(q+1\right)\left(1+{\mu }_p\left(1+{\sigma }^3\right)\right).\hfill \end{array}$

(8)

At the limit $\left(q\to 1\right)$ , the non-extensive distribution converges to the traditional Maxwell-Boltzmann distribution. In this work, we assume that the non-thermal plasma is characterized by q−index 1/3 < q < 1 for the super-extensive plasma. In the next section, we will focus on reducing equations (1)-(4) and equation (7) to a single nonlinear equation capturing the nonlinear behavior of IAWs in the plasma.

3. Nonlinear wave structure

To analyze the nonlinear behavior of IASWs within this plasma system, we employ the RPT ¹ to derive a FDZK equation. Following this approach, we introduce the following stretching transformations for the independent variables^43,57:

$X={\epsilon }^{1/2}x\text{\hspace{0.17em};\hspace{0.17em}}Y={\epsilon }^{1/2}y\text{\hspace{0.17em};\hspace{0.17em}}Z={\epsilon }^{1/2}\left(z-v_{p}t\right)\text{\hspace{0.17em}and\hspace{0.17em}}T={\epsilon }^{3/2}t,$

(9)

where 0 < ɛ < 1 is the strength of nonlinearity and v _p is the phase velocity of IASW, which can be extracted from the first-order approximation in ɛ. This choice guarantees the existence of coherent stable structures of nonlinear evolution equations, as established in relativistically enhanced environments and in magnetized plasmas, where the compensation of nonlinearity, dispersion, and transverse diffraction precisely requires this separation of electrons.^67,68 Additionally, this scaling makes it possible to analyze transversal stability without altering the longitudinal equilibrium’s order, making comparisons with reconstructions based on physical learning and recent numerical diagnostics in multi-species magnetized environments easier.^69,70 Accordingly, the state variables are expanded as ⁵⁵ :

$\begin{array}{c}{n}_i=1+\epsilon N_i^{(1)}+{\epsilon }^2{N}_i^{(2)}+{\epsilon }^3{N}_i^{(3)}+\cdots ,\hfill \\ u_{\mathit{ix}}={\epsilon }^{3/2}{U}_{\mathit{ix}}^{(1)}+{\epsilon }^2{U}_{\mathit{ix}}^{(2)}+{\epsilon }^{5/2}{U}_{\mathit{ix}}^3+\cdots ,\hfill \\ u_{\mathit{iy}}={\epsilon }^{3/2}{U}_{\mathit{iy}}^{(1)}+{\epsilon }^2{U}_{\mathit{iy}}^{(2)}+{\epsilon }^{5/2}{U}_{\mathit{iy}}^3+\cdots ,\hfill \\ u_{\mathit{iz}}=\epsilon U_{\mathit{iz}}^{(1)}+{\epsilon }^2{U}_{\mathit{iz}}^{(2)}+{\epsilon }^3{U}_{\mathit{iz}}^{(3)}+\cdots ,\hfill \\ \varphi =\epsilon {\varphi }^{(1)}+{\epsilon }^2{\varphi }^{(2)}+{\epsilon }^3{\varphi }^{(3)}+\cdots ,\hfill \\ \nu ={\epsilon }^{3/2}{\nu }_0,\hfill \\ S={\epsilon }^2{S}^{(2)}.\hfill \end{array}$

(10)

Inserting the above scaling ansatz into the evolution equations (1)-(4) and equation (7) and isolating the terms arising in lowest order $\left(\sim {\epsilon }^{3/2}\right)$ , yields:

$\begin{array}{c}{U}_{\mathit{ix}}^{(1)}=-\frac{1}{{\omega }_{\mathit{ci}}+2{\mathrm{\Omega }}_0}\left(1+{\alpha }_1{\sigma }_i\right)\frac{\partial }{\partial Y}{\varphi }^{(1)},\hfill \\ U_{\mathit{iy}}^{(1)}=\frac{1}{{\omega }_{\mathit{ci}}+2{\mathrm{\Omega }}_0}\left(1+{\alpha }_1{\sigma }_i\right)\frac{\partial }{\partial X}{\varphi }^{(1)},\hfill \\ U_{\mathit{iz}}^{(1)}=\frac{1}{v_p}\left({\alpha }_1{\sigma }_i+1\right){\varphi }^{(1)}=v_p{N}_i^{(1)}.\hfill \end{array}$

(11)

To lowest order $\left(\sim \epsilon \right)$ , Poisson’s equation results in:

$N_i^{(1)}={\alpha }_1{\varphi }^{(1)}.$

(12)

Uing the latter expression, one can derive the pulse propagation speed v _p ,

$v_p=\sqrt{{\sigma }_i+\frac{1}{{\alpha }_1}},$

(13)

with α₁ defined in equation (8) and σ _i the ratio ion-electron temperature. The numerical values chosen for the plasma parameters in this study (e.g., σ _i ∼ 0.1 − 0.9, μ _p ∼ 0.2 − 0.6, ω _ci ∼ 0.3, γ ∼ 0.01 − 0.05) are representative of warm, collisional, magnetized electron-positron-ion plasmas encountered in both planetary magnetospheres and corotating regions of the ionosphere, where Coriolis and Lorentz forces compete, as modeled in geophysical fluid dynamics studies. ⁷¹ Plasma parameters determine the phase velocity of IAWs through the coefficient α₁. Notably, our analysis reveals that external periodic forcing, collisional effects, and Coriolis forces do not influence the phase velocity of IAWs. Figure 1 demonstrates the dependence of phase speed on the nonextensive distribution parameter q, illustrating how this fundamental wave characteristic varies with different degrees of plasma nonthermality.

Figure 1.

Evolution of phase velocity v _p versus q−index q: $\left(a\right)$ for several values of ratio ion-electron temperature σ _i with μ _p = 0.4 and $\left(b\right)$ for several values of positron concentration μ _p with σ _i = 0.3, other parameters fixed are: σ = 0.5.

Our analysis reveals that the phase velocity of IAWs decreases monotonically with increasing q-index of the nonextensive distribution. Furthermore, the phase velocity exhibits a positive correlation with the ion-to-electron temperature ratio while showing an inverse dependence on positron concentration. Specifically, increasing positron density reduces the phase velocity, thereby stabilizing electrostatic perturbations, whereas higher ion temperatures enhance wave propagation by increasing phase velocity. The phase velocity decreases with the nonextensive parameter q because a harder nonthermal tail enhances screening, reducing the acoustic restoring force, while higher ion temperature σ _i adds pressure support, increasing the phase speed. Notably, in Maxwellian plasmas $\left(q\to 1\right)$ , the phase velocity reaches significantly lower values compared to superthermal plasma conditions, demonstrating the crucial role of nonthermal effects in wave dynamics.

Collecting the next higher order $\left(\sim {\epsilon }^{4/2}\right)$ contribution from the ion momentum equation and higher order $\left(\sim {\epsilon }^2\right)$ contribution from Poisson’s equation are respectively:

$\begin{array}{c}{U}_{\mathit{ix}}^{(2)}=\frac{v_p\left(1+{\alpha }_1{\sigma }_i\right)}{{\left({\omega }_{\mathit{ci}}+2{\mathrm{\Omega }}_0\right)}^2}\frac{{\partial }^2{\varphi }^{(1)}}{\partial X\partial Z},\hfill \\ U_{\mathit{iy}}^{(2)}=\frac{v_p\left(1+{\alpha }_1{\sigma }_i\right)}{{\left({\omega }_{\mathit{ci}}+2{\mathrm{\Omega }}_0\right)}^2}\frac{{\partial }^2{\varphi }^{(1)}}{\partial Y\partial Z},\hfill \end{array}$

(14)

$-{\alpha }_2{\varphi }^{{(1)}^2}-{\alpha }_1{\varphi }^{(2)}+N_i^{(2)}+\frac{{\partial }^2}{\partial X^2}{\varphi }^{(1)}+\frac{{\partial }^2}{\partial Y^2}{\varphi }^{(1)}+\frac{{\partial }^2}{\partial Z^2}{\varphi }^{(1)}-S^{(2)}=0\cdot$

(15)

Similarly, the subsequent higher order $\left(\sim {\epsilon }^{5/2}\right)$ contribution from the ion continuity equation and z-component of momentum equation are respectively:

$\frac{\partial }{\partial T}{N}_i^{(1)}-v_p\left(\frac{\partial }{\partial Z}{N}_i^{(2)}\right)+\left(\frac{\partial }{\partial Z}{N}_i^{(1)}\right)U_{\mathit{iz}}^{(1)}+N_i^{(1)}\left(\frac{\partial }{\partial Z}{U}_{\mathit{iz}}^{(1)}\right)+\frac{\partial }{\partial X}{U}_{\mathit{ix}}^{(2)}+\frac{\partial }{\partial Y}{U}_{\mathit{iy}}^{(2)}+\frac{\partial }{\partial Z}{U}_{\mathit{iz}}^{(2)}=0,$

(16)

$\frac{\partial }{\partial T}{U}_{\mathit{iz}}^{(1)}-v_p{N}_i^{(1)}\frac{\partial }{\partial Z}{U}_{\mathit{iz}}^{(1)}+U_{\mathit{iz}}^{(1)}\frac{\partial }{\partial Z}{U}_{\mathit{iz}}^{(1)}+N_i^{(1)}\frac{\partial }{\partial Z}{\varphi }^{(1)}-v_p\frac{\partial }{\partial Z}{U}_{\mathit{iz}}^{(2)}+{\sigma }_i\frac{\partial }{\partial Z}{N}_i^{(2)}+\frac{\partial }{\partial Z}{\varphi }^{(2)}+{\nu }_0{U}_{\mathit{iz}}^{(1)}=0.$

(17)

Solving equations (15)-(17) by using equations (11) and (14), the following nonlinear equation emerges:

$\frac{\partial }{\partial T}{\varphi }^{(1)}+A{\varphi }^{(1)}\left(\frac{\partial }{\partial Z}{\varphi }^{(1)}\right)+B\frac{\partial }{\partial Z}\left(\frac{{\partial }^2}{\partial X^2}{\varphi }^{(1)}+\frac{{\partial }^2}{\partial Y^2}{\varphi }^{(1)}\right)+C\left(\frac{{\partial }^3}{\partial Z^3}{\varphi }^{(1)}\right)+\gamma {\varphi }^{(1)}-\mathrm{D}\left(\frac{\partial S^{(2)}}{\partial Z}\right)=0,$

(18)

where A is the nonlinear term, B and C the dispersion parameters, γ is the damping term and the coefficient D is present due to the external periodic force. These coefficients are given by:

$\begin{array}{c}A=C\left({\alpha }_1^2\left(3+2{\sigma }_i{\alpha }_1\right)-2{\alpha }_2\right),\hfill \\ B=C\left(1+\frac{{\left({\alpha }_1{\sigma }_i+1\right)}^2}{{\left({\omega }_{\mathit{ci}}+2{\mathrm{\Omega }}_0\right)}^2}\right),\hfill \\ C=D=\frac{1}{2{\alpha }_1^{3/2}\sqrt{{\alpha }_1{\sigma }_i+1}}=\frac{1}{2v_p{\alpha }_1^2},\hfill \\ \gamma =\frac{{\nu }_0}{2}.\hfill \end{array}$

(19)

Equation (18) represents the FDZK equation, which governs the nonlinear dynamics of IAWs in a magnetized, rotating plasma with ion-neutral collisions. When we simplify the system by neglecting ion temperature effects $\left({\sigma }_i=0\right)$ , collisional damping $\left(\gamma =0\right)$ , and plasma rotation $\left({\mathrm{\Omega }}_0=0\right)$ , our results reduce to forms consistent with Ref. 55, differing primarily in the choice of distribution function (q-nonextensive versus Maxwellian).

3.1. Pulse solution

To qualitatively model IAW propagation, we seek a solitary wave solution of the FDZK equation (18), which shares structural similarities with the KdV equation but represents a non-integrable system due to the presence of external periodic forcing and ion-neutral collisions (manifested through ∂ _Z E ≠ 0, where E is the system energy). To obtain an analytical solution, we simplify the problem by considering the collision-free (γ = 0) and force-free (D = 0) limit, introducing the transformed coordinate:

$\zeta =l_{x}X+l_{y}Y+l_{z}Z+vT,$

(20)

with l _x , l _y and l _z the direction cosines of wave vector such that l _x ²+ l _y ² + l _z ² = 1, v the constant propagation speed of the soliton. Therefore the standard ZK equation is giving by:

$\frac{\partial }{\partial T}{\varphi }^{(1)}+A{\varphi }^{(1)}\left(\frac{\partial }{\partial Z}{\varphi }^{(1)}\right)+B\frac{\partial }{\partial Z}\left(\frac{{\partial }^2}{\partial X^2}{\varphi }^{(1)}+\frac{{\partial }^2}{\partial Y^2}{\varphi }^{(1)}\right)+C\left(\frac{{\partial }^3}{\partial Z^3}{\varphi }^{(1)}\right)=0.$

(21)

Taking account the independent variable defined in equation (20) into equation (21), the conserved energy density takes the form:

$\frac{1}{2}{\left[\frac{d{\varphi }^{(1)}}{d\zeta }\right]}^2+V\left({\varphi }^{(1)}\right)=0,$

(22)

where $V\left({\varphi }^{(1)}\right)$ define the pseudo-potential expressed as:

$V\left({\varphi }^{(1)}\right)=-\frac{1}{2}\frac{v/l_z}{l_z^{2}C+B\left(1-l_z^2\right)}\left(1-\frac{A}{3v}{\varphi }^{(1)}\right){\varphi }^{{(1)}^2},$

(23)

with $l_z=\cos \left(\theta \right)$ , where θ is the oblique propagation angle of IASW occurs in plasma.

Figure 2 demonstrates the influence of the ion-to-electron temperature ratio σ _i and positron concentration μ _p on the pseudo-potential V(ϕ). The potential profile reveals a characteristic compressive solitary wave structure, manifested as a single potential well. Our analysis shows that increasing σ _i induces a significant reduction in both the potential depth (Figure 2(a)) and the corresponding solitary wave amplitude. This inverse relationship between ion temperature and wave amplitude suggests that thermal effects in the ion population tend to suppress nonlinear wave structures in the plasma medium. Figure 2(b) demonstrates that the pseudo-potential V(ϕ) reaches its maximum depth at μ _p = 0.6, indicating this positron concentration yields the tallest soliton structure. Our analysis reveals a critical potential value ${\varphi }_c^{(1)}$ in the range $(0.1<{\varphi }_c^{(1)}<0.15)$ where $V({\varphi }_c^{(1)})$ remains constant across different μ _p values. Analytical integration of Equation (21) yields the soliton solution, which can be expressed as:

${\varphi }^{(1)}\left(\zeta \right)={\varphi }_m^{(1)}{\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^2\left(\zeta /W\right),$

(24)

where ${\varphi }_m^{(1)}$ denotes the amplitude and W the width of soliton respectively, defined as:

$\begin{array}{c}{\varphi }_m^{(1)}=\frac{3v}{A{l}_z},W={\left(\frac{C{l}_z^2+B\left(1-l_z^2\right)}{v/4l_z}\right)}^{1/2}.\hfill \end{array}$

(25)

Figure 2.

Variation of pseudo-potential V versus electrostatic potential ϕ⁽¹⁾ (2a) for several values of σ _i with μ _p = 0.6 and (2b) for several values of μ _p with σ _i = 0.3. The other fixed parameters are: q = 0.4, σ = 0.1, ω _ci = 0.3, Ω ₀ = 0.2, θ = 30° and v = 0.05.

Assuming first the impact of collisions and the external force taken at null (S⁽²⁾ = 0), equation (18) takes the following form:

$\frac{\partial }{\partial T}{\varphi }^{(1)}+A{\varphi }^{(1)}\left(\frac{\partial }{\partial Z}{\varphi }^{(1)}\right)+B\frac{\partial }{\partial Z}\left(\frac{{\partial }^2}{\partial X^2}{\varphi }^{(1)}+\frac{{\partial }^2}{\partial Y^2}{\varphi }^{(1)}\right)+C\left(\frac{{\partial }^3}{\partial Z^3}{\varphi }^{(1)}\right)+\gamma {\varphi }^{(1)}=0,$

(26)

The approximate solution of equation (26) is derived by applying the conservation of momentum,^43,58,72 resulting in:

$E={\int }_{-\infty }^{+\infty }{\left({\tilde{\varphi }}^{(1)}\right)}^{2}d\tilde{\zeta }\text{\hspace{0.17em}and\hspace{0.17em}}\frac{\partial E}{\partial T}=-2\gamma E.$

(27)

From equation (24), we derive the approximate solution of the damped ZK equation expressed by:

${\tilde{\varphi }}^{(1)}={\tilde{\varphi }}_m^{(1)}{\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^2\left(\tilde{\zeta }/\tilde{W}\right),$

(28)

where ${\tilde{\varphi }}_m^{(1)}$ , $\tilde{W}$ are the amplitude and width of damped soliton respectively both dependent on time. Thus substituting equation (28) into equation (27), and applying the condition γ = 0, we determine the amplitude and width of the soliton defined by:

$\begin{array}{c}{\tilde{\varphi }}_m^{(1)}={\varphi }_m^{(1)}{e}^{-\gamma T},\tilde{W}=W{e}^{\frac{1}{2}\gamma T},\hfill \end{array}$

(29)

with ${\varphi }_m^{(1)}$ and W defined in equation (25). Thus, the approximate solution of equation (26) is:

${\tilde{\varphi }}^{(1)}={\varphi }_m^{(1)}{e}^{-\gamma T}{\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^2\left(\frac{\left(l_{x}X+l_{y}Y+l_{z}Z-v{e}^{-\gamma T}\right)}{W}{e}^{-\frac{1}{2}\gamma T}\right).$

(30)

To incorporate the external periodic force in equation (18), we model the source term S⁽²⁾ as a linear function of Z, given by $S^{(2)}=f_0/DZ\sin \left(\omega T\right)$ , where f₀ represents the force amplitude and ω its frequency. Inserting this expression into equation (18) yields the FDZK equation:

$\frac{\partial }{\partial T}{\varphi }^{(1)}+A{\varphi }^{(1)}\left(\frac{\partial }{\partial Z}{\varphi }^{(1)}\right)+B\frac{\partial }{\partial Z}\left(\frac{{\partial }^2}{\partial X^2}{\varphi }^{(1)}+\frac{{\partial }^2}{\partial Y^2}{\varphi }^{(1)}\right)+C\left(\frac{{\partial }^3}{\partial Z^3}{\varphi }^{(1)}\right)+\gamma {\varphi }^{(1)}=f_0\sin \left(\omega T\right),$

(31)

with S⁽²⁾ ≠ 0. It is evident from equation (30) that the approximate solution of the FDZK equation (31) can be written as:

${\tilde{\varphi }}^{(1)}={\varphi }_m^{(1)}\left(T\right)e^{-\gamma T}{\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^2\left(\frac{\left(l_{x}X+l_{y}Y+l_{z}Z-v\left(T\right)T{e}^{-\gamma T}\right)}{W(T)}{e}^{-\frac{1}{2}\gamma T}\right),$

(32)

where ${\varphi }_m^{(1)}\left(T\right)=3/A{l}_z\tilde{v}$ and $W\left(T\right)={\left(C{l}_z^2+B\left(1-l_z^2\right)/\tilde{v}/4l_z\right)}^{1/2}$ .

The main challenge now is to determine the expression of soliton speed time-dependent, which is denoted $\tilde{v}$ . To examine the influence of external periodic force on the propagation of IAWs, we assume the law of conservation of angular momentum. Thus, in the absence of collisions and periodic force effects, the Hamiltonian system is conserved and we can write:

$E={\int }_{-\infty }^{+\infty }{\left({\tilde{\varphi }}^{(1)}\right)}^{2}d\tilde{\zeta }.$

(33)

Substituting equation (32) in equation (33), we get the following expression:

$E=\frac{4}{3}{\left({\varphi }^{(1)}\left(T\right)e^{-\gamma T}\right)}^{2}W\left(T\right)e^{\frac{1}{2}\gamma T}.$

(34)

On the other hand, differentiation of equation (33) gives:

$\frac{dE}{dT}=2{\int }_{-\infty }^{+\infty }{\tilde{\varphi }}^{(1)}\frac{\partial {\tilde{\varphi }}^{(1)}}{\partial T}d\tilde{\zeta }.$

(35)

Using equation (31) to isolate $\partial {\tilde{\varphi }}^{(1)}/\partial T$ and substituting into equation (35), we obtain the following equation:

$\frac{dE}{dT}=2f_0\sin \left(\omega T\right){\int }_{-\infty }^{+\infty }{\tilde{\varphi }}^{(1)}d\tilde{\zeta }=4f_0\sin \left(\omega T\right)\left({\varphi }^{(1)}\left(T\right)e^{-\gamma T}\right)W\left(T\right)e^{\frac{1}{2}\gamma T}.$

(36)

Combining equation (34) and equation (36) where we are replaced ${\varphi }^{(1)}\left(T\right)$ and $W\left(T\right)$ by their value, we get the following differential equation:

$\frac{d}{dT}\tilde{v}-\gamma \tilde{v}=\frac{2}{3}A{l}_z{e}^{\gamma T}\sin \left(\omega T\right).$

(37)

The integration of equation (37) gives the speed of IAWs time-dependent as:

$\tilde{v}=\left(-\frac{2}{3\omega }A{l}_z{f}_0\cos \left(\omega T\right)+v{e}^{-\gamma T}\right)e^{\gamma T},$

(38)

where v is the initial velocity when γ = 0 and f ₀ = 0. When we assume γ = 0, equation (38) matches with Eq. $\left(31\right)$ in Ref. 55. Finaly, the time-dependent wave solution of the FdZK equation (18) is:

${\tilde{\varphi }}^{(1)}={\tilde{\varphi }}_m^{(1)}{\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^2\left(\tilde{\zeta }/\tilde{W}\right),$

(39)

with amplitude ${\tilde{\varphi }}_m^{(1)}$ , width $\tilde{W}$ and independent variable $\tilde{\zeta }$ time-dependent expressed by:

$\begin{array}{c}{\tilde{\varphi }}_m^{(1)}=-\frac{2}{\omega }{f}_0\cos \left(\omega T\right)+\frac{3v}{A{l}_z}{e}^{-\gamma T},\hfill \\ \tilde{W}={\left(\frac{C{l}_z^2+B\left(1-l_z^2\right)}{\left(-\frac{2}{3\omega }A{l}_z{f}_0\cos \left(\omega T\right)+v{e}^{-\gamma T}\right)/4l_z}\right)}^{1/2},\hfill \\ \tilde{\zeta }=\left(l_{x}X+l_{y}Y+l_{z}Z-T\left(-\frac{2}{3\omega }A{l}_z{f}_0\cos \left(\omega T\right)+v{e}^{-\gamma T}\right)\right).\hfill \end{array}$

(40)

3.2. Numerical analysis

We now numerically investigate the propagation characteristics of IASWs by systematically varying key plasma parameters, including the ion-to-electron temperature ratio σ _i , the nonextensive distribution index q, the collision frequency γ, the external periodic force amplitude f₀, and the plasma rotation frequency Ω₀.

Figure 3 demonstrates the dependence of soliton amplitude on the nonextensive q-index for various values of σ _i , f₀ and ω. As shown in the different panels, IAWs decrease with increasing q-parameter due to enhanced dispersive effects that counteract nonlinear steepening. This behavior is also observed in panel 3 $\left(a\right)$ with increasing ion-to-electron temperature ratio σ _i . In contrast, the soliton amplitude exhibits a positive correlation with the periodic force amplitude f₀ (Figure 3 $\left(b\right)$ ) and frequency ω (Figure 3 $\left(c\right)$ ), indicating that external forcing enhances nonlinear wave structures by injecting energy that amplifies the wave. These findings are consistent with previous results reported by Ozah and Deka, ⁵⁵ confirming the dual role of nonthermal effects and external perturbations in governing IAW dynamics.

Figure 3.

$\left(a\right)$ Evolution of ${\tilde{\varphi }}_m^{(1)}$ versus q for several values of σ _i with f ₀ = 0.2, ω = 10, μ _p = 0.2 and T = 4. $\left(b\right)$ Variation of ${\tilde{\varphi }}_m^{(1)}$ versus q for several values of f ₀ with σ _i = 0.6, ω = 20 and σ = 0.1. $\left(c\right)$ Variation of ${\tilde{\varphi }}_m^{(1)}$ versus q with σ _i = 0.6, and f ₀ = 0.1. Other fixed parameters: T = 3, θ = 15°, v = 0.05, σ = 0.3, μ _p = 0.6 and γ = 0.03.

Figure 4 presents the dependence of soliton width $\tilde{W}$ on the nonextensive q-index for varying (a) σ _i , f ₀ and Ω ₀ , and (b) time evolution for different q values. Panels 4 $\left(a\right)$ - $\left(c\right)$ demonstrate that $\tilde{W}$ decreases monotonically with increasing σ _i , f₀, and Ω₀, indicating reduced dispersion effects at higher parameter values that compress the solitary structure to maintain the nonlinear-dispersive balance. Similarly, Figure 4(d) reveals an inverse relationship between q-index and wave width, with Maxwellian plasmas (q → 1) exhibiting significantly narrower solitons than their super-extensive counterparts. Conversely, the temporal analysis in Figure 4(d) shows progressive width expansion over time.

Figure 4.

$\left(a\right)$ Variation of soliton width $\tilde{W}$ versus q−index for different values of ion temperature σ _i with f ₀ = 0.2 and θ = 0°. $\left(b\right)$ Variation of soliton width $\tilde{W}$ versus q−index for different values of periodic force f ₀ with Ω ₀ = 0.2, v = 0.01, ω = 15, θ = 30°. $\left(c\right)$ Variation of soliton width $\tilde{W}$ versus q−index for different values of rotation frequency Ω ₀ with f ₀ = 0.1. $\left(d\right)$ Variation of soliton width $\tilde{W}$ versus time T for different values of q−index with Ω ₀ = 0.2 and f ₀ = 0. Other fixed parameters: μ _p = 0.2, σ = 0.1, ω _ci = 0.3, Ω ₀ = 0.1, σ _i = 0.5, ω = 10, v = 0.05, θ = 15°, T = 2 and γ = 0.01.

Figure 5 demonstrates how increasing the ion-neutral collision frequency γ affects the amplitude modulation of IASWs. The plots indicate that higher collision rates result in more rapid damping of wave oscillations, leading to a faster decay of the amplitude modulation envelope over time. It is observed that the wave maintains its periodic modulation pattern but with progressively diminishing intensity as γ increases, illustrating how collisional friction systematically dissipates wave energy. This confirms that ion-neutral collisions act as an effective damping mechanism in plasmas, gradually suppressing both the carrier wave and its modulations regardless of external forcing conditions.

Figure 5.

Time-series for different values of frequency collision γ. Along with σ _i = 0.9, μ _p = 0.6, σ = 0.3, ω = 15, θ = 30°, f ₀ = 0.2, v = 0.05 and q = 0.5.

Figure 6 illustrates the time-series profiles $\left({\tilde{\varphi }}_m^{(1)}\right)$ under varying strengths of the external periodic force f₀, revealing a competition between forcing and collisional damping. For weak forcing (f₀ ≪ 1, panels a-b), the waves maintain stable, near-constant amplitudes with minimal oscillations, as collisional damping dominates and gradually dissipates energy. However, for stronger forcing (f₀ > 0.1, panels c-d), the periodic drive induces pronounced amplitude modulations, creating oscillatory peaks and dips in the wave envelope. At the same time, collisions still cause an overall decay. This result can be interpreted as evidence that strong forcing disrupts the soliton balance, giving rise to modulated and irregular structures. This interplay leads to a transition from coherent solitary waves (at low f₀) to irregular aperiodic structures (at high f₀), demonstrating how external forcing can disrupt wave stability in collisional plasmas. The figure underscores that the balance between periodic energy injection and damping dictates whether waves propagate stably or degrade into modulated, dissipative patterns. Figure 7 illustrates the temporal evolution of soliton amplitude under varying frequencies ω of the periodic force, revealing a distinct transition where amplitude modulation diminishes and aperiodic oscillations emerge before decaying completely as ω increases. This demonstrates that damping effects overpower periodic excitation and fundamentally alter the behavior of the solution. Specifically, the oscillatory pattern gives way to monotonic decay when ω exceeds 30. The transition from coherent to aperiodic oscillations with higher driving frequency ω indicates detuned excitation where the driver becomes inefficient. Complementing this, Figure 8 explores how the initial propagation speed (v) and periodic force frequency (ω) jointly influence IASW dynamics. The time-series profiles reveal that higher initial speeds amplify wave oscillations while increasing ω suppresses them, demonstrating a delicate balance between kinetic energy and external forcing. At low frequencies, the waves exhibit pronounced periodic modulations, but as ω grows, these oscillations decay into smoother profiles. Notably, faster-moving waves (higher v) maintain their structure longer against damping effects, suggesting that propagation speed plays a protective role in wave stability.

Figure 6.

Time-series for different values of external periodic force f ₀ . Along with σ _i = 0.9, μ _p = 0.6, σ = 0.3, ω = 15, θ = 30°, γ = 0.01, v = 0.05 and q = 0.5.

Figure 7.

Time-series for different values of frequency of periodic force ω. Along with σ _i = 0.3, μ _p = 0.2, σ = 0.3, γ = 0.01, θ = 30°, f ₀ = 0.2, v = 0.05 and q = 0.4.

Figure 8.

Time-series for different values of initial speed v and frequency of periodic force ω. Along with σ _i = 0.3, μ _p = 0.2, σ = 0.1, γ = 0.01, θ = 30°, f ₀ = 0.2 and q = 0.6.

Figure 9 displays the variations of IAWs as a function of the space variable ζ for several values of the parameters σ _i , f₀, and q, where Figure 9 $\left(a\right)$ and $\left(b\right)$ reveal that both the amplitude and width of the soliton decrease with increasing ion-electron temperature ratio σ _i and nonextensive index q, indicating that these parameters reduce the system’s energy and contribute to stabilizing solitary waves in the plasma. In contrast to the effect of the periodic force, the soliton’s amplitude increases while its width decreases as f₀ rises, demonstrating that higher f₀ enhances the excitation of IASWs. Additionally, the external periodic force modifies the medium by weakening nonlinear and dispersive effects, as anticipated in Figure 3(b) and 4(b).

Figure 9.

Evolution of soliton ${\tilde{\varphi }}^{(1)}$ against space variable ζ for different values $\left(a\right)$ of σ _i with f ₀ = 0. $\left(b\right)$ of q with f ₀ = 0.2. $\left(c\right)$ of f ₀ with ω = 20, θ = 30°, γ = 0.05 and v = 0.01. Along with σ _i = 0.6, μ _p = 0.6, σ = 0.5, ω = 10, Ω ₀ = 0.1, ω _ci = 0.3, θ = 0°, T = 3, v = 0.05, γ = 0.01, and q = 0.8.

Figures 10 and 11 present 3D plots of soliton evolution concerning the space variable ζ and time variable T. Figure 10(a) demonstrates that the solitary wave propagates undistorted, preserving both its width and amplitude, indicating a stable electrostatic wave with constant energy due to a balance between dispersion and nonlinearity in the plasma. In contrast, Figure 10(b) reveals modified IAW behavior, as the wave spreads during propagation, signifying a highly dispersive medium where energy conservation is lost; here, the solitary waves exhibit decreasing amplitude and increasing width, a consequence of elastic collisions between ions and neutrals that induce energy dissipation and wave damping, ultimately leading to the formation of evanescent IASWs. Figure 11 demonstrates how an external periodic force (f₀ = 0.04) modifies wave propagation in a collisional plasma (γ = 0.03). Unlike the stable soliton in Figure 10(a), this plot reveals periodic amplitude modulations along the time axis, where the wave oscillates between enhanced peaks and suppressed troughs due to constructive and destructive interference with the external force. The persistent but modulated structure shows that while collisions cause gradual damping, the periodic forcing simultaneously injects energy, creating a dynamic equilibrium where the wave neither fully collapses nor propagates undisturbed. This complex interaction produces rogue wave-like behavior, where localized amplitude spikes emerge unpredictably, offering crucial insights into energy localization phenomena in space and laboratory plasmas subject to external perturbations.

Figure 10.

3D plot of electrostatic potential ${\tilde{\varphi }}^{(1)}$ in $\left(\zeta ,T\right)$ plane. $\left(a\right)$ case where f ₀ = 0 and γ = 0. $\left(b\right)$ case where f ₀ = 0 and γ = 0.03. Other fixed parameters: σ _i = 0.6, σ = 0.5, μ _p = 0.6, ω _ci = 0.3, Ω ₀ = 0.1, ω = 15, θ = 30°, q = 0.8 and v = 0.05.

Figure 11.

3D plot of electrostatic potential ${\tilde{\varphi }}^{(1)}$ in $\left(\zeta ,T\right)$ plane where f ₀ = 0.04 and γ = 0.03. Other fixed parameters: σ _i = 0.6, σ = 0.5, μ _p = 0.6, ω _ci = 0.3, Ω ₀ = 0.1, ω = 0.5, θ = 30°, q = 0.8 and v = 0.3.

4. Summary

In this investigation, we’ve studied the dynamical behavior of low-frequency ion-acoustic solitary waves in a slowly rotating magnetized plasma with collisions between ions and neutrals, and electrons/positrons following a q-nonextensive distribution. The setup includes warm ions and an external periodic force added through the Poisson equation. Using standard reductive perturbation, we derived a forced damped Zakharov-Kuznetsov (ZK) equation that ties together nonlinearity from steepening, dispersion from ion temperature and obliquity, damping from collisions, Coriolis tweaks due to rotation, and energy kicks from the periodic drive. In the absence of damping and forcing, the model reduces to the standard ZK equation, for which a closed-form compressive soliton solution and its associated Sagdeev pseudo-potential were obtained, thereby clarifying the role of key plasma parameters in determining the existence and polarity of the solitary structures. For the full forced-damped case with weak effects, perturbative solitons show collisions that eat away at the amplitude (exponential drop) and spread the width, countered by the drive, which sets up oscillations in amplitude whose details depend on the forcing strength f₀ relative to the collision rate. From a numerical standpoint, the impact of various plasma parameters on wave propagation has been displayed in several graphs. The primary outcomes of this work are outlined below:

1. Numerical analysis reveals that phase velocity of the IAWs diminishes with increasing both q−nonextensive index and the positron concentration μ _p (better screening), and climbs with the ion-to-electron temperature ratio σ _i (stiffer restoring force), indicating that nonthermal tails and additional light species enhance screening while ion heating strengthens the restoring force for ion-acoustic oscillations.

2. The nonlinear and dispersive coefficients exhibit a complementary sensitivity to q−nonextensive index, the ion-to-electron temperature ratio σ _i , and the rotation frequency: nonextensivity and rotation generally weaken dispersion, whereas higher ion temperature tends to enhance it, leading to narrower, more localized solitary profiles when nonlinearity and dispersion remain in balance.

3. The soliton width decreases with the enhancement of σ _i , f ₀ , Ω ₀ , and q.

4. It was observed that the variation in the soliton amplitude results in amplitude modulation for specific values of the periodic force intensity f ₀ and the damping coefficient γ.

5. The amplitude and width of soliton decrease as the ion-electron temperature σ _i and nonextensive index q increase but as for increases, the width decreases while the amplitude increases.

6. The collisions between ions and neutrals have effects on damping the propagation of IAWs.

7. The amplitude modulation vanishes, and the aperiodic oscillation of the solitons appears and decays strictly as the frequency of the periodic force and constant speed increase.

It is revealed that the q−nonextensive force potential f ₀ , ion temperature σ _i , frequency rotation Ω ₀ , damped coefficient γ, frequency of periodic force ω and time parameter T affect significantly the nonlinear features $\left(\text{amplitude\hspace{0.17em}and\hspace{0.17em}width}\right)$ of the ion acoustic solitary waves in a collisional plasma. The results of the present research may have relevance in laboratory and astrophysical plasmas where nonextensive q−distributed electrons and positrons exist.

5. Future work

Given the central role of the FDZK model in the present study and the rapidly growing interest in fractional models for wave propagation and transport, several natural directions emerge for future work. A first avenue is to generalize the present magneto-rotating, collisional, nonextensive electron-positron-ion plasma model to its fractional counterpart by introducing memory effects either through Caputo-type time-fractional derivatives, space-fractional operators, or fully space-time fractional forms in the governing equations. In such extensions, the resulting fractional forced damped ZK equation could be tackled using advanced analytical–numerical approaches, with the Tantawy technique^73–79 playing a central role in constructing highly accurate and stable approximations of fractional solitary and cnoidal-wave solutions. Recent work has already established the effectiveness of the Tantawy technique for families of fractional Burgers and KdVB equations, as well as for fractional KdV-type ion-acoustic models, which strongly suggests that the same framework can be adapted to investigate fractional FDZK dynamics in rotating, nonextensive plasmas.

Another promising line of research is to extend the present configuration to include additional physical effects that are known to be important in space and laboratory plasmas but were neglected here for tractability. Examples include pressure anisotropy, relativistic degeneracy, and strong coupling effects in dense astrophysical settings, multi-temperature ion populations, or more realistic collision operators that go beyond simple linear damping. Coupling the FDZK or fractional FDZK framework to stochastic forcing or random fluctuations would also be valuable for mimicking turbulent environments and for studying how noise interacts with nonlinearity, dispersion, and fractional memory to shape the statistics of extreme events. In this context, the Tantawy technique could be adapted to treat inhomogeneous or stochastic fractional wave equations, building on recent progress where the method has been successfully applied to a range of fractional Burgers-type, Fokker–Planck-type, and diffusion models with plasma-physics applications.

From a more phenomenological perspective, it would be highly instructive to use the present model as a baseline for interpreting observations and numerical data from specific space and laboratory systems. For instance, one may seek to map particular ranges of the nonextensive index q, rotation frequency, and collisionality to measured conditions in planetary magnetospheres, pulsar environments, or strongly driven laboratory plasmas, and then use the FDZK and fFDZK predictions to identify regimes where compressive ion-acoustic solitary waves, modulated structures, or rogue-wave-like events are expected. This program would naturally benefit from combining the Tantawy technique with data assimilation or inverse methods, where fractional parameters and nonextensive indices are inferred from experimental time series or spacecraft observations by fitting the model’s solitary or modulated solutions to the measured waveforms.