Fractional dissipative low-frequency ion-acoustic solitary waves in a nonextensive collisional plasma: An application of the Tantawy technique

Abstract

This work investigates the linear and nonlinear behavior of low-frequency fractional ion-acoustic waves in a collisional, classical electron-ion plasma using a fluid description. This model consists of inertial ions and inertialess electrons, obeying the nonextensive distribution, with dissipation introduced via ion-neutral collisions. The influence of electron nonextensivity and the ion-neutral collisional parameters on the linear dispersion and damping characteristics of the ion-acoustic waves is analyzed in detail. To explore the nonlinear regime, the governing fluid equations are reduced, via the reductive perturbation technique under the assumption of weak ion-neutral collisionality, to an integer-damped Korteweg-de Vries (KdV) equation. This equation is completely non-integrable and does not admit exact closed-form soliton solutions; thus, the Ansatz method is employed to derive a semi-analytical approximation to investigate the characteristics of dissipative ion-acoustic solitons in the model under consideration. This equation is further generalized to a fractional damped KdV equation by introducing a suitable time-fractional operator, which constitutes the main novelty of the present study. The resulting fractional damped KdV equation is then analyzed analytically using the Tantawy technique, yielding a time-dependent approximate solution that captures the evolution of fractional dissipative ion-acoustic solitons. A comparative numerical analysis is carried out to contrast the dynamics of conventional (integer-order) and fractional dissipative solitons and to quantify the roles of electron nonextensivity and collisionality. The results show that, in the presence of dissipation, the soliton amplitude decays while its width increases with time, and that the nonextensive parameter does not affect the decay rate, which is controlled solely by the dissipative parameter. These findings provide insight into nonlinear wave damping and energy transport in laboratory plasmas and in space environments, such as the ionosphere and the solar wind, where nonextensive electron populations are frequently observed.

Introduction

Nonlinear wave phenomena within plasmas have long been of considerable theoretical and experimental interest, as they are relevant in astrophysical, space, and laboratory environments. Solitary waves or solitons, among all of these phenomena, represent localized, strong structures that result from the delicate balance of dispersion and nonlinearity.^1,2 Solitons, described by various differential equations such as the Korteweg-de Vries (KdV) equation (1), maintain their shape and speed even after interactions with other waves, and have been widely observed in plasmas including, but not limited to ion-acoustic, electron-acoustic, and dust-acoustic modes. ³ However, real plasma systems rarely behave as perfectly conservative media in reality. They often include dissipative processes such as viscosity and resistivity, as well as collisions with neutrals. These processes tend to degrade the wave energy. In such contexts, dissipative solitons arise. Dissipative solitons are localized nonlinear excitations (solitary waves) that maintain a stable shape despite the presence of energy dissipation in the medium. ⁴ Unlike non-dissipative solitons within conservative systems (which balance nonlinearity with dispersion alone), dissipative solitons emerge in systems where nonlinearity, dispersion, external driving, and dissipation all reach equilibrium simultaneously.

Dissipative solitary waves have been extensively investigated in plasma physics^5–11 as well as in nonlinear optics.^12–14 In fiber-optic communications, solitons are particularly valuable due to their ability to maintain a stationary profile, enabling them to serve as robust carriers of information. For reliable data transmission in optical fibers, these localized wave structures must preserve their integrity over long distances, despite the inherent dissipative effects of the medium. Unlike solitons in conservative, energy-preserving systems, dissipative solitons do not exhibit a perfectly stationary shape. Their profiles evolve during propagation, undergoing variations in amplitude, width, and velocity, and they ultimately decay over time. Given that most physical systems are intrinsically lossy, sustaining stable solitonic structures in such media requires continuous compensation for energy losses, typically through external driving or amplification mechanisms. In plasma, solitary waves can undergo dissipation and damping due to collisions between charged particles and uncharged (neutral) particles. Furthermore, damping effects may arise from interparticle collisions, Landau damping, and kinematic viscosity resulting from fluid shear stresses induced by the inertia of plasma motion. ¹⁵ Researchers demonstrated that electron and ion collisions with neutrals and dust, along with dust-charging effects, cause dissipation. Krapak and Morfill ¹⁶ reported that Langmuir waves undergo damping primarily due to electron collisions. The study considers several elastic processes, including electron-dust Coulomb collisions, electron-neutral collisions, and dust-electron and dust-neutral collisions. Ion acoustic (IA) dissipative solitary waves have been extensively studied to understand nonlinear wave propagation in realistic plasmas where energy dissipation cannot be ignored. Early work by Washimi and Taniuti ³ derived the KdV equation for collisionless ion acoustic solitons in ideal plasmas. However, to include dissipative effects such as ion viscosity, resistivity, and ion-neutral collisions, the KdV Burgers (KdVB) equation was developed, ¹⁷ predicting shock-like and asymmetric solitary structures. Subsequent analyses ¹⁸ have shown how dissipation broadens and dampens solitary wave profiles, producing monotonic or oscillatory shocks. Laboratory experiments ¹⁹ have confirmed the existence of IA shocks modified by collisional dissipation. Recent research has focused on the role of dissipation and non-thermal particle collections on the dynamics of solitary wave structures in both astrophysical and experimental plasma settings. In this context, Sultana and Kourakis ²⁰ examined the electron scale dissipative solitary structures in superthermal plasma, wherein a damped KdV (dKdV) equation was derived via RPT. Their results reveal that in superthermal plasmas, solitons travel faster and attain higher amplitudes compared to those in Maxwellian plasmas. Khan et al. ¹⁰ studied dissipative dust acoustic solitons in superthermal plasma, wherein a dKdV equation was derived. The study focused on how dust content, dust-neutral collisions, and superthermality influence the dynamics of dissipative dust acoustic solitons. Sultana et al. ²¹ studied dissipative high-frequency envelope solitons in three-species collisional non-thermal plasmas, demonstrating how collision frequency and particle distributions affect wave localization. Farooq et al. ²² expanded the analysis to linear and nonlinear regimes of collisional ion acoustic waves (IAWs) in magnetically confined rotating plasmas, showing that wave kinematics depend sensitively on rotation rate, magnetic field intensity, and the degree of electron nonthermality. Furthermore, in Ref. 23, authors looked into how superthermally energetic positrons and variable ion masses alter wave features in collisional plasma, tying the results to ionospheric plasma phenomena.

Fractional differential equations (FDEs) have emerged as powerful tools for modeling complex physical systems that exhibit memory and hereditary properties, which are not adequately captured by classical integer-order differential equations. FDE is the generalization of classical differential equations, applying the methods of fractional calculus, which extend integral and derivative operations to non-integer (real or complex) orders. With the introduction of fractional-order derivatives, one can describe nonlocal interdependencies and anomalous transport processes observed in regions such as viscoelasticity, fluid dynamics, electromagnetism, and plasma physics. There has been a great application of FDE^24,25 in analyzing the evolution wave equations (EWEs), such as the KdV, Burgers, and Schrödinger nonlinear equations, in their fractional forms to reveal the physical principles that underlie many physical processes. They have been successfully applied in engineering structures, biological tissues, and plasmas, particularly in describing dissipative, subdiffusive transport, and non-equilibrium processes. For instance, fractional modeling in plasma physics has enhanced the understanding of wave propagation, damping, and nonlinear structures, such as solitons, in collisional plasmas, where traditional models often fall short. Thus, the growing interest in FDEs reflects their unmatched potential in uncovering complex dynamics and offering improved predictions in systems that are otherwise too intricate for classical differential equations. Unlike classical models, FDEs incorporate temporal fractional derivatives and spatial fractional derivatives to characterize long-range correlations and fractal-like structures, thereby providing more accurate and flexible solutions than classical models for modeling natural phenomena. In plasma physics, the use of fractional calculus enables the description of dissipative effects as well as non-Markovian behaviors, which are of significant importance in complex media where collisional or memory effects are not negligible.

Among the numerous applications of fractional calculus in nonlinear plasma theory, none is more celebrated than the construction of fractional analogs to classical nonlinear wave equations, such as the fractional KdV (FKdV) equation. This is a type of equation that describes the time evolution of weakly nonlinear, weakly dispersive waves in plasma systems. These equations play a crucial role in describing ion acoustic solitons in collisional or nonlocal plasmas. The FKdV equation allows for a broader way of generalizing them simply by changing the ordinary time and or space derivatives with their fractional versions to incorporate dissipative mechanisms, wave damping, and memory effects. Several analytical and numerical techniques have been developed to study such fractional models, including the Adomian decomposition method, Laplace transform method, Homotopy analysis method, and Variational iteration method. These approaches enable the construction of approximate or semi-analytical solutions, providing deep insights into the dynamics and stability of solitonic structures in fractional-order systems.^26,27 A proper analytical method for evaluating fractional evolution equations related dynamics and related others is the Tantawy technique (TT), which has recently drawn interest. The method’s accuracy in capturing nonlinear and dissipative behavior in plasma systems was demonstrated by Almuqrin et al., ²⁸ when they used it to the fractional Burgers’ equation and fractional Fokker-Planck-type equations. Using the Yang transform, El-Tantawy et al. ²⁹ expanded its use to the fourth-order fractional Cahn–Hilliard equation and contrasted it with other semi-analytical methods. Additionally, El-Tantawy et al. ³⁰ used the technique to examine fractional IA solitons in electronegative plasmas with Cairns-distributed electrons in the context of plasma modeling, providing novel insights on the dynamics of FKdV solitary waves. These investigations demonstrate the reliability and adaptability of the TT in solving fractional nonlinear wave equations, establishing it as a valuable approach in contemporary theoretical plasma physics.

The velocity distribution of plasma particles has a great impact on the nonlinear behavior of plasma waves. Plasma kinetic theory usually supposes that the velocity distribution of plasma species follows the Maxwell–Boltzmann (MB) distribution. It originates from Boltzmann–Gibbs (BG) statistical mechanics, and it refers to systems within thermodynamic equilibrium. However, this approach falls short even for systems that exhibit long-range interactions and non-equilibrium stationary states, such as plasmas and gravitational systems. In this situation, the distribution deviates from the standard Maxwellian distribution; and particles have a change in energy called superthermal particles. Such particles have been found naturally in the solar wind ³¹ and in different magnetosphere.^32,33 To address this limitation, Tsallis introduced a generalized statistical framework, Tsallis (nonextensive) statistics, which extends the BG entropy by incorporating a nonextensivity parameter q. This parameter characterizes deviations from equilibrium: for q > 1, the plasma contains more high-energy (superthermal) particles, while q < 1, it has more low-speed particles than the Maxwellian case. As q → 1, the Tsallis distribution reduces to the classical MB form. Many theoretical and experimental studies have utilized Tsallis statistics to investigate various types of waves, such as IA, electron acoustic, dust acoustic, and dust ion-acoustic waves, within nonextensive plasma systems. Often, these studies focus on nonextensive effects in either the electron population, the ion population, or both.^34–37 Arbitrary ion acoustic solitary waves (IASWs) were studied in 34 with electrons exhibiting nonextensive behavior. Due to the electron nonextensivity, the coexistence of compressive and rarefactive IASWs was observed. The propagation of IASWs was investigated in magnetized electron positron ion plasma, where nonextensive electron distributions coexisted with Maxwellian positrons, as presented in 38. Electron acoustic solitary waves in unmagnetized plasma with electrons featuring the Cairns-Tsallis distribution have been studied in 39. Recently, the existence and evolution of IASWs in electron positron ion plasma under the Tsallis nonextensive distribution of electrons and positrons was studied in 40.

In most of the existing work on fractional evolution equations, the emphasis has been on treating relatively “friendly” fractional KdV-type models whose integral form is more amenable to standard analytical techniques. In contrast, the present study addresses a fully nonintegrable, stiff damped FKdV equation that arises self-consistently from a collisional fluid model with Tsallis q-nonextensive electrons, where both ion-neutral collisions and electron nonextensivity are treated on equal footing. This combination makes the problem significantly more challenging than the FKdV cases usually considered, yet more realistic for laboratory and space plasmas, where non-Maxwellian electrons and collisional dissipation coexist. Methodologically, the work departs from earlier FKdV studies in two main ways. First, the integer-order dKdV equation is not postulated, but derived systematically via the reductive perturbation technique from the underlying fluid equations, which allows the nonlinear, dispersive, and damping coefficients to be expressed explicitly in terms of the collisional parameter and the Tsallis nonextensivity index q. This direct link between the microscopic electron distribution and the macroscopic soliton parameters (amplitude, width, and decay rate) is, to the best of our knowledge, new for dissipative ion-acoustic solitons in Tsallis plasmas. Second, instead of treating the fractional extension only at a formal level, the integer dKdV is generalized to the damped FKdV equation and then analyzed in detail using the TT, which yields higher-order time-fractional series solutions together with a careful residual and absolute-error assessment. This provides a controlled, semi-analytical description of fractional dissipative solitons in a regime where exact solutions are unavailable.

The following schematic provides a simplified overview of the analysis process in this study: (1) Introduced the fluid plasma model → (2) Make a normalization to the fluid governing equations → (3) For nonlinear analysis, RPT scaling → (4) Consider perturbation expansion → (5) Order-by-order balance → (6) Integer dKdV → (7) Fractional operator leads to the dFKdV → (8) Tantawy mapping and fractional approximations. A summary of the abbreviations utilized is presented in Table 1 for clarity and readability.

Table 1.

Table of abbreviations.

Abbreviation	Full Form
KdV	Korteweg-de Vries
dKdV	Damped Korteweg-de Vries
FKdV	Fractional Korteweg-de Vries
EWEs	Evolution wave equations
FDEs	Fractional differential equations
IAWs	Ion-acoustic waves
RPT	Reductive perturbation technique
TT	Tantawy technique
TFDO	Time fractional derivative operator

Physical model and fluid equations

The plasma model is made of nonextensive inertialess electrons and inertial ions to explain the nonlinear behavior of fractional dissipative IASWs in an unmagnetized collisional plasma. Ions provide inertia, and electrons supply the restoring force. By applying the hydrodynamic model, wherein the following normalized continuity and momentum equations describe the dynamics of the ion fluid

$\left\{\begin{array}{c}{\partial }_t{n}_i+{\partial }_x(n_i{u}_i)=0,\hfill \\ {\partial }_t{u}_i+u_i{\partial }_x{u}_i=-{\partial }_x\phi -\nu u_i,\hfill \\ {\partial }_x^2\phi -n_e+n_i=0,\hfill \end{array}\right.$

(1)

where the indices i and e refer to ions and electrons, respectively, u _i represents ion fluid velocity normalized by ${(T_e/m_i)}^{\frac{1}{2}}$ . The electrostatic potential φ is normalized by (eφ/T _e ). Inverse plasma frequency ${\omega }_{pi}^{-1}=\sqrt{\left(m_i/4\pi n_{i0}{e}^2\right)}$ and Debye length ${\lambda }_D=\sqrt{T_e/4\pi n_{i0}{e}^2}$ are used to normalize the space variable (x) and time variable (t), respectively. ν = ν _i /ω _pi indicates the normalized collisional frequency. Here, n _e shows the normalized electron number density, which is assumed to follow a nonextensive distribution

$n_e={\left[1+(q-1)\phi \right]}^{\frac{q+1}{2(q-1)}}.$

(2)

For φ ≪ 1, equation (2) can be expanded as follows

$n_e=1+c_1\phi +c_2{\phi }^2+\cdots ,$

(3)

with

$c_1=\frac{\left(q+1\right)}{2}\text{\hspace{0.17em}\&\hspace{0.17em}}{c}_2=\frac{\left(q+1\right)\left(3-q\right)}{8}.$

For q → 1, the Maxwellian limit is recovered as expected.

Linear analysis and dispersion relation

In order to study the linear properties of the dissipative IAWs in electron-ion plasma, the system of equations (1) is linearized by introducing small deviations (perturbation) from the equilibrium states

$\left\{\begin{array}{c}{n}_i=1+n_{i1},\hfill \\ u_i=0+u_{i1},\hfill \\ \phi =0+{\phi }_1.\hfill \end{array}\right.$

(4)

These perturbations lead to the so-called harmonic perturbation of the form ∼ exp ( $i\left(kx-\omega t\right)$ , where ω and k being the normalized wave frequency and the wave number, respectively. This leads to the following linear dispersion relation

$\omega (\omega +i\nu )=\frac{k^2}{c_1+k^2},$

(5)

and by rearranging this equation, we finally get

$\omega =-\frac{i\nu }{2}\pm \sqrt{\frac{k^2}{c_1+k^2}-\frac{{\nu }^2}{4}}.$

(6)

It can be seen from equation (6) that the linear wave dispersive characteristics are influenced by the presence of the collisional parameter ν and nonextensive parameter q via c₁. The effect is shown in Figures 1(a) and (b). As stated in equation (6), Figure 1(a) shows the wave frequency ω versus wave number k. It is important to point out that the imaginary part of ω describes the wave damping rate, which is constant against k. From Figure 1(a), one can observe that the wave frequency (real part) decreases with the strength of the collisional parameter ν. Analytical evaluation of equation (6) reveals the existence of a threshold wave number, dependent on ν, below which the wave becomes overdamped and fails to propagate, specifically, for ν = 0.02 and ν = 0.05. Thus, it can be concluded that the wave propagates only upon satisfying the condition

$\mid k\mid >\sqrt{\frac{c_1{\nu }^2}{4-{\nu }^2}}\equiv k_{0d},$

(7)

which defines the critical wave number k_0d: an overdamped regime is shown below this threshold, where oscillatory behavior stops and the wave frequency and its derivatives turn purely imaginary. In the limit of weak ion collisions (i.e., ν ≪ ω _pi ), equation (6) becomes

$\omega =\sqrt{\frac{k^2}{c_1+k^2}-\frac{{\nu }^2}{4}}.$

(8)

Figure 1.

(a) A dispersion relation shows how the real part of the angular frequency ω for the nonextensive parameter q = 2.5 depends on the wave number k and (b) the threshold wavenumber k_0d, as given in equation (7), is plotted as a function of the collisional parameter ν for different values of the non-extensive index q.

It is necessary to note that the phase velocity approaches in the long wavelength limit, or for k < ω

$v_{ph}=\frac{\omega }{k}\approx \sqrt{\frac{1}{c_1}}.$

(9)

For k < k_0d, the phase speed vanishes, and the wave propagation becomes impossible. The analysis of propagating structures should therefore be limited to the region above k_0d.

Nonlinear analysis and the dKdV equation

Here, the RPT is applied to derive an evolutionary wave equation (EWE) that governs the nonlinear structures (electrostatic dissipative IASWs) in the current plasma model. According to this technique, the following stretched coordinates for the variables $\left(x,t\right)$ are introduced

$X={ϵ}^{\frac{1}{2}}\left(x-V_{ph}t\right)\text{\hspace{0.17em}\&\hspace{0.17em}}T={ϵ}^{\frac{3}{2}}t\text{,}$

(10)

where ϵ is a tiny perturbation parameter (0 < ϵ ≤ 1) measuring the wave amplitude and V _ph is the phase velocity (to be determined through compatibility conditions). The RPT applies to nonlinear waves propagating at velocities close to the IA sound velocity. Based on the stretching (10), the following differential operators are obtained

$\left\{\begin{array}{c}{\partial }_x={ϵ}^{\frac{1}{2}}{\partial }_X,\hfill \\ {\partial }_t={ϵ}^{\frac{3}{2}}{\partial }_T-V_{ph}{ϵ}^{\frac{1}{2}}{\partial }_X.\hfill \end{array}\right.$

(11)

The dependent plasma quantities $\left(u_i,n_i,\phi \right)$ , are expanded about their equilibrium values in a power series of ϵ using the RPT, as follows

$\left\{\begin{array}{c}{n}_i=1+ϵn_{i1}+{ϵ}^2{n}_{i2}+{ϵ}^3{n}_{i3}+\cdots ,\hfill \\ u_i=ϵu_{i1}+{ϵ}^2{u}_{i2}+{ϵ}^3{u}_{i3}+\cdots ,\hfill \\ \phi =ϵ{\phi }_1+{ϵ}^2{\phi }_2+{ϵ}^3{\phi }_3+\cdots .\hfill \end{array}\right.$

(12)

For a weak damping, we consider v = ϵ^3/2ν₀. Making use of equation (11) and (12), along with v = ϵ^3/2ν₀ (assuming weak damping) in system (1) and collecting the lowest order terms of ϵ, that is, coefficients of ϵ^3/2, we have

$n_{i1}=\frac{u_{i1}}{V_{ph}^2}\text{\&\hspace{0.17em}}{u}_{i1}=\frac{{\phi }_1}{v_{ph}}.$

(13)

Similarly, by collecting terms of order (ϵ) from Poisson equation, we get

$n_{i1}=c_1.$

(14)

By solving equations (13) and (14), the following expression is obtained

$V_{ph}=\frac{1}{\sqrt{c_1}}.$

(15)

Equation (15) determines the soliton speed with the IA speed, as defined in equation (9). Proceeding to the next order in ϵ, and eliminating the second order perturbation quantities in conjunction with equation (15), the following dKdV equation is obtained

${\partial }_Tϝ+Aϝ{\partial }_Xϝ+B{\partial }_X^3ϝ+Cϝ=0,$

(16)

with the following nonlinear, dispersion, and damping coefficients, respectively, as

$A=\left(\frac{3}{2v_{ph}}-\frac{c_2}{v_{ph}^3}\right),B=\frac{V_{ph}^3}{2}\text{\hspace{0.17em}\&\hspace{0.17em}}C=\frac{v_0}{2}\text{,}$

(17)

where ϝ ≡ φ₁.

Equation (16) governs the rate of increase of the leading-order electrostatic potential disturbance ϝ.

It is expected that coherent (soliton) structures are formed as a result of the mutual balance between dispersion and nonlinearity. Therefore, it is reasonable to look into how the nonextensive index (via the nonextensive parameter q) affects the nonlinearity and dispersion coefficients A and B. These behaviors are illustrated in the Figures 2(a) and (b), subsequently. It can be seen that, for higher values of the superextensivity (larger values of q), the nonlinear coefficient A increases in absolute value, whereas the dispersive coefficient B decreases. This imbalance between nonlinearity and dispersion implies that, under superextensive conditions, the system’s capacity to maintain a stable soliton structure is compromised. Consequently, an initially localized soliton-like pulse may lose its coherence, leading to energy dispersion into smaller-scale structures or the emergence of irregular oscillatory behavior.

Figure 2.

Variation of (a) the nonlinear coefficient A vs the non-extensive index q and (b) The dispersion coefficient B vs the non-extensive index q.

The dKdV equation (16) is known to be non-integrable, so an exact closed-form solution cannot be obtained in the presence of the damping coefficient “C.” For this reason, it has been studied using various approximate and quasi-analytical approaches in order to capture the nonlinear phenomena it describes. Among these, one of the most important results at weak damping, widely reported in the literature via the Ansatz method, is the following dissipative soliton solution^41–43

$ϝ(X,T){=ϝ}_0(T){\text{sech}}^2\left(\frac{\eta -\frac{A}{3}{{\int }_0^Tϝ}_0(t)dt}{L(T)}\right),$

(18)

where ϝ₀(T), $L\left(T\right)$ , and $U\left(T\right)$ are, respectively, the amplitude, width, and velocity of the dissipative solitons as time-dependent functions, which are given by

$\begin{array}{l}{ϝ}_0(T)=N_0\exp \left(-\frac{4C}{3}T\right),\hfill \\ L(T)=\sqrt{\frac{12B}{{Aϝ}_0(T)}}=\sqrt{\frac{12B}{A{N}_0}}\exp \left(\frac{2C}{3}T\right),\hfill \\ U\left(T\right)=\frac{A}{3}{\partial }_T{{\int }_0^Tϝ}_0(T)=U_0\exp \left(-\frac{4C}{3}T\right).\hfill \end{array}$

(19)

where N₀ = 3U₀/A and U₀ indicate the initial amplitude and velocity of the dissipative solitons, respectively.

The Tantawy technique (TT) for modeling ion-acoustic dissipative FKDV-solitons

This section provides a clear and detailed step-by-step tutorial on the TT for analyzing the damped FKdV equation. For more information about this method, we recommend reviewing Refs. 44–51. Before applying the TT to analyze fractional EWEs, it is essential to convert the integer-order dKdV equation (16) into damped FKdV. To achieve this, the approach outlined in Refs. 52,53 is employed to transform the dKdV equation (16) into its fractional version as follows

$D_T^{\delta }ϝ+A_dϝ{\partial }_Xϝ+B_d{\partial }_X^3ϝ+Cϝ=0,$

(20)

with the following initial condition (IC) to the damped soliton

$ϝ(X,0)\equiv {\mathrm{\Phi }}_0=N_0{\text{sech}}^2\left(\frac{X}{L}\right),$

(21)

where $L=\sqrt{4B/U_0}$ and N₀ = 3U₀/A, are, respectively, the width and amplitude of the undamped integer KdV-soliton and U₀ indicates the pulse speed. Here, $D_{\tau }^{\delta }$ represents the time fractional derivative operator (TFDO) in the framework of Caputo sense and 0 < δ < 1.

The TFDO in the framework of Caputo sense can be defined as follows^54,56

$D_T^{\delta }ϝ\left(X,T\right)=\left\{\begin{array}{c}\frac{1}{\mathrm{\Gamma }\left(m-\delta \right)}\underset{0}{\overset{T}{\int }}{\left(T-t\right)}^{m-\delta -1}{ϝ}^{\left(m\right)}\left(X,t\right)dt,\hspace{0.17em}\forall m-1<\delta \hspace{0.17em}<m\hspace{0.17em}\&\hspace{0.17em}m\in \mathbb{N},\hfill \\ {\partial }_T^mϝ\left(X,T\right)\text{,\hspace{0.17em}}\forall \delta =m\text{\hspace{0.17em}\&\hspace{0.17em}}m\in \mathbb{N}.\hfill \end{array}\right.$

(22)

According to the definition (22) the following relation is fulfilled

$D_T^{\delta }{T}^{\beta }=\frac{\mathrm{\Gamma }(\beta +1)}{\mathrm{\Gamma }(\beta +1-\delta )}{T}^{\beta -\delta }.$

(23)

To analyze and solve the damped FKdV equation (20) using the TT, the following fundamental procedures are outlined:

Step (1) This technique assumes the analytical approximate solution to any fractional differential equation in the subsequent Ansatz form

$ϝ={\mathrm{\Phi }}_0+\sum _{i=1}^{\infty }{T}^{i\delta }{ϝ}_i.$

(24)

where ${\mathrm{\Phi }}_0\equiv {\mathrm{\Phi }}_0\left(X\right)$ represents the any IC to equation (20) as a function of the spatial variable only “X” and ${ϝ}_i\equiv {ϝ}_i\left(X\right)$ are unknown functions that need to be determined.

• Step (2) By inserting the Ansatz (24) into equation (20) and for a finite number (e.g., m = 3) of the convergent approximation (24), we have

$\begin{array}{ll}\hfill & D_{\tau }^{\delta }\left({\mathrm{\Phi }}_0+T^{\delta }{ϝ}_1+T^{2\delta }{ϝ}_2+T^{3\delta }{ϝ}_3+\cdots \right)\hfill \\ \hfill & +A\left({\mathrm{\Phi }}_0+T^{\delta }{ϝ}_1+T^{2\delta }{ϝ}_2+T^{3\delta }{ϝ}_3+\cdots \right)\hfill \\ \hfill & \times \hspace{0.17em}{\partial }_X\left({\mathrm{\Phi }}_0+T^{\delta }{ϝ}_1+T^{2\delta }{ϝ}_2+T^{3\delta }{ϝ}_3+\cdots \right)\hfill \\ \hfill & +B{\partial }_X^3\left({\mathrm{\Phi }}_0+T^{\delta }{ϝ}_1+T^{2\delta }{ϝ}_2+T^{3\delta }{ϝ}_3+\cdots \right)\hfill \\ \hfill & +\hspace{0.17em}C\left({\mathrm{\Phi }}_0+T^{\delta }{ϝ}_1+T^{2\delta }{ϝ}_2+T^{3\delta }{ϝ}_3+\cdots \right)=0.\hfill \end{array}$

(25)

• Step (3) Utilizing definition (23) in equation (25) yields

$\begin{array}{ll}\hfill & K_0{ϝ}_1+K_1{ϝ}_2{T}^{\delta }+K_2{ϝ}_3{T}^{2\delta }+\cdots \hfill \\ \hfill & +A\left({\mathrm{\Phi }}_0+T^{\delta }{ϝ}_1+T^{2\delta }{ϝ}_2+T^{3\delta }{ϝ}_3+\cdots \right)\hfill \\ \hfill & \times \hspace{0.17em}{\partial }_X\left({\mathrm{\Phi }}_0+T^{\delta }{ϝ}_1+T^{2\delta }{ϝ}_2+T^{3\delta }{ϝ}_3+\cdots \right)\hfill \\ \hfill & +B{\partial }_X^3\left({\mathrm{\Phi }}_0+T^{\delta }{ϝ}_1+T^{2\delta }{ϝ}_2+T^{3\delta }{ϝ}_3+\cdots \right)\hfill \\ \hfill & +\hspace{0.17em}C\left({\mathrm{\Phi }}_0+T^{\delta }{ϝ}_1+T^{2\delta }{ϝ}_2+T^{3\delta }{ϝ}_3+\cdots \right)=0,\hfill \end{array}$

(26)

with

$K_i=\frac{\mathrm{\Gamma }\left(\left(i+1\right)\delta +1\right)}{\mathrm{\Gamma }\left(i\delta +1\right)}\hspace{0.17em}\forall \hspace{0.17em}i=\mathrm{0,1,2,3},\cdots .$

• Step (4) Collecting the coefficients of the same power of T ^iδ ∀ i = 1, 2, 3, ⋯ , yields

$P_0+P_1{T}^{\delta }+P_2{T}^{2\delta }+P_3{T}^{3\delta }+\cdots =0,$

(27)

with

$\begin{array}{l}\hfill P_0=K_0{ϝ}_1+A{\mathrm{\Phi }}_0{\mathrm{\Phi }}_0^{\prime }+B{\mathrm{\Phi }}_0^{\left(3\right)}+C{\mathrm{\Phi }}_0,\\ \hfill P_1=K_1{ϝ}_2+A\left({ϝ}_1{\mathrm{\Phi }}_0^{\prime }+{\mathrm{\Phi }}_0{ϝ}_1^{\prime }\right)+B{ϝ}_1^{\left(3\right)}+C{ϝ}_1,\\ \hfill P_2=K_2{ϝ}_3+A\left({ϝ}_2{\mathrm{\Phi }}_0^{\prime }+{\mathrm{\Phi }}_0{ϝ}_2^{\prime }+{ϝ}_1{ϝ}_1^{\prime }\right)+B{ϝ}_2^{\left(3\right)}+C{ϝ}_2,\\ \hfill & ⋮.\hfill \end{array}$

(28)

• Step (5) Equating the coefficients P _i ∀ i = 0, 1, 2, ⋯ , to zero and solving this system, we get the following implicit forms for ϝ₁, ϝ₂, and ϝ₃

$\begin{array}{ll}\hfill {ϝ}_1& =\frac{-1}{K_0}\left[\left(A{\mathrm{\Phi }}_0^{\prime }+C\right){\mathrm{\Phi }}_0+B{\mathrm{\Phi }}_0^{\left(3\right)}\right],\hfill \\ \hfill {ϝ}_2& =\frac{-1}{K_1}\left[\left(A{\mathrm{\Phi }}_0^{\prime }+C\right){ϝ}_1+A{\mathrm{\Phi }}_0{ϝ}_1^{\prime }+B{ϝ}_1^{\left(3\right)}\right],\hfill \\ \hfill {ϝ}_3& =-\frac{1}{K_3}\left[\left(A{\mathrm{\Phi }}_0^{\prime }+C\right){ϝ}_2+A{\mathrm{\Phi }}_0{ϝ}_2^{\prime }+A{ϝ}_1{ϝ}_1^{\prime }+B{ϝ}_2^{\left(3\right)}\right],\hfill \\ \hfill & ⋮.\hfill \end{array}$

(29)

• Step (6) By solving system (29) simultaneously together, the following explicit values for ϝ₁, ϝ₂, and ϝ₃, are obtained

$\begin{array}{ll}\hfill {ϝ}_1& =-\frac{1}{{\mathrm{\Gamma }}_1}\left(A{\mathrm{\Phi }}_0{\mathrm{\Phi }}_0^{\prime }+B{\mathrm{\Phi }}_0^{\left(3\right)}+C{\mathrm{\Phi }}_0\right),\hfill \end{array}$

(30)

$\begin{array}{ll}\hfill {ϝ}_2& =\frac{1}{{\mathrm{\Gamma }}_2}\left(A^2{\mathrm{\Phi }}_0^2{\mathrm{\Phi }}_0^{\prime \prime }+{\mathrm{\Phi }}_0{S}_1+B{S}_2\right),\hfill \end{array}$

(31)

$\begin{array}{ll}\hfill {ϝ}_3& =-\frac{1}{{\mathrm{\Gamma }}_3{\mathrm{\Gamma }}_1^2}\left(A^3{\mathrm{\Gamma }}_1^2{\mathrm{\Phi }}_0^3{\mathrm{\Phi }}_0^{\left(3\right)}+A^2{\mathrm{\Phi }}_0^2{S}_3+{\mathrm{\Phi }}_0{S}_4+B{S}_5\right),\hfill \end{array}$

(32)

where ${\mathrm{\Gamma }}_i=\mathrm{\Gamma }\left(i\delta +1\right)\forall i=\mathrm{1,2,3},\cdots$ , and the values of coefficients S₁, S₂, S₃, S₄, and S₅, are given in Appendix (I).

Step (7) The obtained values of ϝ₁, ϝ₂, and ϝ₃ given in equations (30)–(32) are general values for any IC “Φ₀” to the damped FKdV equation (20) and can be used to construct various approximations for different types of nonlinear structures (e.g., solitons, cnoidal waves, etc.) described by the damped FKdV equation (20). By considering the IC given in equation (21), we can generate the following approximation to the dissipative FKdV-soliton

$\begin{array}{ll}\hfill {ϝ}_1& =\frac{N_0{\text{sech}}^2\left(\mathrm{\Theta }\right)}{{\mathrm{\Gamma }}_1{L}^3}{Y}_1,\hfill \end{array}$

(33)

$\begin{array}{ll}\hfill {ϝ}_2& =\frac{N_0{\text{sech}}^2\left(\mathrm{\Theta }\right)}{{\mathrm{\Gamma }}_2{L}^6}{Y}_2,\hfill \end{array}$

(34)

$\begin{array}{ll}\hfill {ϝ}_3& =\frac{N_0{\text{sech}}^2\left(\mathrm{\Theta }\right)}{{\mathrm{\Gamma }}_2{L}^9}{Y}_3,\hfill \end{array}$

(35)

$\begin{array}{ll}\hfill & ⋮,\hfill \end{array}$

where Θ = X/L and the values of coefficients Y₁, Y₂, Y₃, ⋯ , are given in Appendix (II).

Step (8) By inserting the obtained values of ϝ₁, ϝ₂, and ϝ₃ into the convergent Ansatz (24), the following analytical dissipative FKdV-soliton approximation is obtained

$\begin{array}{ll}\hfill ϝ& ={\mathrm{\Phi }}_0+{ϝ}_1{T}^{\delta }+{ϝ}_2{T}^{2\delta }+{ϝ}_3{T}^{3\delta }+\cdots \hfill \\ \hfill & =N_0{\text{sech}}^2\left(\mathrm{\Theta }\right)+\frac{N_0{\text{sech}}^2\left(\mathrm{\Theta }\right)}{{\mathrm{\Gamma }}_1{L}^3}{Y}_1{T}^{\delta }\hfill \\ \hfill & +\frac{N_0{\text{sech}}^2\left(\mathrm{\Theta }\right)}{{\mathrm{\Gamma }}_2{L}^6}{Y}_2{T}^{2\delta }+\frac{N_0{\text{sech}}^2\left(\mathrm{\Theta }\right)}{{\mathrm{\Gamma }}_2{L}^9}{Y}_3{T}^{3\delta }+\cdots .\hfill \end{array}$

(36)

Parametric investigation

The primary objective of this work is to investigate how dissipation, or the collisional parameter, affects pulse’s structural behavior. To accomplish this, consider dissipation as negligible by taking the limit v₀ → 0. In this case, equation (16) simplifies to the KdV equation, which admits a solitary wave solution of the form

$ϝ(X,T)=N_0{\text{sech}}^2\left(\frac{\xi -U_{0}T}{L}\right).$

(37)

The dissipative effect on pulse-type IASWs will be analyzed using the solitary wave solution provided in equation (37) as an initial condition.

The numerical analysis begin by examining the stability of a single pulse traveling in a highly superextensive plasma (q = 2.5). Assuming no dissipation, that is, ν₀ = 0, the soliton solution provided in equation (37) is utilized as the initial condition in this instance. While considering the impact of dissipation by taking ν₀ = 0.02 and q = 2.5. The graph reveals that the pulse’s amplitude gradually drops as it travel over time as shown in Figures 3(a) and (b), respectively. Moreover, the pulse amplitude decreases gradually with increasing both the nonextansive parameter q and the collisional parameter ν₀. Physically, this could be attributed to the fact that solitary pulses lose energy over time and, with increasing collision rate in the plasma, during propagation. As the amplitude weakens, the nonlinear steepening becomes less effective in balancing dispersion, causing the pulse to broaden and causes its width to increase with time. Thus dissipation causes the amplitude of the solitary pulses to decrease and their width to change over time, unlike the stable solitons found in collisionless plasmas. Moreover, the superextensive character of the electrons (q = 2.5) modifies the dispersion, makes the wave more sensitive to damping, meaning amplitude decays faster. In second numerical plotting, it can be observe that the soliton amplitude in a moderately extensive plasma (q = 2.0) decreases over time as it propagates in the dissipative plasma medium ν₀ = 0.02 (as shown in Figure 3). With pulse solutions for ν₀ = 0 and q = 1 as boundary conditions, the identical situation for a quasi-Maxwellian plasma (q = 1) has also been examined (see blue dotted curve). As seen in Figure 3, the properties of the ion-acoustic pulses are considerably altered, and it is found that as they travel through the dissipative plasma, their amplitude (width) decreases (increases). Compared to its counterpart in superextensive (q = 2.5) plasma, the pulse in this quasi-Maxwellian scenario has a greater amplitude and the narrowest width, which can be seen in Figure 3.

Figure 3.

The profile of dissipative IASWs as described by equation (18) is examined against $\left(T,q,{\nu }_0\right)$ : (a) the profile against $\left(T,q\right)$ with ν₀ = 0.05 and (b) the profile against $\left(T,{\nu }_0\right)$ with q = 2.

Figures 4(a) and (b), illustrate, respectively, the time evolution of dissipative soliton amplitude ϝ₀(T) as given in equation (19) and its width L(T) as given in equation (19) for some values of the nonextensive parameter q. The parameters used are U₀ = 0.5 and ν₀ = 0.02. Both figures show that the amplitude and width for the superextensive plasma case q = 2.5 are smaller than that of q = 1.0 (Maxwellian case). Also, for both cases, the dissipative soliton amplitude ϝ₀(T) decays, while the soliton width L(T) increases, which aligns with the expected behavior because narrower solitons are always taller, as indicated by the analytical solution. Remarkably, the characteristic decay time of the soliton turns out to be independent of q and relies solely on the dissipative coefficient ν₀, consistent with analytical predictions.

Figure 4.

The amplitude ϝ₀(T) and width L(T) of the dissipative IASWs versus $\left(T,q\right)$ for v₀ = 0.02: (a) the amplitude ϝ₀(T) and (b) the width L(T).

The introduction of fractional dynamics via the damped FKdV equation adds an additional layer of complexity and realism, particularly in plasmas, where memory and nonlocal transport are essential for accurately describing the characteristic dynamics of moving nonlinear waves. The TT is applied to the damped FKdV equation with an initial solitary wave condition, yields higher-order approximate solutions that describe fractional dissipative solitons as a time-fractional series. To understand the dynamic behavior of fractional dissipative solitons in the current plasma model and how memory influences these waves, the approximation (36) is numerically investigated against the fractional parameter δ, as shown in Figures 5 and 6 for C = 0.02 and C = 0.05, respectively. The numerical evaluation of these approximations shows that the fractional order parameter δ significantly affects the dissipative soliton profile, especially at longer times. Moreover, it is observed that as δ increases from values much less than unity toward the integer limit, the amplitude of the fractional dissipative soliton decreases more rapidly, and its spatial profile becomes smoother and more diffused. This behavior indicates that more substantial fractional effects (larger δ in the present formulation) can enhance effective damping or dispersive spreading, thereby capturing anomalous attenuation patterns that classical integer-order models cannot reproduce. This behavior could not detect using the integer solutions to traditional differential equations. Therefore, this result may explain some of the anomalous behavior observed in experiments or space observations that are inconsistent with the exact theoretical solutions. To verify the accuracy of the derived approximations, a graphical comparison between the fractional approximations (36) and the solution (18) is examined, as shown in Figure 7. Additionally, the maximum residual errors $E\left(X,T\right)$ to the second-order (i.e., sϝ₃ = 0) and third-order (i.e., ϝ₃ ≠ 0) approximations (36) in the domain $\mathrm{\Omega }\in \left[X_i,X_f\right]\times \left[T_i,T_f\right]=\left[-\mathrm{15,15}\right]\times \left[\mathrm{0,10}\right]$ are, respectively, calculated as follows

$\left\{\begin{array}{c}{\left.\mathcal{E}\left(X,T\right)\right|}_{\delta =0.1}={\max }_{\mathrm{\Omega }}\left|D_T^{\delta }ϝ+A_dϝ{\partial }_Xϝ+B_d{\partial }_X^3ϝ+Cϝ\right|=0.0000965193,\hfill \\ {\left.\mathcal{E}\left(X,T\right)\right|}_{\delta =0.5}={\max }_{\mathrm{\Omega }}\left|D_T^{\delta }ϝ+A_dϝ{\partial }_Xϝ+B_d{\partial }_X^3ϝ+Cϝ\right|=0.000647341,\hfill \\ {\left.\mathcal{E}\left(X,T\right)\right|}_{\delta =1}={\max }_{\mathrm{\Omega }}\left|D_T^{\delta }ϝ+A_dϝ{\partial }_Xϝ+B_d{\partial }_X^3ϝ+Cϝ\right|=0.00445249,\hfill \end{array}\right.$

(38)

and

$\left\{\begin{array}{c}{\left.\mathcal{E}\left(X,T\right)\right|}_{\delta =0.1}={\max }_{\mathrm{\Omega }}\left|D_T^{\delta }ϝ+A_dϝ{\partial }_Xϝ+B_d{\partial }_X^3ϝ+Cϝ\right|=0.0000901291,\hfill \\ {\left.\mathcal{E}\left(X,T\right)\right|}_{\delta =0.5}={\max }_{\mathrm{\Omega }}\left|D_T^{\delta }ϝ+A_dϝ{\partial }_Xϝ+B_d{\partial }_X^3ϝ+Cϝ\right|=0.000857029,\hfill \\ {\left.\mathcal{E}\left(X,T\right)\right|}_{\delta =1}={\max }_{\mathrm{\Omega }}\left|D_T^{\delta }ϝ+A_dϝ{\partial }_Xϝ+B_d{\partial }_X^3ϝ+Cϝ\right|=0.00489437.\hfill \end{array}\right.$

(39)

Figure 5.

The fractional dissipative solitons approximation ϝ(X, T) (36) is examined: (a) 3D-ϝ(X, T) at δ = 0.1, (b) 3D-ϝ(X, T) at δ = 0.7, (c) 3D-ϝ(X, T) at p = 1, and (d) 2D-ϝ(X, 10) at different values for δ. Here, q = 2.5, U₀ = 0.1, and C = 0.02.

Figure 6.

The fractional dissipative solitons approximation ϝ(X, T) (36) is examined: (a) 3D-ϝ(X, T) at δ = 0.1, (b) 3D-ϝ(X, T) at δ = 0.7, (c) 3D-ϝ(X, T) at p = 1, and (d) 2D-ϝ(X, 10) at different values for δ. Here, q = 2.5, U₀ = 0.1, and C = 0.1.

Figure 7.

The generated approximation (36) at δ = 1 is compared to the approximate solution (18) for the integer case δ = 1. Here, q = 2.5, U₀ = 0.1, and C = 0.02.

Moreover, the absolute error to the second-order (i.e., ϝ₃ = 0) and third-order (i.e., ϝ₃ ≠ 0) approximations is numerically estimated, as presented in Tables 2 and 3 for C = 0.02 and C = 0.1, respectively

$\begin{array}{ll}\hfill L_2& ={\left|\text{Approx.\hspace{0.17em}}\left(36\right)-\text{solution\hspace{0.17em}}(18\hspace{0.17em})\right|}_{{ϝ}_3=0},\hfill \\ \hfill L_3& ={\left|\text{Approx.\hspace{0.17em}}\left(36\right)-\text{solution\hspace{0.17em}}(18\hspace{0.17em})\right|}_{{ϝ}_3\ne 0}\text{.}\hfill \end{array}$

Table 2.

The absolute error for second-order approximation L₂ and third-order approximation L₃ as given in equation (36) at δ = 1. Here, q = 2.5, U₀ = 0.1, and C = 0.02.

X	2 nd − Approx. (36)	3 rd − Approx. (36)	Solution (18)	10−4 × L2	10−4 × L3
−15	0.0000297148	0.0000297148	0.0000305143	0.00799463	0.00799466
−12	0.000228718	0.000228718	0.000233596	0.0487775	0.0487777
−9	0.00175469	0.00175469	0.00178229	0.275997	0.275999
−6	0.0131281	0.0131281	0.0132585	1.30342	1.30343
−3	0.0819984	0.0819984	0.0822897	2.91281	2.91289
0	0.202583	0.202583	0.202448	1.34981	1.34992
3	0.0828613	0.0828613	0.0831508	2.89529	2.89534
6	0.013302	0.013302	0.0134331	1.31099	1.31099
9	0.00177863	0.00177863	0.00180647	0.278451	0.278451
12	0.00023185	0.00023185	0.000236778	0.0492742	0.0492741
15	0.0000301219	0.0000301219	0.0000309301	0.0080818	0.0080818

Table 3.

The absolute error for second-order approximation L₂ and third-order approximation L₃ as given in equation (36) at δ = 1. Here, q = 2.5, U₀ = 0.1, and C = 0.1.

X	2 nd − Approx. (36)	3 rd − Approx. (36)	Solution (18)	10−4 × L2	10−4 × L3
−15	0.0000294781	0.000029478	0.0000336383	0.0416028	0.041603
−12	0.000226896	0.000226896	0.000251998	0.251019	0.251021
−9	0.00174071	0.00174071	0.00188115	1.40436	1.40438
−6	0.0130236	0.0130236	0.0136787	6.55103	6.55116
−3	0.0813472	0.0813471	0.082788	14.4075	14.4085
0	0.200969	0.200969	0.2003	6.6867	6.68695
3	0.0821989	0.0821989	0.083631	14.3205	14.3204
6	0.0131959	0.0131959	0.0138548	6.5883	6.58831
9	0.00176445	0.00176445	0.00190612	1.41667	1.41667
12	0.000230003	0.000230003	0.000255357	0.253542	0.253542
15	0.0000298819	0.0000298819	0.000034087	0.0420509	0.0420509

It should be noted that the absolute error is computed by comparing the generated approximate solution (36) with the approximate solution (18), as this problem is completely non-integrable and has no exact solution. The analysis results showed that the derived approximations are characterized by high accuracy and more stable across the whole study domain. This, in turn, enhances the efficiency of the used technique. Consequently, these results give us confidence in analyzing a wider range of more complex problems by accounting for various physical effects, such as wave curvature. This is one of the significant challenges facing many researchers. However, the promising results of the technique demonstrate its ability to overcome all the obstacles researchers encounter in analyzing such complex problems.

From a physical standpoint, fractional analysis clarifies anomalous behaviors of nonlinear plasma waves observed in laboratory and space experiments, in which recorded damping rates, spatial profiles, or relaxation times diverge from predictions based on classical local models. In these cases, memory effects, nonlocal interactions, and complex collisional processes can significantly alter the medium’s response in time and space. The damped FKdV framework is a simple way to combine these effects. The present results indicate that fractional dynamics can adjust both the rate and pattern of soliton decay and broadening, offering additional degrees of freedom to reconcile theory with observations. For example, the observed sensitivity of the fractional soliton profile to δ suggests that appropriately chosen fractional orders can mimic the impact of unresolved microphysical processes, such as turbulence or correlated collisions, on large-scale nonlinear structures.

Finally, the TT is quite flexible in practice: it can be applied to analyze a broad range of nonlinear fractional evolutionary differential equations without any challenges. Additionally, this technique is distinguished by its superior ability to analyze a wide range of fractional evolutionary differential equations at minimal computational cost, without requiring high-performance equipment.

Conclusions

The study has examined the propagation of low-frequency dissipative IAWs in an unmagnetized, collisional plasma composed of inertial ions and nonextensive electrons, with dissipation arising from ion-neutral collisions. Starting from a fluid model, the linear analysis established a dispersion relation in which the collisional frequency ν sets the damping rate and, together with the nonextensive parameter q, determines a threshold wavenumber below which waves become overdamped and fail to propagate. In the weakly nonlinear regime, the RPT has been applied to derive an integer damped KdV (dKdV) equation whose coefficients explicitly depend on electron nonextensivity, thereby linking the microscopic electron distribution to the macroscopic properties of dissipative IASWs. The obtained important results can be summarized in the following points:

The analytical solution obtained for the integer-order dKdV equation, under the assumption of weak damping, shows that the dissipative soliton amplitude and velocity decay with time while the width increases, illustrating the characteristic broadening of dissipative solitary waves. Both analytical expressions and numerical simulations demonstrate that the decay rate is governed solely by the dissipative parameter v and is independent of the nonextensive index q. However, the nonextensive index q controls the soliton’s initial amplitude and width.

It was observed that quasi-Maxwellian plasmas (q = 1) support taller and narrower pulses that remain more robust against dissipative spreading, while superextensive cases (q ≈ 2.5) yield lower-amplitude pulses. These broader structures are more sensitive to damping, with intermediate values of the nonextensive index q exhibiting behavior between these two limits.

To incorporate memory and nonlocal effects, the dKdV equation has been transformed into a damped fractional KdV (FKdV) equation by introducing a Caputo time-fractional derivative. Thereafter, the TT has been applied to analyze this model and construct higher-order analytical approximations for fractional dissipative solitons with the help of the initial soliton solution of the integer undamped KdV equation.

The resulting solutions highlight a marked sensitivity of the fractional dissipative soliton profile to the fractional order parameter δ. It was found that as the fractionality δ increases, the amplitude of the fractional dissipative soliton shrinks and its spatial structure becomes more diffused, especially at longer times. Analyses of residual and absolute errors indicated that the derived fractional approximate solutions are highly accurate and stable across the parameter ranges investigated, confirming the effectiveness of the TT for analyzing fractional nonlinear wave problems and other complex problems that were previously difficult to diagnose, particularly for junior physicists.

These results suggest that fractional modeling, combined with nonextensive electron statistics and collisional dissipation, provides a flexible and physically meaningful framework for describing nonlinear dissipative IA structures in realistic plasmas. In particular, the approach can help interpret nonlinear wave damping and energy transport in laboratory plasmas and in space environments such as the ionosphere and solar wind, where non-Maxwellian, nonextensive electron populations and collisional or memory effects are often significant. The demonstrated ability of the damped FKdV model and the TT to reproduce subtle amplitude–width evolutions and anomalous damping patterns motivates their application to more complex geometries and additional physical effects, such as magnetic fields, curvature, or multi-species compositions, in future work.

Future work: Many early-career researchers find it difficult to work with stiff, nonlinear evolution equations, even though such equations are central to modeling a broad spectrum of physical and engineering systems. The mathematical complexity often leads to severe technical obstacles in both analysis and computation. To make progress, it is common practice to introduce simplifying assumptions or constraints that render the governing equations more tractable. Yet these idealizations can inadvertently create a gap between theoretical predictions and empirical observations in experiments or real-world applications.

When reliable analytical or semi-analytical tools are available that allow one to solve these equations without sacrificing essential physical effects, it is important to use them so that the problem can be treated in a more realistic setting. For instance, in nonlinear wave studies, many media (involving plasmas) are better described in curved (nonplanar) geometries than in strictly planar ones. Thus, incorporating curvature effects into the present problem, or any other problems, would naturally lead to a nonplanar/damped FKdV equation. This problem is again stiff and fully non-integrable, and its detailed analysis poses a substantial challenge. A promising direction for future work is to apply TT to such nonplanar, damped FKdV models. This approach offers a flexible framework that can be adapted to a wide class of FDEs, making it accessible to researchers who may not be specialists in advanced nonlinear analysis. By relying on a method that accounts for the full complexity of the model, including curvature and dissipation, one can aim for theoretical descriptions that remain closely aligned with experimental reality, rather than relying on overly restrictive simplifying assumptions.