Front. Phys. Frontiers in Physics Front. Phys. 2296-424X Frontiers Media S.A. 1604640 10.3389/fphy.2025.1604640 Physics Original Research Elzaki homotopy perturbation method for modeling fractional ion-acoustic solitary and shock waves in a non-maxwellian plasma Alhejaili et al. 10.3389/fphy.2025.1604640 Alhejaili Weaam 1 Khan Adnan 2 Al-Johani Amnah S. 3 El-Tantawy Samir A. 4 5 * 1 Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia 2 Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Pakistan 3 Mathematics Department, Faculty of Science, University of Tabuk, Tabuk, Saudi Arabia 4 Department of Physics, Faculty of Science, Al-Baha University, Al-Baha, Saudi Arabia 5 Department of Physics, Faculty of Science, Port Said University, Port Said, Egypt

Edited by: Chun-Hui He, Xi’an University of Architecture and Technology, China

Reviewed by: Abdelhalim Ebaid, University of Tabuk, Saudi Arabia

Liaqat Ali, Southern University of Science and Technology, China

 *Correspondence: Samir A. El-Tantawy, tantawy@sci.psu.edu.eg, samireltantawy@yahoo.com 02 09 2025 2025 13 1604640 04 04 2025 18 07 2025 Copyright © 2025 Alhejaili, Khan, Al-Johani and El-Tantawy. 2025 Alhejaili, Khan, Al-Johani and El-Tantawy

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It is known that the family of nonlinear Korteweg-de Vries-type (KdV) equations is widely used in modeling many realistic phenomena that occur in nature, such as the propagation of solitons, shock waves, multiple solitons, cnoidal waves, and periodic waves in seas and oceans, plasma physics, fluid mechanics, and electronic circuits. Motivated by these applications, we proceed to analyze the time-fractional forms of this family, including the planar quadratic nonlinear fractional KdV (FKdV) and planar cubic nonlinear fractional modified KdV (FmKdV) using Elzaki Homotopy perturbation method (HPTM). By implementing this method, we can derive some highly accurate approximations to both FKdV and FmKdV equations. Using the suggested method, the nonlinear planar FKdV equation is solved and analytical FKdV-soliton approximation is obtained. For the nonlinear planar FmKdV equation, two general formulas are derived depending on the polarity of the cubic nonlinearity coefficient “ C ”. At C > 0 , the mKdV-soliton is used as an initial solution, and an analytical FmKdV-soliton approximation is generated. On the contrary, for C < 0 , the nonlinear planar FmKdV equation does not support solitons but instead supports shock waves. Using the suggested approach, a general formula for the FmKdV-shock wave approximation is derived. As a practical application of the derived approximations, the fluid-governed equations of a collisionless and unmagnetized plasma composed of inertial cold ions and inertialess Cairns-Tsallis distributed electrons are reduced to both the FKdV and the FmKdV equations to study the properties of fractional ion-acoustic waves and gain a deeper understanding of their dynamical behavior. The derived approximations for both nonlinear planar FKdV and FmKdV equations are not limited to plasma physics and its applications but extend to the simulation of many nonlinear phenomena described by these equations, as the derived approximations are general.

 Elzaki transform caputo operator homotopy perturbation method nonlinear fractional (modified) KdV equations fractional solitons and shock waves a non-maxwellian plasma PNURSP2025R229 Princess Nourah Bint Abdulrahman University10.13039/501100004242 section-at-acceptance Interdisciplinary Physics 1 Introduction

Fractional differential equations are a generalization of integer differential equations. Thus, fractional calculus (FC) is a more comprehensive version of classical integer-order calculus. FC investigates integrals and derivatives of fractional order [1]. Over the past 30 years, fractional calculus has been regarded as a valuable tool for addressing sustainable and complex issues due to its numerous benefits, including nonlocality, heritability, high dependability, and analyticity. FC was employed extensively and effectively to characterize a wide range of phenomena that arise in various fields, including engineering, physics, economics, and science [2–7]. Many physical systems can be more precisely represented by the formulation of fractional derivatives, as evidenced by recent investigations [8–11]. As a result, FDEs are widely used in many fields like physics (e.g., interstellar matter’s ability to absorb light) [12], chemistry [13, 14], biology [15], water treatment model [16], modeling COVID-19 pandemic [17], science and engineering [18], and many other applications [19–24] including visco-elasticity, electrical circuits, fractional multipoles, electroanalytical chemistry, entropy theory, image processing, fluid mechanics, and modeling plasma waves [25–28].

Solving nonlinear fractional differential equations (FDEs) presents computing challenges due to the nonlocal characteristics of fractional derivatives. Many researchers have recently examined FDEs from various perspectives. They have developed and used numerical simulation techniques as part of their study to solve these equations and accurately predict their behavior [29–31]. Consequently, numerous practical approaches employed to investigate FDEs, including the Adomian decomposition method (ADM) [32, 33], Variational Iteration Method (VIM) [34], Spectral Method [35], Homotopy analysis method (HAM) [36], new iterative method (NIM) [37, 38], Differential transform method [39, 40], residual power series method [41–43], Chebyshev plynomial method [44], Haar wavelet collocation method [45], Homotopy perturbation method (HPM) [46–50], the Tantawy Technique [25–27], among others [51, 52]. All of these methods have effectively produced approximations for various types of fractional differential equations, which have also proven successful in modeling a broad range of physical and engineering phenomena.

Nonlinear physical systems have significantly advanced the study of nonlinear equations for traveling wave solutions. Nonlinear wave dynamics have been studied in many scientific and engineering domains. These include hydrodynamics, solid-state physics, fiber optics, geological sciences, and plasma physics. Nonlinear wave theory is a recent mathematical study that often investigates asymptotic conditions (e.g., fluctuating over several scales, significant amplitude, high frequency) that are not readily accessible by numerical simulations. In addition, nonlinear wave theory is crucial to investigating actual water waves, light-matter interactions, optical fiber transmission, earthquakes, galaxy formation, traffic flow, and the steepening of short gravity waves over long wave crests. The Korteweg–de Vries (KdV)-type equations and their family are among the most significant evolutionary wave equations, extensively utilized to explain and model various nonlinear structures that arise and propagate in many physical and engineering systems. For instance, this family were used for modeling nonlinear structures in various practical fields, including electronic circuits [53, 54], fluid mechanics [55, 56], shallow water waves [57–60], plasma physics [61], and many others. For example, in the framework of the planar KdV equation, the overtaking collisions of Alfvén solitons have been investigated in a low beta collisionless magnetoplasma composed of electron and ion fluids [62]. Also, the propagation of nonlinear electron-acoustic CWs (EACWs) in a homogeneous magnetoplasma comprising fluid cold electrons and inertialess nonthermal electrons, as well as stationary ions, has been investigated in the framework of the planar KdV equation [63]. Moreover, the non-fractional form of this equation has been used to analyze many other phenomena that propagate in various plasma models, whether directly, such as describing phenomena that propagate at the phase velocity [64], or indirectly, such as describing the wave that propagates at the group velocity (e.g., dark solitons), by transforming it to the nonlinear Schrödinger equation (NLSE) [65]. The integer-order forms of this family has been widely used to study the propagation and interaction of solitary waves (SWs) and cnoidal waves (CWs) in various plasma models. However, some theoretical results obtained using the integer forms of these equations may differ slightly from some observed data. Thus, one way to overcome this deviation is to treat these phenomena in fractional forms. Therefore, in this work, we focus our efforts on analyzing this family in its fractional form using some effective methods, which may reveal the mystification surrounding specific experiments or space observations. The general forms for nonlinear time-fractional quadratic nonlinearity KdV equation [26] and cubic nonlinearity modified KdV (mKdV) equation [27, 66, 67] are, respectively, given by D t p B + A B ∂ η B + B ∂ η 3 B = 0 , and D t p B + C B 2 ∂ η B + B ∂ η 3 B = 0 , where B ≡ B η , t , and B indicates the coefficient of the dispersion term, whereas A and C represent the coefficients of the quadratic and cubic nonlinear terms, respectively. These coefficient depend on the physical model under study, such as the plasma model, which will be discussed later as one of the applications of this family.

The goal of the study is to analyze the fractional planar nonlinear KdV-type equations, including quadratic nonlinearity planar fractional KdV (FKdV) Equation 1 and cubic nonlinearity planar fractional mKdV (FmKdV) Equation 2 and derive some analytical approximations to model nonlinear ion-acoustic waves (IAWs) in a collisionless, unmagnetized plasma composed of inertial cold ions and inertialess Cairns-Tsallis distributed electrons [68–70]. It is well-known that FKdV Equation 1 does not support shock waves, but it does support solitary and periodic waves. In the current study, we will focus on fractional solitary waves (SWs), with the possibility of also studying fractional periodic waves, as we will derive a general formula for the fractional approximation as a function of the initial solution. Through this formula, fractional periodic waves can also be studied. On the other hand, the FmKdV Equation 2 can support both solitary and shock waves, depending on the sign of the cubic nonlinearity coefficient “ C ”. For more clarification, the FmKdV Equation 2 can be divided into two parts as follows:

• For fractional soliton:

D t p B + C B 2 ∂ η B + B ∂ η 3 B = 0 , with the initial condition (IC) B η , 0 = B max s e c h η w , where B max = 6 u 0 / C and w = B / u 0 are, respectively, the peak KdV-soliton amplitude and width.

• For fractional shock waves:

 D t p B − C B 2 ∂ η B + B ∂ η 3 B = 0 , With the IC B η , 0 = B max ⁡ tanh η w , where B max = 3 u 0 / C and w = 2 B / u 0 are, respectively, the peak mKdV-shock wave amplitude and width.

Now, Elzaki HPM (EHPM) can be implemented for analyzing these fractional Equation 1 in order to model the IAWs in the mentioned plasma model. Note that this approach is considered a combination between Elzaki transform (ET) [71] and the conventional HPM [72, 73]. EHPTM, a synthesis of ET and HPM, was first employed by Mohamed et al. [74] to solve initial value problems both analytically and numerically. Based on the numerous applications of this approach and its efficacy in analyzing various evolutionary wave equations (EWEs), thus, this method will be employed to investigate the different types of fractional IAWs (fractional solitary and shock waves) inside the aforementioned plasma model.

 2 Preliminaries

Here, we briefly overview a few fractional calculus concepts, traits, and results.

The Riemann–Liouville’s (RL) fractional integral operator is expressed as [75, 76] D t − p B η , t ≡ J t p B η , t = 1 Γ p ∫ 0 t t − τ p − 1 B η , τ d τ , 0 < p ≤ 1 , J t 0 B η , t = B η , t with the following properties J t p t m = Γ m + 1 Γ m + p + 1 t m + p   ∀ m > − 1 , D t p t m = Γ m + 1 Γ m − p + 1 t m − p , where m ∈ N + & p ∈ R + .

The Caputo fractional derivative operator (FDO) is expressed as [75, 76] D t p B η , t = 1 Γ m − p ∫ 0 t ∂ τ m B η , τ t − τ p + 1 − m d τ , m − 1 < p < m , ∂ τ m B η , τ , m = p , with the following properties J t p D t p B η , t = B η , t − ∑ k = 0 m − 1 ∂ τ k B η , 0 + t k k ! , f o r t > 0 , a n d m − 1 < p ≤ m ∈ N . D t p J t p B η , t = B η , t .

Elzaki transform (ET) for the function B η , t is expressed as [77] E B η , t = P u = u ∫ 0 ∞ B η , t e − t u d t , t ≥ 0 .

If E B η , t = P ( u ) then ET to the Caputo FDO reads [78] E D τ p B η , t = u − p P u − ∑ i = 0 m − 1 u 2 − p + k ∂ t i B η , t t = 0 , ∀ m − 1 < p < m .

3 Elzaki homotopy perturbation method (EHPM) for analyzing FPDES

Here, EHPM is employed for analyzing the following general FPDE: D t p B = F + G , ∀ 0 < p ≤ 1 , with the initial condition (IC) B η , 0 = B 0 η , where D t p = ∂ p ∂ t p symbolizes the Caputo FDO of order p and B ≡ B η , t , while F ≡ F B and G ≡ G B symbolize the linear and nonlinear terms, respectively.

To analyze problem (12) using the EHPM, the following brief points are introduced:

Step 1: Taking ET to Equation 12 yields E D t p B = E F + G .

Step 2: Using ET to the Caputo FDO as given in Equation 11 in Equation 13, we get u − p E B − ∑ k = 0 m − 1 u 2 − p + k ∂ t i B η , t t = 0 = E F + G , which leads to E B = ∑ k = 0 m − 1 u 2 + k ∂ t i B η , t t = 0 + u p E F + G , or P u = u 2 B 0 η + u p E F + G ,

Step 3: Taking the inverse ET to Equation 16 implies B = E − 1 u 2 B 0 η + E − 1 u p E F + G = B 0 η + E − 1 u p E F + G .

Step 4: The approximate solution according to the HPM is given by the following convergent series solution: B = ∑ m = 0 ∞ ε m B m , where ε ∈ [ 0,1 ] represents the homotopy parameter.

Step 5: In the following manner, the nonlinear term is decomposed G = ∑ m = 0 ∞ ε m H m B , where H m indicates He’s polynomials and it is defined by H m B = 1 m ! ∂ m ∂ ε m G ∑ m = 0 ∞ ε m B m ε = 0 , where m = 0,1,2 , ⋯   .

Step 6: Inserting Equations 18, 19 into Equation 17 yields ∑ m = 0 ∞ ε m B m = B 0 η + ε E − 1 u p E F ∑ m = 0 ∞ ε m B m + ∑ m = 0 ∞ ε m H m B .

Step 7: Collecting the coefficients of various order of ε implies ε 0 : B 0 = B 0 η , 0 , ε 1 : B 1 = E − 1 u p E F B 0 + H 0 B , ε 2 : B 2 = E − 1 u p E F B 1 + H 1 B , ⋮ ε m : B m = E − 1 u p E F B m − 1 + H m − 1 B , m > 0 , m ∈ N ,

Step 8: For ε → 1 , the following convergent approximate solution is obtained B = B 0 + B 1 + B 2 + ⋯ = lim ε → 1 ∑ m = 0 ∞ ε m B m .

4 Plasma applications and test examples

This section is considered for examining and analyzing some fractional EWEs, such as the planar FKdV and FmKdV equations, which are critical differential equations for analyzing various nonlinear phenomena in numerous physical systems, including fluids, optical fibers, communications, seawater, oceans, and plasma physics, which is characterized by a plethora of nonlinear phenomena. Here, we apply EHPTM to analyze the proposed models and attempt to derive highly accurate analytical approximations for these models.

4.1 Fluid plasma model

Since both quadratic nonlinearity KdV and cubic nonlinearity mKdV equations are among the essential EWEs that are widely used to study various nonlinear phenomena (such as solitons, cnoidal waves, shock waves, and so on) in various plasma systems, thus, we can take a realistic application model of a multicomponent plasma and then derive these equations by employing the reductive perturbation technique (RPT). For this purpose, we consider the propagation of nonlinear ion-acoustic waves (IAWs) in a collisionless, unmagnetized plasma composed of inertial cold ions and inertialess Cairns-Tsallis distributed electrons. In this model, the ion mass is responsible for providing inertia, while the electron thermal pressure is responsible for providing the restoring force. The fluid-governed equations in the normalized form are given by [68–70]. ∂ τ n + ∂ x n u = 0 , ∂ τ u + u ∂ x u + ∂ x ϕ = 0 , ∂ x 2 ϕ − n e + n = 0 .

In this context, n and n e represent the normalized number densities of ions and electrons, respectively, u denotes the normalized ion fluid velocity, ϕ signifies the normalized electrostatic potential, and the symbols x and τ refer to the normalized spatial and temporal variables, respectively.

The normalized number density of the electrons according to the Cairns-Tsallis distribution reads n e = 1 + a 1 ϕ + a 2 ϕ 2 1 + q − 1 ϕ q + 1 2 q − 1 , = α 0 + α 1 ϕ + α 2 ϕ 2 + α 3 ϕ 3 + ⋯   , with a 1 = − 16 q α 3 − 14 q + 15 q 2 + 12 α ,  a 2 = a 1 1 − 2 q , α 1 = a 1 + 1 2 1 + q ,  α 2 = 1 8 3 + 8 a 2 + 2 q − q 2 + 4 a 1 1 + q , α 3 = 1 + q 48 15 + 24 a 2 − 14 q + 3 q 2 − 6 a 1 − 3 + q .

The acceptable physical values of the parameters q , α read: 0.6 < q ≤ 1 and 0 ≤ α < 0.25 .

The reductive perturbation technique is utilized to examine the propagation of nonlinear electrostatic waves in the current plasma model. According to this technique, the independent space-time variables x , τ are stretched as: η = ε 1 2 x − λ τ & t = ε 3 2 τ , while the dependent quantities Φ ≡ n , u , ϕ are expanded as follows: Φ = Φ 0 + ∑ m = 1 ∞ ε m Φ m , where Φ 0 ≡ 1,0,0 and Φ m ≡ n m , u m , ϕ m .

By inserting both the mentioned stretching and expansion into Equations 23–25, and after straightforward calculations, the following planar KdV equation is obtained [70]. ∂ t B + A B ∂ η B + B ∂ η 3 B = 0 , where B ≡ ϕ 1 and the coefficients A , B read A = B 3 α 1 2 − 2 α 2 &  B = 1 2 α 1 3 . It is well-known that the polarity of the nonlinear waves described by the KdV Equation 26 depends on the sign of the nonlinearity coefficient A . If A > 0 , compressive pulses may arise and propagate in the current model. Nonetheless, when A < 0 , rarefactive pulses may arise in this model. Moreover, if A > 0 for specific values to the relevant plasma parameters and A < 0 for others, both compressive and rarefactive structures may arise and propagate within the examined model. In this scenario, at some critical values for the relevant plasma parameters such as the nonthermal parameter α , the coefficient A becomes null, resulting in the KdV equation’s inability to accurately represent the nonlinear structures that may arise and propagate within the current model. To address this problem, we seek higher-order nonlinearity and then derive another evolutionary wave equation (EWE), which is called the cubic nonlinearity mKdV equation. To derive the mKdV equation, we first determine the critical value of the nonthermal parameter α c as follows [70]. α c = 1 4 3 q 2 − 3 q − 1 + 1 3 q 2 q − 1 3 q − 1 2 . Now, by using the new stretching: η = ε x − λ τ & t = ε 3 τ with the same expansion (25) in Equations 23–25, and after straightforward calculations, the following planar mKdV equation is obtained [70]. ∂ t B + C B 2 ∂ η B + B ∂ η 3 B = 0 , with C = 3 2 α 1 − 3 2 α 1 − α 3 α 1 3 . To examine the influence of fractionality on the dynamics of nonlinear wave propagation characterized by the planar KdV and mKdV Equations 26, 27, it is necessary to transform these equations from their integer representations to their fractional counterparts. To do this, we will follow the same methodology explained in detail in Refs. [79–81], which ultimately arrive at the following fractional forms: D t p B + A B ∂ η B + B ∂ η 3 B = 0 ,   ∀ 0 < p ≤ 1 , and D t p B + C B 2 ∂ η B + B ∂ η 3 B = 0 ,   ∀ 0 < p ≤ 1 , where D t p B stands for the Caputo FDO to the function B and for order p .

 4.2 Example (I): planar nonlinear FKdV equation

In this section, we proceed to analyze the following planar nonlinear FKdV Equation 70. D t p B + A B ∂ η B + B ∂ η 3 B = 0 ,   ∀ 0 < p ≤ 1 , with the IC B η , 0 ≡ B 0 = B max s e c h 2 η w , By setting p = 1 , the following exact soliton solution to Equation 30 is obtained B = B max s e c h 2 η − u 0 t w , where B max = 3 u 0 / A and w = 4 B / u 0 are, respectively, the maximum amplitude of the KdV-soliton and u 0 indicates the nonlinear wave velocity in the relative frame.

To analyze problem (30) using the HPTM, we start from Equation 37 in addition to the following brief points:

Step 1: Applying ET on Equation 30 yields E D t p B = − E A B ∂ η B + B ∂ η 3 B .

Step 2: Using ET to the Caputo FDO as given in Equation 11 in Equation 33, we get u − p E B − ∑ k = 0 m − 1 u 2 − p + k ∂ t i B η , t t = 0 = − E A B ∂ η B + B ∂ η 3 B , which leads to E B = ∑ k = 0 m − 1 u 2 + k ∂ t i B η , t t = 0 − E A B ∂ η B + B ∂ η 3 B , or E B = u 2 B 0 η − u p E A B ∂ η B + B ∂ η 3 B ,

Step 3: Taking the inverse ET to Equation 36 implies B = E − 1 u 2 B 0 η − E − 1 u p E A B ∂ η B + B ∂ η 3 B = B 0 η − E − 1 u p E A B ∂ η B + B ∂ η 3 B .

Step 4: The approximate solution according to the HPM is given by the following convergent series solution: B = ∑ m = 0 ∞ ε m B m , where ε ∈ [ 0,1 ] represents the homotopy parameter.

Step 5: In the following manner, the nonlinear term A B ∂ η B is decomposed A B ∂ η B = ∑ m = 0 ∞ ε m H m B , where H m B indicates He’s polynomials and it is defined by H m B = 1 m ! ∂ m ∂ ε m A ∑ m = 0 ∞ ε m B m ∂ η ∑ m = 0 ∞ ε m B m ε = 0 , which leads to H 0 B = A B 0 ∂ η B 0 , H 1 B = A B 0 ∂ η B 1 + B 1 ∂ η B 0 , H 2 B = A B 2 ∂ η B 0 + B 1 ∂ η B 1 + B 0 ∂ η B 2 , H 3 B = A B 3 ∂ η B 0 + B 2 ∂ η B 1 + B 1 ∂ η B 2 + B 0 ∂ η B 3 , ⋮ .

Step 6: Inserting Equations 38, 39 into Equation 37 yields ∑ m = 0 ∞ ε m B m = B 0 η − ε E − 1 u p E B ∂ η 3 ∑ m = 0 ∞ ε m B m + ∑ m = 0 ∞ ε m H m B .

Step 7: Collecting the coefficients of various order of ε , we get the following the following successive approximations:

• For O ε 0 , the zeroth-order approximation is obtained as follows: B 0 = B 0 η , 0 = B max s e c h 2 η w ,

• For O ε , the 1 s t − order approximation is obtained as follows: B 1 = − E − 1 u p E B ∂ η 3 B 0 + H 0 B = 2 B max W 0 ⁡ tanh η w sech 4 η w w 3 t p Γ p + 1 , with W 0 = A w 2 B max + 2 B cosh 2 η w − 5 .

• For O ε 2 , the 2 n d − order approximation is obtained as follows: B 2 = − E − 1 u p E B ∂ η 3 B 1 + H 1 B = 2 B max W 1 sech 8 η w w 6 t 2 p Γ 2 p + 1 , with W 1 = A w 2 B max I 1 + I 2 , where the coefficients I 1 and I 2 are given in Supplementary Appendix SI.

• For O ε 3 , the 3 r d − order approximation is obtained as follows: B 3 = − E − 1 u p E B ∂ η 3 B 2 + H 2 B = 4 B max W 2 ⁡ tanh η w sech 10 η w w 9 × t 3 p Γ p + 1 2 Γ 3 p + 1 , with W 2 = A w 2 B max Γ p + 1 2 I 3 + I 4 + Γ 2 p + 1 I 5 + I 6 + B 3 Γ p + 1 2 I 7 , where the coefficients I 3 - I 7 are given in Supplementary Appendix SI.

• For O ε m , the m t h − order approximation is obtained as follows: B m = − E − 1 u p E ( B ∂ η 3 B m − 1 + H m − 1 B , m > 0 , m ∈ N .

Step 8: For ε → 1 , the following convergent approximate solution in finite series form up to the third-order approximation is obtained B = B 0 + B 1 + B 2 + B 3 = B max s e c h 2 η w + 2 B max I 0 ⁡ tanh η w sech 4 η w w 3 t p Γ p + 1 + 2 B max sech 8 η w A w 2 B max I 1 + I 2 w 6 t 2 p Γ 2 p + 1 + 4 B max S ⁡ tanh η w sech 10 η w w 9 × t 3 p Γ p + 1 2 Γ 3 p + 1 .

It is clear that both approximation (46) aligns perfectly with the solution derived from the Tantawy technique, as discussed in Refs. [26].

To study how fractionality affects the dynamics of the ion-acoustic FKdV-solitons in the current plasma model, the following values of plasma parameters are considered: for compressive solitons α , q = 0.1 , 0.95 , which leads to A , B = 0.250545 , 0.994067 and for rarefactive solitons α , q = 0.1 , 0.7 , which leads to A , B = − 1.39867 , 5.19566 . The derived approximation (46) is analyzed at different values of the fractionality p to investigate the influence of fractionality on the behavior and dynamics of compressive and rarefactive ion-acoustic FKdV-solitons that arise and propagate in the plasma model under investigation as shown in Figures 1, 2, respectively. These figures illustrate how fractionality affects the dynamics of soliton propagation in various physical systems, particularly in plasma physics. We also compared the derived approximation (46) for the compressive and rarefactive ion-acoustic FKdV-solitons with the exact solution (32) for the integer case, as illustrated in Figures 3, 4, respectively, to verify the accuracy of the generated approximations. We additionally computed the absolute error of the derived approximations to the compressive and rarefactive ion-acousticFKdV-solitons, as presented in Tables 1, 2, respectively. The analysis results demonstrate the efficiency of the used methods in analyzing various complicated EWEs and deriving high-accuracy approximations.

The profile of compressive ion-acoustic FKdV-soliton according to the approximation (46) is investigated against the fractional-order parameter p : (a) 3D-graph in the η , t -plane for p = 0.1 , (b) 3D-graph in the η , t -plane for p = 0.8 , (c) 3D-graph in the η , t -plane for p = 1 , and (d) 2D-graph for t = 30 and different values of p .

Four panels depict graphs of a function \( \mathcal{B} \) versus \( \eta \) and \( t \). Panels (a), (b), and (c) show 3D surface plots with varying shapes and color gradients. Panel (d) displays a 2D plot with three lines representing different \( p \) values: 0.1 (dotted red), 0.8 (dashed blue), and 1.0 (solid black), showing distinct peak shapes.

The profile of rarefactive ion-acoustic FKdV-soliton according to the approximation (46) is investigated against the fractional-order parameter p : (a) 3D-graph in the η , t -plane for p = 0.1 , (b) 3D-graph in the η , t -plane for p = 0.8 , (c) 3D-graph in the η , t -plane for p = 1 , and (d) 2D-graph for t = 30 and different values of p .

Four panels display data visualizations. The top left, top right, and bottom left panels show three-dimensional surface plots with axes labeled \(\eta_a\), \(\eta_b\), \(\eta_c\), \(t\), and \(\mathcal{B}\). The bottom right panel contains a two-dimensional line graph with a legend showing three lines labeled \(p = 0.1\), \(p = 0.8\), and \(p = 1.0\) plotted over \(\eta_d\) and \(\mathcal{B}\). The top right plot appears similar to the left but with varied axis labels.

A comparison between the generated approximation (46) for compressive FKdV-soliton and the exact solution (32) at p = 1 , (a) 3D-graph for the two solutions in the η , t -plane and (b) 2D-graph for the two solutions at t = 1 .

A 3D surface plot on the left shows a rainbow-colored wave pattern with axes labeled B, ηₐ, and t. On the right, a line graph compares EHPM and Exact data, with similar curves peaking at the center.

1 , (a) 3D-graph for the two solutions in the η , t -plane and (b) 2D-graph for the two solutions at t = 1 .

A 3D surface plot on the left shows a multicolored wave-like structure with axes labeled \( \eta \), \( t \), and \( \mathcal{B} \). On the right, a 2D line graph compares two curves: a dashed blue line labeled "Exact" and a dotted red line labeled "EHPM," both depicting a downward curve centered at zero on the \( \eta \) axis and against the \( \mathcal{B} \) axis.

The absolute error for the generated approximations for compressive ion-acoustic FKdV-soliton is estimated at p , t = 1,1 .

η EHPM (46) Exact (32) R ∞
−20 0.0081309 0.0081309 0.333389 × 1 0 − 9
−15 0.0391875 0.0391875 1.28208 × 1 0 − 9
−10 0.179673 0.179673 0.193241 × 1 0 − 9
−5 0.661615 0.661615 24.008 × 1 0 − 9
0 1.19709 1.19709 50.4814 × 1 0 − 9
5 0.689904 0.689904 24.0795 × 1 0 − 9
10 0.190464 0.190464 0.122219 × 1 0 − 9
15 0.0417088 0.0417088 1.29312 × 1 0 − 9
20 0.00866146 0.00866146 0.337397 × 1 0 − 9

The absolute error for the generated approximations for rarefactive ion-acoustic FKdV-soliton is estimated at p , t = 1,1 .

η EHPM (46) Exact (32) R ∞
−30 −0.0127841 −0.0127841 0.114761 × 1 0 − 10
−25 −0.0248238 −0.0248238 0.0872928 × 1 0 − 10
−20 −0.0468352 −0.0468352 0.211078 × 1 0 − 10
−15 −0.083724 −0.083724 1.03017 × 1 0 − 10
−10 −0.136046 −0.136046 1.54542 × 1 0 − 10
−5 −0.189736 −0.189736 0.745381 × 1 0 − 10
0 −0.214479 −0.214479 3.31056 × 1 0 − 10
5 −0.1915 −0.1915 0.775011 × 1 0 − 10
10 −0.138331 −0.138331 1.54111 × 1 0 − 10
15 −0.0855512 −0.0855512 1.03816 × 1 0 − 10
20 −0.0479963 −0.0479963 0.215437 × 1 0 − 10
25 −0.0254795 −0.0254795 0.0863833 × 1 0 − 10
30 −0.0131326 −0.0131326 0.114951 × 1 0 − 10
4.3 Example (II): planar cubic nonlinear FmKdV equation

Here, we proceed to analyze the following planar cubic nonlinear FmKdV equation [70]. D t p B + C B 2 ∂ η B + B ∂ η 3 B = 0 ,   ∀ 0 < p ≤ 1 , with IC B η , 0 ≡ B 0 = f , where f ≡ f η any IC for the planar integer mKdV, such as solitary and shock waves. For solitons to propagate in any plasma model or other physical system, the nonlinearity coefficient signal must be positive, i.e., C > 0 . Otherwise, i.e., for C < 0 , shock waves can propagate instead of solitons. Accordingly, the following IC for soliton is considered: f = B max s e c h η w , which, the following exact soliton solution to Equation 47 for C > 0 and p = 1 , is introduced B = B max s e c h η − u 0 t w , where B max = 6 u 0 / C and w = B / u 0 are, respectively, the peak KdV-soliton amplitude and width, while u 0 indicates the nonlinear wave velocity in the relative frame.

To derive the shock wave solution for Equation 47 in its integer case, we rewrite it in the following new form based on the negative sign of the cubic nonlinear coefficient C : D t p B − C B 2 ∂ η B + B ∂ η 3 B = 0 ,   ∀ 0 < p ≤ 1 ,

Note that we separate the negative sign from the nonlinearity coefficient; therefore, this coefficient must take positive values during the analysis.

Now, by applying tanh method to Equation 51, the following exact shock wave solution at p = 1 is obtained B = B max ⁡ tanh η − u 0 t w , where B max = 3 u 0 / C and w = 2 B / u 0 are, respectively, the maximum amplitude of the mKdV-shock waves and u 0 indicates the nonlinear wave velocity in the relative frame. In this case, the following IC for the shock waves is introduced f = B max ⁡ tanh η w , To analyze problems (47) and (51) using EHPM, we start from Equation 58 in addition to the following brief points:

Step 1: Applying ET on Equations 47, 51 yields E D t p B = − E ± C B 2 ∂ η B + B ∂ η 3 B ,

Note that the positive sign refers to Equation 47, which supports solitons, while the negative sign indicates Equation 51, which supports shock waves.

Step 2: Using ET to the Caputo FDO as given in Equation 11 in Equation 54, we have u − p E B − ∑ k = 0 m − 1 u 2 − p + k ∂ t i B η , t t = 0 = − E ± C B 2 ∂ η B + B ∂ η 3 B , which leads to E B = ∑ k = 0 m − 1 u 2 + k ∂ t i B η , t t = 0 − E ± C B 2 ∂ η B + B ∂ η 3 B , or E B = u 2 B 0 η − u p E ± C B 2 ∂ η B + B ∂ η 3 B ,

Step 3: Taking the inverse ET to Equation 57 yields B = E − 1 u 2 B 0 η − E − 1 u p E ± C B 2 ∂ η B + B ∂ η 3 B = B 0 η − E − 1 u p E ± C B 2 ∂ η B + B ∂ η 3 B .

Step 4: The approximate solution according to the HPM is denoted by the subsequent convergent series solution: B = ∑ m = 0 ∞ ε m B m , where ε ∈ [ 0,1 ] represents the homotopy parameter.

Step 5: In the following manner, the nonlinear term ± C B 2 ∂ η B is decomposed ± C B 2 ∂ η B = ∑ m = 0 ∞ ε m H m B , where H m B indicates He’s polynomials and it is defined by H m B = 1 m ! ∂ m ∂ ε m ± C ∑ m = 0 ∞ ε m B m 2 ∂ η ∑ m = 0 ∞ ε m B m ε = 0 , which leads to H 0 B = ± C B 0 2 ∂ η B 0 , H 1 B = ± C B 0 2 ∂ η B 1 + 2 B 0 B 1 ∂ η B 0 , H 2 B = ± C B 0 2 ∂ η B 2 + 2 B 0 B 1 ∂ η B 1 + B 1 2 + 2 B 0 B 2 ∂ η B 0 , H 3 B = ± C B 0 2 ∂ η B 3 + 2 B 0 B 1 ∂ η B 2 + B 1 2 + 2 B 0 B 2 ∂ η B 1 + 2 B 0 B 3 + B 1 B 2 ∂ η B 0 , ⋮ .

Step 6: Inserting Equations 59, 60 into Equation 58 yields ∑ m = 0 ∞ ε m B m = B 0 η − ε E − 1 u p E B ∂ η 3 ∑ m = 0 ∞ ε m B m + ∑ m = 0 ∞ ε m H m B .

Step 7: Collecting the coefficients of various order of ε , we get the following the following successive approximations:

• For O ε 0 , the zeroth-order approximation is obtained as follows: B 0 = B 0 η , 0 = f .

By considering the ICs for the solitary and shock waves as given in Equations 49, 53, respectively, we can get the explicit values for the zeroth-order approximations to the two nonlinear structures: B 0 Solitons = B max s e c h η w , B 0 Shocks = B max ⁡ tanh η w .

• For O ε , the 1 s t − order approximation is obtained as follows: B 1 = − E − 1 u p E B ∂ η 3 B 0 + H 0 B = − ± C f 2 f ′ + B f 3 t p Γ p + 1 .

By considering the ICs for the solitary and shock waves as given in Equations 49, 53, respectively, we can get the explicit values for the 1 s t − order approximations to the two nonlinear structures as follows: B 1 Solitons = B max Q 1 ⁡ tanh η w s e c h 3 η w 2 w 3 t p Γ p + 1 , B 1 Shocks = B max R 1 s e c h 4 η w w 3 t p Γ p + 1 , with Q 1 = B cosh 2 η w − 11 + 2 C w 2 B max 2 , R 1 = − 2 B cosh 2 η w − 2 + C w 2 B max 2 ⁡ sinh 2 η w .

• For O ε 2 , the 2 n d − order approximation is obtained as follows: B 2 = − E − 1 u p ⁡ E B ∂ η 3 B 1 + H 1 B = ± 2 B C f 3 f ′ ′ 2 + 5 f ′ f 3 + B ± 12 C f ′ 2 f ′ ′ + B f 6 + 4 C 2 f 3 f ′ 2 + C 2 f 4 f ′ ′ ± 2 B C f 2 f 4 t 2 p Γ 2 p + 1 .

By considering the ICs for the solitary and shock waves as given in Equations 49, 53, respectively, we can get the explicit values for the 2 n d − order approximations to the two nonlinear structures as follows: B 2 Solitons = B max Q 2 s e c h 7 η w 32 w 6 t 2 p Γ 2 p + 1 , B 2 Shocks = − B max R 2 ⁡ tanh η w s e c h 6 η w w 6 t 2 p Γ 2 p + 1 , with Q 2 = 8 C w 2 B max 2 S 1 + S 2 , R 2 = − 2 B C w 2 B max 2 J 1 + C 2 w 4 B max 4 J 2 + J 3 , where the coefficients S 1 , S 2 , J 1 , J 2 , and J 3 , are given in Supplementary Appendix SII.

• For O ε 3 , the 3 r d − order approximation is obtained as follows: B 3 = − E − 1 u p E B ∂ η 3 B 2 + H 2 B = − ∑ i = 1 6 Y i t 3 p Γ p + 1 2 Γ 3 p + 1 , where the coefficients Y 1 − Y 6 are given in Supplementary Appendix SIII.

By considering the ICs for the solitary and shock waves as given in Equations 49, 53, respectively, we can get the explicit values for the 1 s t − order approximations to the two nonlinear structures as follows: B 3 Solitons = B max Q 3 ⁡ tanh η w s e c h 9 η w 128 w 9 t 3 p Γ p + 1 2 Γ 3 p + 1 , B 3 Shocks = − B max R 3 s e c h 10 η w 8 w 9 t 3 p Γ p + 1 2 Γ 3 p + 1 , with Q 3 = 4 C w 2 B max 2 B 2 S 3 + S 4 + 4 C w 2 B max 2 S 5 + S 6 + S 7 + B 3 Γ p + 1 2 S 8 , R 3 = J 4 + C w 2 B max 2 4 B 2 J 5 + J 6 + C w 2 B max 2 J 7 + J 8 + J 9 , where the coefficients S 3 - S 8 and J 4 - J 9 , are given in Supplementary Appendix SIV.

Step 8: For ε → 1 , the following convergent approximate solutions in finite series form up to the third-order approximation are obtained:

• Soliton solution up to 3 r d − order approximation: B = B 0 + B 1 + B 2 + B 3 = B max s e c h η w + B max Q 1 ⁡ tanh η w s e c h 3 η w 2 w 3 t p Γ p + 1 + B max Q 2 s e c h 7 η w 32 w 6 t 2 p Γ 2 p + 1 + B max Q 3 ⁡ tanh η w s e c h 9 η w 128 w 9 t 3 p Γ p + 1 2 Γ 3 p + 1 .

• Shock wave solution up to 3 r d − order approximation: B = B 0 + B 1 + B 2 + B 3 = B max ⁡ tanh η w + B max R 1 s e c h 4 η w w 3 t p Γ p + 1 − B max R 2 ⁡ tanh η w s e c h 6 η w w 6 t 2 p Γ 2 p + 1 − B max R 3 s e c h 10 η w 8 w 9 t 3 p Γ p + 1 2 Γ 3 p + 1 .

It is clear that the generated soliton approximation (74) using EHPM is identical to the derived soliton approximation using the Tantawy technique, as discussed in Refs. [27].

To investigate how fractionality affects the dynamics of the ion-acoustic FmKdV-solitons in the current plasma model, the value of the nonextensive parameter is considered: q = 0.95 , which leads to C , B = 0.359292 , 1.1968 . The generated soliton approximation (74) is analyzed at different values of the fractionality p to examine the impact of fractionality on the behavior and dynamics of ion-acoustic FmKdV-solitons that emerge and propagate inside the plasma model under consideration, as illustrated in Figure 5. This graphic illustrates how fractionality impacts the propagation dynamics of FmKdV-solitons in various physical systems, particularly in plasma physics. Additionally, we compared the derived FmKdV-soliton approximation (74) with the exact mKdV-soliton solution (50) for the integer case, as depicted in Figure 6, to validate the accuracy of the generated approximations. Moreover, the absolute error of the generated FmKdV-soliton approximation is numerically calculated, as shown in Table 3. The analysis results demonstrate the effectiveness of the methods employed in examining diverse, strong nonlinearity EWEs and producing high-precision approximations.

The profile of compressive ion-acoustic FmKdV-soliton according to the approximation (74) is investigated against the fractional-order parameter p : (a) 3D-graph in the η , t -plane for p = 0.1 , (b) 3D-graph in the η , t -plane for p = 0.8 , (c) 3D-graph in the η , t -plane for p = 1 , and (d) 2D-graph for t = 30 and different values of p .

Three 3D surface plots and one 2D plot. The 3D plots (a, b, c) depict surfaces with axes labeled as \(\eta\), \(t\), and \(\mathcal{B}\), featuring colored grids with a wave-like pattern. The 2D plot (d) shows curves labeled with \(p = 0.1\), \(p = 0.8\), and \(p = 1.0\), illustrating variations in \(\mathcal{B}\) against \(\eta\).

A comparison between the generated approximation (74) for compressive FmKdV-soliton and the exact solution (50) at p = 1 , (a) 3D-graph for the two solutions in the η , t -plane and (b) 2D-graph for the two solutions at t = 1 .

Three-dimensional graph on the left shows a multicolored surface plot with axes labeled eta sub a, t, and script B, depicting a wave-like shape. On the right, a two-dimensional plot compares two curves, labeled EHPM in red dots and Exact in blue dashes, with both peaking at the center.

The absolute error for the generated approximations for compressive ion-acoustic FmKdV-soliton is estimated at p , t = 1,1 .

η EHPM (74) Exact (50) R ∞
−21 0.00580147 0.00580147 0.172643 × 1 0 − 9
−18 0.0138083 0.0138083 0.410103 × 1 0 − 9
−15 0.0328621 0.0328621 0.965087 × 1 0 − 9
−12 0.0781592 0.0781592 2.14931 × 1 0 − 9
−9 0.185243 0.185243 3.20257 × 1 0 − 9
−6 0.430543 0.430543 12.7532 × 1 0 − 9
−3 0.904263 0.904263 79.5494 × 1 0 − 9
0 1.29173 1.29173 187.897 × 1 0 − 9
3 0.941599 0.941599 79.1974 × 1 0 − 9
6 0.454575 0.454575 13.3662 × 1 0 − 9
9 0.196144 0.196144 3.18679 × 1 0 − 9
12 0.0828016 0.0828017 2.17008 × 1 0 − 9
15 0.0348172 0.0348172 0.975994 × 1 0 − 9
18 0.0146301 0.0146301 0.414849 × 1 0 − 9
21 0.00614675 0.00614675 0.174649 × 1 0 − 9

The profile of the fractional compressive FmKdV-shock waves according to the generated approximation (75) is examined against the fractional parameter p as elucidated in Figure 7. We also presented a comparison between both the derived fractional compressive FmKdV-shock wave approximation (75) and the exact shock wave solution (52) at p = 1 , as shown in Figure 8, to verify the accuracy and stability of the derived approximations along the whole study domain. Furthermore, the absolute error of the compressive and rarefactive FmKdV-shock wave approximation (75) is estimated as shown in Tables 3, 4, respectively.

The profile of compressive FmKdV-shock waves according to the approximation (75) is investigated against the fractional-order parameter p : (a) 3D-graph in the η , t -plane for p = 0.1 , (b) 3D-graph in the η , t -plane for p = 0.8 , (c) 3D-graph in the η , t -plane for p = 1 , and (d) 2D-graph for t = 30 and different values of p .

Four graphs are displayed: three 3D surface plots and one 2D plot. The 3D plots (a, b, c) show surfaces transitioning from purple to red, with axes labeled η, B, and t. The 2D plot (d) presents curves for p values 0.1, 0.8, and 1.0, illustrating how B changes with η.

A comparison between the generated approximation (75) for compressive FmKdV-shock waves and the exact solution (52) at p = 1 , (a) 3D-graph for the two solutions in the η , t -plane and (b) 2D-graph for the two solutions at t = 1 .

3D surface plot on the left shows variable \(\mathscr{B}\) as a function of \(\eta_a\) and \(t\), with colors ranging from purple to red. The right shows a 2D plot, comparing EHPM and Exact values of \(\mathscr{B}\) against \(\eta_b\), with both lines closely matched, blue for Exact and red for EHPM.

The absolute error for the generated approximations for compressive FmKdV-shock waves is estimated at p , t = 1,1 and B , C = ( 1,1 ) .

η EHPM (75) Exact (52) R ∞
−30 −0.547721 −0.547721 0.00274558 × 1 0 − 10
−25 −0.547707 −0.547707 0.025685 × 1 0 − 10
−20 −0.547573 −0.547573 0.239867 × 1 0 − 10
−15 −0.546326 −0.546326 2.205 × 1 0 − 10
−10 −0.534792 −0.534792 17.3808 × 1 0 − 10
−5 −0.437597 −0.437597 7.03169 × 1 0 − 10
0 0.0122454 0.0122454 4.08166 × 1 0 − 10
5 0.446146 0.446146 5.02629 × 1 0 − 10
10 0.535886 0.535886 17.1328 × 1 0 − 10
15 0.546445 0.546445 2.16666 × 1 0 − 10
20 0.547586 0.547586 0.235625 × 1 0 − 10
25 0.547708 0.547708 0.0252298 × 1 0 − 10
30 0.547721 0.547721 0.00269673 × 1 0 − 10
5 Conclusion

In this study, two of the most fundamental nonlinear evolutionary wave equations, which are widely used in various physical and engineering applications, have been analyzed. These equations are called the nonlinear planar fractional KdV (FKdV) and fractional modified KdV (FmKdV) equations, and they examined using Elzaki homotopy perturbation method (EHPM). For the quadratic nonlinear planar FKdV equation, a general formula up to the third order has been derived as a function of initial condition. After that, the soliton solution has been used as an initial solution, and an analytical fractional soliton approximation up to the third order has been generated. On the other hand, the cubic nonlinear planar FmKdV equation was divided into two parts: For the first part, if the cubic nonlinearity coefficient is positive, in this case, the FmKdV equation supports solitons and does not support shock waves. For this case, a general formula has been derived using the proposed approach as a function of initial condition. Subsequently, the soliton solution has been used as an initial solution, and an analytical fractional soliton approximation has been derived up to the third order. For the second form of the cubic nonlinear planar FmKdV equation, if the cubic nonlinearity coefficient is negative, in this case, the FmKdV equation does not support solitons, but rather shock waves. Using the proposed technique, a general formula has been derived as a function of initial condition. As a practical application to the obtained results, the fluid-governed equations for a collisionless and unmagnetized plasma composed of inertial cold ions and inertialess Cairns-Tsallis distributed electrons have been reduced to both the FKdV and FmKdV equations. After that, the effect of the fractional parameter on the dynamic behavior of the propagation of ion-acoustic waves in the plasma model under study has been investigated. We also performed a graphical comparison between all derived approximations and the exact solutions for the integer cases, i.e., at p = 1 , showing complete agreement between all derived approximations and the counterpart exact solutions. We also computed the absolute error for all derived approximations and found that they demonstrate good accuracy, thereby enhancing the effectiveness of the employed technique.

The derived approximations demonstrated that the provided approach can successfully and precisely solve problems with strong nonlinearity. We may conclude from the results that the used method is accurate in simulating the nonlinear structures (solitons and shock waves) in plasma physics and other scientific fields. The suggested results provide a comprehensive and valuable examination of the behavior of these waves. Several authors, particularly those working in nonlinear sciences, can benefit from the results in evaluating and interpreting their experimental and observational data.

 6 Future work

This investigation has examined both the nonlinear planar FKdV and FmKdV equations. However, in numerous instances, the nonplanar and damped cases are more realistic for describing nonlinear phenomena in various plasma models. Consequently, in forthcoming studies, we will apply the two proposed approaches, in addition to the Tantawy technique [25–28], to analyze various nonlinear fractional EWEs that are extensively utilized in modeling numerous nonlinear phenomena in different plasma models, such as the nonplanar/damped FKdV-type equations [82, 83], the nonplanar/damped fractional Kawahara-type equations [84–87], the nonplanar/damped fractional Schrödinger-type equations [88–90], etc.

 Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

 Author contributions

WA: Formal Analysis, Investigation, Methodology, Writing – review and editing. AK: Investigation, Methodology, Writing – original draft. AA-J: Formal Analysis, Supervision, Validation, Writing – review and editing. SE-T: Formal Analysis, Investigation, Methodology, Software, Supervision, Writing – review and editing.

 Funding

The author(s) declare that financial support was received for the research and/or publication of this article. The authors express their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2025R229), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

The authors express their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2025R229), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

 Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

 Generative AI statement

The author(s) declare that no Generative AI was used in the creation of this manuscript.

 Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

 Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2025.1604640/full#supplementary-material

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