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Learn more: PMC Disclaimer | PMC Copyright Notice Sci Rep . 2026 Mar 4;16:12176. doi: 10.1038/s41598-026-41053-w Search in PMC Search in PubMed View in NLM Catalog Add to search Space–time variable-order fractional analysis of nonlinear longitudinal wave propagation in magneto-electro-elastic materials Muhammad Asim Khan Muhammad Asim Khan 1 School of Mathematical Sciences, Universiti Sains Malaysia, Gelugor, Pulau Pinang Malaysia Find articles by Muhammad Asim Khan 1 , Majid Khan Majahar Ali Majid Khan Majahar Ali 1 School of Mathematical Sciences, Universiti Sains Malaysia, Gelugor, Pulau Pinang Malaysia Find articles by Majid Khan Majahar Ali 1, ✉ , Saratha Sathasivam Saratha Sathasivam 1 School of Mathematical Sciences, Universiti Sains Malaysia, Gelugor, Pulau Pinang Malaysia Find articles by Saratha Sathasivam 1, ✉ , Naglaa F Soliman Naglaa F Soliman 2 Department of Information Technology, College of Computer and Information Sciences, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh, 11671 Saudi Arabia Find articles by Naglaa F Soliman 2 , Abeer D Algarni Abeer D Algarni 2 Department of Information Technology, College of Computer and Information Sciences, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh, 11671 Saudi Arabia Find articles by Abeer D Algarni 2 Author information Article notes Copyright and License information 1 School of Mathematical Sciences, Universiti Sains Malaysia, Gelugor, Pulau Pinang Malaysia 2 Department of Information Technology, College of Computer and Information Sciences, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh, 11671 Saudi Arabia ✉ Corresponding author. Received 2026 Jan 1; Accepted 2026 Feb 17; Collection date 2026. © The Author(s) 2026 Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ . PMC Copyright notice PMCID: PMC13077061 PMID: 41781431 Abstract This work presents a space-time variable-order (V-O) fractional framework for the analytical investigation of nonlinear longitudinal wave propagation in magneto-electro-elastic (MEE) materials. The model contains Caputo-type V-O derivatives, which account for spatial and temporal memory effects and nonlocal interactions inherent to coupled mechanical, electrical, and magnetic fields. On this basis, exact traveling-wave solutions of the resulting nonlinear fractional wave equation were derived using an exponential expansion methodology, in the form of periodic, kink-type, and solitary waves. The effect of the V-O parameters on the wave amplitude, dispersion characteristics, and stability behavior is systematically analyzed. Stability and bifurcation analyses show parameter-dependent transition of both stable and unstable regimes, whereas insertion of random perturbations illustrates the development of chaotic dynamics. The proposed V-O model offers increased modeling flexibility and parameter-dependent stability regimes in the reduced system compared to constant-order (C-O) fractional formulations. This enhancement provides increased analytical flexibility at the theoretical level for coupled field interactions in MEE media. These results suggest that the V-O fractional modeling represents a valid tool for the analysis of complex nonlinear electromechanical phenomena and could lead to a better understanding of the wave dynamics at a theoretical level in multifunctional material systems. Keywords: Variable-order fractional operators, Magneto-electro-elastic materials, Caputo fractional derivative, Traveling wave solutions, Nonlinear fractional longitudinal wave equation, Wave transformation methods Subject terms: Engineering, Mathematics and computing, Physics Introduction In recent years, fractional differential equations and V-O extensions have become more and more important for modeling complex physical systems that include memory effects, spatial heterogeneity, and nonlocal interactions. The reason for their growing popularity is that they can better model real-world dynamic phenomena with greater accuracy than traditional integer-order models, especially systems whose response to the current state is also affected by their history. The V-O geometry models have been effectively applied in a range of fields, such as nonlinear optics, fluid mechanics, plasma physics, signal processing, and advanced materials science 1 , 2 . As a result, a wide range of analytical and semi-analytical procedures has been developed to analyze these models, offering both precise and approximate solutions based on the complexity of the equations involved. The alloy and composite forms of MEE materials have received a lot of research attention within the fields of nanotechnology and micro-technology due to their outstanding physicochemical qualities 3 . Such materials usually contain magnetostrictive, piezoelectric, and ferroelectric materials, thus allowing the strong interaction between mechanical, electrical, and magnetic fields. Moreover, their nature of being dielectric, nonlinear, thermal, and frequency-responsive makes them well suited in the applications they can be used in: sensing, actuation, and intelligent structural systems 4 . Therefore, gaining insight into the dynamic behavior observed under different operating conditions is essential for developing future technologies. Among numerous physical phenomena, the propagation of longitudinal waves, including acoustic, seismic, and ultrasonic waves, through magneto-electro-elastic circular rods offers an effective technique of examining the interaction between mechanical deformation and electromagnetic fields. A longitudinal wave moving through an MEE rod puts stress and strain on the rod and changes the magnetic field. This kind of interaction is called the piezoelectric effect and magnetostrictive coupling. Such interactions have significant changes to the effective stiffness, density, dispersion properties, and speed of the wave and create complicated dynamic behavior that is not well resolved using classical models. In the absence of external body forces, the longitudinal wave propagation in an MEE circular rod is governed by the classical wave equation 5 1 where , with denoting the elastic modulus and the material density. When the intrinsic magnetic and electric body forces associated with MEE coupling are incorporated, the governing equation takes the nonlinear dispersive form 2 where the parameter represents dispersion effects induced by the transverse Poisson effect and intrinsic magneto-electric coupling. Although Eq. ( 2 ) is suitable to capture nonlinear dispersion processes, its application is limited by assumptions that include local interactions and instant material response. Experimental and theoretical studies have shown that MEE materials have a strong time-dependence memory effect and spatial heterogeneity, both of which cannot be accurately modeled by classical integer-order derivatives. As a result, the introduction of the concept of fractional calculus becomes a physically relevant generalization of the model. Specifically, V-O fractional derivatives offer even more flexibility by allowing the differentiation order to be variable with space and time, thus being able to model the changing material behavior. Several fractional operators can be used to describe memory and nonlocal effects, including the Riemann–Liouville, two-scale fractal and Atangana–Baleanu derivatives 6 , 7 . The Riemann–Liouville fractional derivative 2 The Riemann–Liouville derivative requires fractional-order initial conditions with limited physical interpretability for wave propagation problems 6 , while two-scale fractal derivatives are primarily suited for fractal or multiscale geometries rather than classical wave dynamics 8 . Similarly,the Atangana–Baleanu Caputo (ABC) fractional derivative of a function , with is defined as 9 : 3 where satisfies the condition . Although the Atangana–Baleanu derivative employs a non-singular kernel advantageous for diffusion-type processes 10 , its analytical applicability to exact solutions and V-O wave formulations remains limited. Therefore, the Caputo V-O fractional derivative is adopted due to its compatibility with physically meaningful integer-order initial conditions and its well-established analytical structure, which supports traveling-wave transformations and the -expansion method. This choice makes it particularly suitable for modeling space–time-dependent memory effects in MEE materials. While Caputo-type derivatives may exhibit numerical discretization sensitivity, this issue does not arise here since the analysis is purely analytical. In this study, we utilize the V-O Caputo fractional derivative 2 , which is well suited for physical modeling due to its compatibility with classical initial conditions and the fact that the fractional derivative of a constant is zero. The V-O Caputo fractional derivative of a sufficiently smooth function is defined as 4 and the corresponding property for the algebraic term is given by 5 These properties make the Caputo V-O derivative particularly appropriate for wave propagation problems involving memory and nonlocal effects. Following established approaches in V-O fractional modeling, the integer-order derivatives in Eq. ( 2 ) are replaced by Caputo-type V-O fractional derivatives according to 6 which leads to the space–time V-O fractional longitudinal wave equation (FLWE) 7 The model involves variable fractional orders and , representing temporal and spatial differentiation, respectively. The temporal order captures history-dependent memory effects, while the spatial order accounts for nonlocal interactions associated with material heterogeneity and long-range coupling. The magneto-electric coupling coefficient M is incorporated within the same V–O fractional framework, ensuring physical and dimensional consistency. The simultaneous use of temporal and spatial V-O fractional derivatives in Eq. ( 7 ) aligns with the fractional spatio-temporal relation proposed by Wang and He 8 . This relation indicates that balanced wave dynamics require the utilization of coupled space-time fractional operators. Using fractional derivatives in only one domain can lead to nonphysical wave velocities or dispersion behavior. Accordingly, the present formulation adopts coupled orders and to describe physically admissible longitudinal wave propagation in MEE materials. Within this framework, the temporal V-O Caputo derivative models relaxation and memory effects arising from electromechanical interactions, whereas the spatial V-O derivative governs nonlocal dispersion linked to microstructural heterogeneity. As a result, the longitudinal wave motion represents an adaptive process in which memory and nonlocality evolve with the material state. This V-O formulation explicitly incorporates dynamic heterogeneity and history dependence, providing a flexible and physically consistent framework for modeling nonlinear wave propagation in MEE materials. Solitons are localized, shape-preserving wave structures that play a significant role in the study of nonlinear wave propagation. They have a wide range of applications in modern science and engineering. Solitons can be useful in long-range signal transmission, energy-efficient communication, the controlled dynamics of plasma, and the mitigation of natural processes such as erosion and flooding. Consequently, a variety of analytical and computational techniques have been developed to investigate soliton solutions of nonlinear evolution equations (NLEEs). These include rational sine-Gordon expansion 11 , Painlevé analysis 12 , sub-equation and auxiliary equation methods 13 , 14 , modified tanh and exponential expansion techniques 15 , 16 , generalized -type approaches 17 , Hirota and Wronskian methods 18 , as well as Kudryashov, Riccati, and Jacobi elliptic function methods 19 – 21 . Recent advances in algebraic and transformation-based techniques continue to expand the scope of exact soliton analysis 22 – 24 . Despite significant progress in C-O fractional models, the role of space-time V-O fractional operators in nonlinear longitudinal wave propagation within MEE structures remains insufficiently explored. In particular, the combined effects of V-O memory, dispersion behavior, and stability characteristics have not yet been fully understood. In this study, we utilize the -expansion method to obtain approximate traveling-wave solutions associated with the space-time V-O FLWE that governs MEE circular rods. This approach has proven effective for handling nonlinear fractional wave models and offers analytical clarity in capturing the influence of variable fractional parameters. The present study aims to develop clear soliton solutions, explain their stability properties, and describe the effect that V-O effects have on the wave propagation in MEE materials. The formulation of the proposed space–time V-O fractional model is based on the following physically motivated assumptions, which ensure both mathematical consistency and physical validity of the governing equation: (i) The material properties of the MEE medium vary smoothly in space and time, allowing the fractional orders and to be continuous, bounded, and restricted to the interval (0, 1]. (ii) Memory and nonlocal effects are sufficiently pronounced that classical integer–order formulations become inadequate for accurately capturing the observed wave dynamics, while their spatial and temporal variations remain gradual, thereby justifying the use of space–time V-O Caputo fractional derivatives. (iii) Longitudinal wave motion dominates the dynamics, and transverse effects are incorporated only through effective nonlinear dispersion and magneto–electro–mechanical coupling parameters. (iv) External body forces are neglected, and the analysis focuses on intrinsic interactions among mechanical, electric, and magnetic fields within the MEE structure. Under these assumptions, the resulting V-O fractional longitudinal wave model remains physically meaningful while accurately capturing spatial heterogeneity, temporal memory, and nonlocal interactions inherent to advanced MEE materials. The manuscript is organized as follows. Section “Introduction” introduces the Caputo fractional derivative. Section "The Proposed exp(−φ(ξ))-expansion method" provides the methodology for the analysis. Section "Application of the proposed method" applies the proposed method to the V-O FLWE, while Section “Physical interpretations” elaborates on the physical implications of the solutions obtained. Similarly, Sections “Discussion” and “Conclusion” examine the stability of equilibrium and chaotic behavior, respectively, and the conclusion is in the last section. The proposed exp( )-expansion method The exp( )-expansion method used in this study represents an enhanced version of the classical exp-function method initially developed for nonlinear evolution equations 25 . Due to its algebraic flexibility, this method effectively addresses polynomial nonlinearities and is compatible with traveling-wave reductions as well as V-O fractional operators. Unlike purely numerical approaches, this method produces explicit analytical solutions that distinctly demonstrate how space–time variable fractional orders affect wave amplitude, localization, and dispersion. This makes it particularly suitable for analyzing nonlinear wave propagation in MEE media. Consider the following nonlinear V-O fractional partial differential equation (FPDE) of orders , , and : 8 where H denotes a polynomial function of u and its V-O fractional derivatives. The FPDE contains highest-order linear and nonlinear terms and admits traveling-wave type solutions. Moreover, for ease of understanding, the symbols and parameters used throughout the analysis are summarized in Table 1 Table 1. Nomenclature of symbols and parameters used in the manuscript. Symbol Description u ( x , t ) Longitudinal displacement field. Traveling-wave profile with . Traveling-wave coordinate used for reduction. Characteristic wave speed, . E Elastic modulus. Mass density. M Dispersion/coupling parameter. Variable fractional order in time. Variable fractional order in space. Caputo-type V-O time-fractional derivative. V-O space-fractional operator. Transformation constants (wave number, frequency, phase shift). Auxiliary function in the -expansion method. Auxiliary-equation parameters/constant. Series coefficients in the solution ansatz. Stability parameters in the planar dynamical system. External perturbation term. Open in a new tab To implement the exp( )-expansion method, a V-O fractional traveling-wave transformation is introduced to reduce Eq. ( 8 ) into an ordinary differential equation (ODE) : 9 where k , l , and are real constants. Thus, the exact solutions obtained in the following sections correspond to the reduced ODE derived from the original space–time V-O FPDE. The V-O fractional traveling-wave transformation used in Eq. ( 9 ), as proposed by Zhengbiao Li 26 , serves as an approximate reduction technique. In general, V-O fractional derivatives do not satisfy a universal chain rule that guarantees an exact equivalence between a V-O FPDE and its reduced ODE. Consequently, the resulting ODE should be interpreted as an approximate representation, valid under smoothness and slow-variation assumptions on the fractional orders. Accordingly, the resulting ODE should be interpreted as an approximate traveling-wave representation of Eq. ( 8 ), valid under appropriate smoothness and slow-variation assumptions on the fractional orders. For the present V-O fractional wave model, the transformation in Eq. ( 9 ) provides a reasonable approximation under the following practical conditions: (i) the fractional orders , , and are continuous and vary smoothly in space and time; (ii) the orders remain within admissible bounds of the adopted V-O Caputo-type definition and do not approach singular endpoints; and (iii) the variation of the fractional orders along the traveling-wave coordinate is sufficiently weak, such that the wave dynamics can be described by a single traveling phase . If the variable orders vary rapidly or exhibit strong spatial–temporal gradients, additional coupling terms associated with the V-O kernels may not be fully captured by the reduced ODE. This may lead to deviations in the predicted wave characteristics, such as amplitude, width, or effective dispersion relations. Consequently, the traveling-wave solutions derived below should be interpreted as approximate analytical solutions of the original V-O fractional model in Eq. ( 8 ), valid primarily within the above regime. Substituting Eq. ( 9 ) into Eq. ( 8 ) yields the following ODE in terms of : 10 The solution of Eq. ( 10 ) is assumed in the finite-series form 11 where the positive integer n is determined using the homogeneous balance principle, and are constants to be determined. The auxiliary function satisfies the first-order ODE 12 with and being arbitrary constants. The solutions of the auxiliary equation ( 12 ) are classified into different cases according to the values of and : Case 1: If and , the solution is given by 13 Case 2: If and , the solution is 14 Case 3: If , the solution is 15 Case 4: If , , and , the solution is given by 16 Case 5: If , , and , the solution is given by 17 Figure 1 illustrates a flowchart summarizing the proposed exp( )-expansion methodology for solving the space–time V-O fractional model. It should be noted that the exp( )-expansion method is primarily applicable to nonlinear evolution equations that admit traveling-wave reductions and polynomial-type nonlinearities. For equations involving strongly non-polynomial or highly nonlocal nonlinear structures, alternative analytical or numerical approaches may be required. Fig. 1. Open in a new tab Flowchart of the proposed exp( )-expansion method. Application of the proposed method In this section, we have analyzed the approximate traveling-wave solutions of the beta space-time FLWE in a MEE circular rod using the -expansion scheme. Initially, the fractional governing model, as presented in Eq. ( 7 ), is converted into an ODE using a wave variable transformation The resulting converted ODE is expressed as: 18 where , , k , and M are constants, and represents the solution of interest. By integrating Eq. ( 18 ) twice and setting the integration constants to zero, we derive a second-order nonlinear differential equation of the form: 19 By balancing the uppermost nonlinear term with the uppermost linear term , we obtain . Therefore, Eq. ( 11 ) becomes: 20 Inserting Eq. ( 20 ) into Eq. ( 19 ) in terms of , and equating the like powers of , we obtain the system of equations given below. 21 22 23 24 25 The system of equations was solved using computer algebra systems, and the following results were obtained: Set 1: 26 Set 2: 27 Set 3: 28 where A , B , and are constants. Now substituting the values of Eq. ( 26 ) into Eq. ( 20 ), we have the solution for Set 1. 29 Now, by substituting the solutions of Eq. ( 12 ), we derive five distinct types of traveling wave solutions for the V-O FLWE, as described in Eq. ( 7 ). In the context of the V-O fractional modified equal width equation, three specific cases are considered for Eq. ( 7 ), which are outlined below. Case 1: If and , the solution is: 30 where Case 2: If and , the solution is given by: 31 Case 3: If , the solution is given by: 32 Case 4: If , , and , the solution is given by: 33 where . Case 5: If , , and , the solution is given by: 34 Similarly, we can find other solutions for Set 2 and Set 3 as well. Remark. It is noted that some solution families, particularly those in Cases 4 and 5, possess rational or polynomial dependence on the wave variable . These structures may induce singular or near-singular behavior for certain parameter choices or extended spatial–temporal domains. Consequently, graphical illustrations are restricted to bounded intervals where the solutions remain finite and physically meaningful. Fig. 6. Open in a new tab Plot of , when ( a ) 3D plot, ( b ) 2D plot, ( c ) Contour plot, ( d ) , and . Physical interpretations The physical interpretation of the results illustrated in Figs. 2 ,3,4,5,6 clarifies the propagation features of longitudinal waves in MEE materials under V-O fractional dynamics. The progression of wave solutions highlights significant physical phenomena, such as wave dispersion, amplitude modulation, and the influence of V-O factors on wave behavior. Moreover, the figures demonstrate that variations in the temporal order primarily affect wave memory and propagation speed, whereas changes in the spatial order directly control wave localization and dispersion. This confirms the distinct physical roles of temporal and spatial fractional operators in the proposed V-O framework. Figure 2 presents the graph of using the parameters . It consists of four subplots: (a) a three-dimensional plot, (b) a two-dimensional plot, (c) a contour plot, and (d) specified parameter values. Let , , , , , and . The solution exhibits a stable wave profile with minimal dispersion over time. Fig. 2. Open in a new tab Plot of for , , , , , , , and : ( a ) three-dimensional plot, ( b ) two-dimensional plot, ( c ) contour plot, and ( d ) comparison for different V-O functions , , , , and , . The spatial and temporal domains are restricted to bounded intervals to avoid nonphysical divergence and singular behavior induced by space–time V-O fractional effects. For visualization purposes, smooth and bounded fractional–order profiles are employed, including trigonometric forms such as which ensure physically meaningful space–time variability and improve the clarity of comparative three–dimensional representations. Figure 3 similarly displays the graph of for the parameter values , , , , , , , and , showcasing a more localized wave structure. The peak intensity is sustained, indicating the presence of solitary waves with minimal energy dissipation. A key aspect of the fractional-order model is its ability to account for the complex interactions among mechanical, electric, and magnetic fields in MEE materials. Moreover, Figs. 4 ,5,6 illustrate the plots of , , and with parameters , , , , , , , and , demonstrating the nonlinear characteristics of wave propagation, where increased fractional orders lead to changes in wave shape and velocity. The results presented in Fig. 4 suggest that the contours represent a dispersive wave train generated by the oscillatory nature of the coupling characteristic of the V-O paradigm. Additionally, the wave response illustrated in Fig. 5 shows a resonant interaction, indicating potential for signal modulation. These examples reveal the key role of V-O fractional derivatives in the description of the peculiarities of waves in the context of the MEE systems. The proposed methodology can support a range of solutions such as periodic, kink, and solitary solitons and, thus, show its ability to describe a wide range of wave phenomena. The findings will be important in the theoretical guidance of materials and technologies that require high sensitivity to the properties of waves. Although the obtained traveling-wave solutions are analytical, their graphical representations are intentionally restricted to finite spatial and temporal intervals. This restriction arises from the explicit dependence of the wave variable on the space–time V-O functions and . For larger values of x and t , V-O functions with exponential or rapidly varying behavior may induce rapid growth of , leading to unbounded amplitudes and loss of physical interpretability. Therefore, the selected plotting ranges correspond to physically admissible regimes of wave propagation rather than nonphysical divergence regions. Fig. 3. Open in a new tab Plot of , when ( a ) 3D plot, ( b ) 2D plot, ( c ) Contour plot, ( d ) , and . Fig. 4. Open in a new tab Plot of , when ( a ) 3D plot, ( b ) 2D plot, ( c ) Contour plot, ( d ) , and . Fig. 5. Open in a new tab Plot of , when ( a ) 3D plot, ( b ) 2D plot, ( c ) Contour plot, ( d ) , and . Effect of variable fractional orders and To explicitly demonstrate the influence of space–time variable fractional orders, a controlled comparative analysis is performed by varying one fractional order while keeping the other fixed, as illustrated in Fig. 7 . This approach allows the individual roles of temporal and spatial memory to be clearly identified. Fig. 7. Open in a new tab The three-dimensional plots used to compare the effects of variable fractional orders include: ( a ) the impact of the temporal order while maintaining a constant spatial order ; and ( b ) the impact of the spatial order while keeping a constant temporal order . The use of trigonometric and exponential-trigonometric profiles increases clarity. First, the temporal fractional order is varied while the spatial order is kept constant. The current study shows that reducing the temporal fractional order negatively impacts the memory effect of the medium, decreases the speed of the wave, and leads to significant modulations in wave amplitude. Conversely, high values of decrease memory retention, which accelerates changes in the waves and results in a smoother spatial gain. The second step in the experimentation involved varying the spatial fractional order while keeping the value of alpha constant. These modifications showed that changes in have a significant effect on the localization of space and dispersion properties. In particular, reduced values of the parameter of are associated with better localization and sharper wave fronts, and higher values are associated with better dispersion and wider wave forms. Such findings are a clear indication that temporal memory and propagation speed are mainly determined by the , whereas nonlocality of space and localization of waves are determined by the . The unique effects mentioned cannot be adequately described by C-O fractional models; therefore, space-time variable fractional operators are necessary. Stability analysis of equilibrium points This section examines the dynamical behavior of the nonlinear system to determine the stability of its equilibrium points. The proposed Eq. ( 7 ) governs longitudinal wave propagation in MEE materials. Stability analysis is conducted by formulating a planar dynamical system based on the governing equation and performing eigenvalue analysis of the corresponding Jacobian matrix, following established methods in nonlinear dynamical systems and bifurcation theory 27 , 28 . The governing equation of the system is 35 Rewriting the equation yields 36 where the parameters are defined as Introducing the transformations and , the system is represented as the planar dynamical system 37 The stability analysis pertains to the reduced planar dynamical system derived from the traveling-wave approximation, rather than the entire space–time V-O FPDE. Equilibrium points The equilibrium points are obtained by setting and . This yields two equilibrium points: Jacobian matrix and stability The Jacobian matrix associated with system ( 37 ) is The stability characteristics depend on the value of the parameter , leading to three distinct cases. Case I: . When , the equilibrium points coincide, since In this case, the Jacobian matrix evaluated at has a double zero eigenvalue, and the equilibrium point is non-hyperbolic. The local dynamics are governed by which produces a distinct phase portrait around the origin, as illustrated in Fig. 8 a. Fig. 8. Open in a new tab Phase portraits of system ( 37 ): ( a ) , where the equilibrium points merge at A (0, 0) forming a degenerate (non-hyperbolic) equilibrium; ( b ) , where A (0, 0) is a saddle point and is a center; ( c ) , where A (0, 0) is a center and is a saddle point. Case II: . For , the eigenvalues at are real with opposite signs, and hence is a saddle point, indicating instability. In contrast, the equilibrium point has purely imaginary eigenvalues and behaves as a center with closed periodic trajectories, representing stable oscillatory motion, as shown in Fig. 8 b. Case III: . When , the equilibrium point possesses purely imaginary eigenvalues and becomes a center corresponding to stable oscillations. Meanwhile, the equilibrium point has real eigenvalues of opposite signs and therefore becomes a saddle point, indicating instability, as depicted in Fig. 8 c. For reproducibility, the phase portraits are generated using with , , and , respectively, and initial conditions chosen in a neighborhood of the equilibrium points. This analysis demonstrates that the stability of the equilibrium points is governed by the parameter . Variations in induce transitions between stable and unstable dynamics, highlighting the bifurcation structure of the system, in agreement with recent studies on bifurcation and dynamical behavior in nonlinear wave equations 27 , 29 . Chaotic behavior Bifurcations may lead to chaotic dynamics, resulting in complex and seemingly unpredictable solutions of nonlinear evolution equations 30 . As demonstrated in the previous section, the unperturbed planar dynamical system ( 37 ) derived from the reduced model is non-chaotic, as confirmed by its equilibrium structure and phase portraits. Chaotic behavior does not arise intrinsically from the original system but emerges only after the introduction of external perturbations. To examine this effect, we consider the perturbed form of system ( 37 ), given by 38 where represents an external perturbation term. Perturbation-induced chaos in reduced dynamical systems has been widely reported in the literature 30 . (i) Gaussian perturbation. Let , then the corresponding two-dimensional and three-dimensional phase portraits are shown in Fig. 9 . This perturbation induces irregular dynamics; however, it is mainly included here for comparative purposes and does not represent periodic external forcing. Fig. 9. Open in a new tab phase portrait for the system ( 38 ), when , and . (ii) Trigonometric perturbation. Motivated by its physical relevance and established effectiveness in generating chaotic dynamics, we next consider the trigonometric perturbation Trigonometric perturbations are known to play a dominant role in producing sustained and physically meaningful chaotic behavior in nonlinear evolution equations and their traveling-wave reductions 31 . The resulting two-dimensional and three-dimensional phase portraits are presented in Fig. 10 . Compared with the Gaussian case, this perturbation yields more pronounced and structured chaotic dynamics, highlighting the strong influence of periodic external forcing. Fig. 10. Open in a new tab Phase portraits of system ( 38 ) for , , and . Overall, the above analysis confirms that the chaotic behavior observed in system ( 38 ) is induced solely by external perturbations. In particular, trigonometric perturbations provide a robust and physically relevant mechanism for chaos generation, while the original unperturbed system remains non-chaotic 30 . Discussion The space–time V-O FLWE has been employed to investigate nonlinear longitudinal wave propagation in MEE materials. Periodic, kink-type, and solitary travelling-wave solutions were obtained using the Caputo V-O formulation in combination with the exp -expansion method. The results demonstrate that the spatially and temporally varying fractional orders and strongly influence wave amplitude, dispersion behavior, and stability characteristics, highlighting the effectiveness of V-O operators in capturing memory and nonlocal effects in complex media. The spatial–temporal variability of and introduces an additional degree of freedom into the model, leading to a continuum of nonlinear wave states rather than the static solitary structures typical of C-O formulations. In this setting, soliton amplitude, phase, and propagation velocity can locally adapt to the prescribed order profiles, reflecting the adaptive nature of wave dynamics in heterogeneous MEE materials. A detailed comparison with C-O results is presented in Table 2 , where the enhanced descriptive capability of the V-O model is evident. Table 2. Comparative features of the present V-O MEE model and the constant–fractional model of Munny Khatun et al. 32 . Aspect Present V-O MEE Model Munny Khatun et al . Fractional order type Space–time variable orders and Constant fractional order (time–space fractional derivative) Wave profiles obtained Periodic, kink-type, and solitary wave solutions with order-dependent modulation Bell, W–shaped, kink, and periodic soliton solutions derived under constant fractional order Dispersion and memory control Independent tuning of temporal memory and spatial dispersion via and Memory and dispersion governed by a single fractional index Stability behavior Parameter-dependent stability regimes influenced by variable fractional orders Stability and sensitivity analyzed with respect to model parameters including Bifurcation nature Order-dependent transition behavior with shifting equilibrium structure Classical bifurcation behavior analyzed under fixed fractional order setting Physical implication Captures spatial heterogeneity and adaptive memory effects in MEE materials Models fractional memory effects in a uniform MEE circular rod with constant order Open in a new tab Parametric analysis reveals that decreasing the temporal order increases waveform amplitude while reducing propagation speed, whereas lowering the spatial order enhances pulse localization and energy concentration. Such behaviors are absent in the C-O fractional model reported by Munny Khatun et al 32 ., where uniform fractional indices limit the ability to capture localized dynamical variations. Furthermore, variations in local parameters are shown to induce transitions among bell-type, kink-type, and breather-type wave structures, indicating richer nonlinear interactions among the elastic, electric, and magnetic fields. The bifurcation and stability analyses confirm that the V-O formulation supports a broader and more physically admissible range of dynamical responses compared to its C-O counterpart. The present study is limited to an analytical investigation based on an approximate travelling-wave reduction of the space–time V-O fractional model. The derived solutions represent approximate analytical structures of the reduced system and do not constitute global solutions of the original fractional partial differential equation. In addition, the analysis is restricted to bounded space–time domains and assumes smoothly and slowly varying fractional orders. Experimental validation is not addressed here; however, such efforts are essential to assess the physical realizability of the proposed model. Future work may extend the formulation to multi-dimensional geometries, incorporate thermal and electromagnetic feedback effects, and validate the theoretical findings through numerical simulations or experimental investigations. Overall, the proposed V-O fractional framework provides a robust theoretical foundation for analyzing nonlinear wave propagation in coupled electromechanical systems. Conclusion This study examined nonlinear longitudinal wave propagation in MEE materials using a space-time V-O FLWE. The main objective was to assess whether Caputo-type V-O fractional operators, combined with an analytical solution technique, can represent spatial heterogeneity and temporal memory effects beyond the capabilities of C-O models.Approximate traveling-wave solutions were obtained using the –expansion method, yielding periodic, kink-type, solitary, and singular wave structures. The results indicate that the temporal fractional order primarily affects memory behavior and propagation speed, while the spatial fractional order governs wave localization and dispersion. Stability and bifurcation analyses of the reduced dynamical system reveal parameter-dependent transitions between stable and unstable regimes. Furthermore, chaotic dynamics arise only in the presence of external perturbations, whereas the unperturbed system remains non-chaotic.The proposed V-O fractional framework offers enhanced analytical flexibility for modeling heterogeneous wave dynamics in MEE materials at a theoretical level. However, the analysis is subject to limitations. The traveling-wave reduction is approximate due to the lack of a general chain rule for V-O fractional derivatives, and the solutions are valid within bounded space-time domains under smoothly varying fractional orders. Moreover, the study is purely analytical and does not include experimental validation. Future research may extend the present formulation to higher-dimensional geometries, incorporate additional physical couplings, and validate the theoretical predictions through numerical or experimental investigations. Acknowledgements This research was supported by the Ministry of Higher Education Malaysia (MOHE) under the Fundamental Research Grant Scheme (FRGS/1/2023/STG06/USM/02/6), Universiti Sains Malaysia, and by the Princess Nourah bint Abdulrahman University Researchers Supporting Project (PNURSP2026R66), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. Author contributions Dr. Muhammad Asim khan: Writing final draft, software, analyzed the results, and analysis. Dr.Majid Khan Majahar Ali : Proofread, revision, supervision. Saratha Sathasivam: Proofread, revision, supervision. Dr. Naglaa F. Soliman: Project administration, Resources, Review & Editing. Dr. Abeer D. Algarni: Project administration, Resources, Review & Editing. Funding This research was supported by the Ministry of Higher Education Malaysia (MOHE) under the Fundamental Research Grant Scheme (FRGS/1/2023/STG06/USM/02/6) and by the Princess Nourah bint Abdulrahman University Researchers Supporting Project (PNURSP2026R66), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. Data availability All data generated or analysed during this study are included in this published article. Declarations Competing interests The authors declare no competing interests. Footnotes Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Contributor Information Majid Khan Majahar Ali, Email: [email protected]. Saratha Sathasivam, Email: [email protected]. References 1. Aibinu, M. & Momoniat, E. Approximate analytical solutions and applications of pantograph-type equations with caputo derivative and variable orders. Appl. Math. Sci. Eng. 31 , 2232091 (2023). 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