Pub Date : 2026-03-01Epub Date: 2025-12-13DOI: 10.1016/j.padiff.2025.101329
Realeboga Dikole
This article investigates nonlinear Dirac equations (NLD) with cubic-type nonlinearities, that is, vector and scalar self-interaction nonlinearities. We present analytical solutions of gap-solitons, which are self-localised, moving or quiescent pulses existing in the band gaps of nonlinear Dirac models. We also perform the linear stability analysis of the gap-soliton bearing systems and find that the gap-solitons possess some regions of instability. We also extend our studies to planar nonlinear Dirac equations and relate them to light propagation in photonic lattices, such as photonic graphene and present their numerical solutions, in particular, the rotationally symmetric localised radial profiles that rotate about the Brillouin zone.
{"title":"Spinor solitons in one-dimensional and planar nonlinear Dirac equations","authors":"Realeboga Dikole","doi":"10.1016/j.padiff.2025.101329","DOIUrl":"10.1016/j.padiff.2025.101329","url":null,"abstract":"<div><div>This article investigates nonlinear Dirac equations (NLD) with cubic-type nonlinearities, that is, vector and scalar self-interaction nonlinearities. We present analytical solutions of gap-solitons, which are self-localised, moving or quiescent pulses existing in the band gaps of nonlinear Dirac models. We also perform the linear stability analysis of the gap-soliton bearing systems and find that the gap-solitons possess some regions of instability. We also extend our studies to planar nonlinear Dirac equations and relate them to light propagation in photonic lattices, such as photonic graphene and present their numerical solutions, in particular, the rotationally symmetric localised radial profiles that rotate about the Brillouin zone.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"17 ","pages":"Article 101329"},"PeriodicalIF":0.0,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-03-01Epub Date: 2026-02-14DOI: 10.1016/j.padiff.2026.101347
Faiza Arif , Adil Jhangeer , F.D. Zaman , F.M. Mahomed
In the present work, we focus on discussing detailed analysis and important features of a (2+1)-dimensional nonlinear time-fractional diffusion-wave equation. The equation models wave phenomena in materials where memory effects play an important role, for example porous mediums and viscoelastic structures. We use the Lie symmetry methods together with the traveling wave reductions to obtain the exact solutions. Some of the solutions are expressed using special functions, like the Mittag-Leffler function and the Lambert W function. These functions describe the role of nonlinearity and fractional-order temporal damping on wave propagation. The graphical representation of the solutions suggest that the parameter α more or less dictates the behavior of the wave by affecting its concentration in space, decay rate, and speed at which it propagates. Further, the system correspond to distinct physical behaviors depending on different values of α, for example, sub-diffusive (0 < α < 1), diffusive , or wave-like motion with memory effects (1 < α ≤ 2). From these observation, it becomes clear that the dynamics of the system strongly gets affected when fractional memory combines with nonlinearity. This effect may have possible applications in modeling transportation processes as well as viscoelastic wave propagation arising in biological systems and porous materials.
{"title":"Lie symmetry and memory-driven dynamics of a (2+1)-dimensional time-fractional nonlinear wave equation in memory media","authors":"Faiza Arif , Adil Jhangeer , F.D. Zaman , F.M. Mahomed","doi":"10.1016/j.padiff.2026.101347","DOIUrl":"10.1016/j.padiff.2026.101347","url":null,"abstract":"<div><div>In the present work, we focus on discussing detailed analysis and important features of a (2+1)-dimensional nonlinear time-fractional diffusion-wave equation. The equation models wave phenomena in materials where memory effects play an important role, for example porous mediums and viscoelastic structures. We use the Lie symmetry methods together with the traveling wave reductions to obtain the exact solutions. Some of the solutions are expressed using special functions, like the Mittag-Leffler function and the Lambert <em>W</em> function. These functions describe the role of nonlinearity and fractional-order temporal damping on wave propagation. The graphical representation of the solutions suggest that the parameter <em>α</em> more or less dictates the behavior of the wave by affecting its concentration in space, decay rate, and speed at which it propagates. Further, the system correspond to distinct physical behaviors depending on different values of <em>α</em>, for example, sub-diffusive (0 < <em>α</em> < 1), diffusive <span><math><mrow><mo>(</mo><mi>α</mi><mo>=</mo><mn>1</mn><mo>)</mo></mrow></math></span>, or wave-like motion with memory effects (1 < <em>α</em> ≤ 2). From these observation, it becomes clear that the dynamics of the system strongly gets affected when fractional memory combines with nonlinearity. This effect may have possible applications in modeling transportation processes as well as viscoelastic wave propagation arising in biological systems and porous materials.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"17 ","pages":"Article 101347"},"PeriodicalIF":0.0,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147397038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-03-01Epub Date: 2026-02-13DOI: 10.1016/j.padiff.2026.101344
Mesfin Mekuria Woldaregay, Tibebu Worku Hunde
A novel approach has been introduced to address time-fractional singularly perturbed parabolic partial differential equations. This method utilizes the L1-Caputo finite difference technique to approximate the fractional derivative term and employs an exact difference scheme for spatial derivative approximation on a Shishkin mesh. Conventional numerical methods in FDM, FEM and Collocation methods relying on uniform meshes often fail to provide accurate solutions due to the presence of boundary layers. The proposed method overcomes this limitation by ensuring the discrete maximum principle, stability bounds, and uniform convergence while effectively resolving boundary layers. Numerical experiments have confirmed the effectiveness of the scheme across various perturbation parameter values and mesh sizes.
{"title":"Exact difference approach on the Shishkin mesh for solving time fractional singularly perturbed parabolic PDE","authors":"Mesfin Mekuria Woldaregay, Tibebu Worku Hunde","doi":"10.1016/j.padiff.2026.101344","DOIUrl":"10.1016/j.padiff.2026.101344","url":null,"abstract":"<div><div>A novel approach has been introduced to address time-fractional singularly perturbed parabolic partial differential equations. This method utilizes the <em>L</em>1-Caputo finite difference technique to approximate the fractional derivative term and employs an exact difference scheme for spatial derivative approximation on a Shishkin mesh. Conventional numerical methods in FDM, FEM and Collocation methods relying on uniform meshes often fail to provide accurate solutions due to the presence of boundary layers. The proposed method overcomes this limitation by ensuring the discrete maximum principle, stability bounds, and uniform convergence while effectively resolving boundary layers. Numerical experiments have confirmed the effectiveness of the scheme across various perturbation parameter values and mesh sizes.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"17 ","pages":"Article 101344"},"PeriodicalIF":0.0,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147397039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-03-01Epub Date: 2026-02-13DOI: 10.1016/j.padiff.2026.101345
Shaher Momani , Fatimah Noor Harun , Rasha Amryeen , Shrideh Al-Omari , Mohammed Al-Smadi
This work concerns the construction of the approximate analytical solutions for the nonlinear complex conformable Ginzburg-Landau equations with external potential using the conformable residual series method. The governing model plays a pivotal role in modeling complex physical phenomena such as Bose-Einstein condensation and building approximate analytical solutions for this model, which is considered a distinctive addition given the scarcity of work presented in the literature in this context. The methodology lies in combines of generalized multivariable power series and residual error function. Convergence analysis is provided to illustrate the theoretical framework of our scheme in handling the projected nonlinear models. For a sake of practical computation, several naturalistic applications for Bose-Einstein condensates are examined involving zero trapping, periodic box, optical lattice, and harmonic potentials. In this orientation, numeric computations and graphical representations are provided to verify the correctness and accuracies of the tested applications. The dynamic behaviors of wave soliton solutions are captured at different parameters in addition to the comparison of acquired wave solutions with previous studies. The overall impact of this work lies in the ease with which the proposed approach can be applied to construct efficient and systematic approximate analytical solutions for complex nonlinear partial differential equations arising in quantum optics, quantum gases, quantum fluids, and other quantum mechanical phenomena.
{"title":"Promoted analytical solutions of conformable Ginzburg-Landau applied in Bose-Einstein condensate with external potentials","authors":"Shaher Momani , Fatimah Noor Harun , Rasha Amryeen , Shrideh Al-Omari , Mohammed Al-Smadi","doi":"10.1016/j.padiff.2026.101345","DOIUrl":"10.1016/j.padiff.2026.101345","url":null,"abstract":"<div><div>This work concerns the construction of the approximate analytical solutions for the nonlinear complex conformable Ginzburg-Landau equations with external potential using the conformable residual series method. The governing model plays a pivotal role in modeling complex physical phenomena such as Bose-Einstein condensation and building approximate analytical solutions for this model, which is considered a distinctive addition given the scarcity of work presented in the literature in this context. The methodology lies in combines of generalized multivariable power series and residual error function. Convergence analysis is provided to illustrate the theoretical framework of our scheme in handling the projected nonlinear models. For a sake of practical computation, several naturalistic applications for Bose-Einstein condensates are examined involving zero trapping, periodic box, optical lattice, and harmonic potentials. In this orientation, numeric computations and graphical representations are provided to verify the correctness and accuracies of the tested applications. The dynamic behaviors of wave soliton solutions are captured at different parameters in addition to the comparison of acquired wave solutions with previous studies. The overall impact of this work lies in the ease with which the proposed approach can be applied to construct efficient and systematic approximate analytical solutions for complex nonlinear partial differential equations arising in quantum optics, quantum gases, quantum fluids, and other quantum mechanical phenomena.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"17 ","pages":"Article 101345"},"PeriodicalIF":0.0,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147397042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-03-01Epub Date: 2026-03-06DOI: 10.1016/j.padiff.2026.101349
Sharanayya Swami , Suresh Biradar , Jagadish V. Tawade , Nitiraj V. Kulkarni , Barno Sayfutdinovna Abdullaeva , Dana Mohammad Khidhir , Nadia Batool , Taoufik Saidani
{"title":"Retraction Notice to “Heat and mass transfer analysis of Williamson nanofluids under the influence of magnetic field and Joule's heating” [Partial Differential Equations in Applied Mathematics 13 (2025) 101148]","authors":"Sharanayya Swami , Suresh Biradar , Jagadish V. Tawade , Nitiraj V. Kulkarni , Barno Sayfutdinovna Abdullaeva , Dana Mohammad Khidhir , Nadia Batool , Taoufik Saidani","doi":"10.1016/j.padiff.2026.101349","DOIUrl":"10.1016/j.padiff.2026.101349","url":null,"abstract":"","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"17 ","pages":"Article 101349"},"PeriodicalIF":0.0,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147394923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-03-01Epub Date: 2025-12-13DOI: 10.1016/j.padiff.2025.101332
G. Shylaja , V. Kesavulu Naidu , B. Venkatesh , S.M. Mallikarjunaiah
This paper presents an explicit integration scheme that incorporates septic-order triangular elements. Cubic arcs are utilized to approximate the curved edges of these elements. This methodology is particularly effective for discretizing curved domains, and its primary objective is the approximation of equations involving differential operators. A 36-node septic-order triangular element with a curved boundary, which consists of one curved edge and two straight edges, is introduced in this study. This element serves as the foundation for the isoparametric coordinate transformation discussed herein. A standard triangle in the local coordinate system is mapped onto the curved triangular element in the global coordinate system by means of a unique point transformation. The curved triangular element is replaced by septic arcs, and the coordinates located on the curved edge are embedded within the parameters that define these arc equations. Each arc consistently represents a distinct cubic arc due to the relationships involved in the parameter calculations. Consequently, the overall numerical approximation is highly accurate. For higher-order curved triangular elements, the finite element method, in conjunction with numerical integration that utilizes curved boundary point transformations (applicable to both the exterior and interior of each curved triangular element), will act as a robust subparametric coordinate transformation and, as a result, a formidable numerical technique. The efficacy of this method is demonstrated through the resolution of three boundary value problems. Numerical outcomes affirm that the proposed technique significantly surpasses existing methods in the approximation of boundary value problems.
{"title":"Septic-order triangular finite elements: An explicit method with cubic arc subparametric transformations","authors":"G. Shylaja , V. Kesavulu Naidu , B. Venkatesh , S.M. Mallikarjunaiah","doi":"10.1016/j.padiff.2025.101332","DOIUrl":"10.1016/j.padiff.2025.101332","url":null,"abstract":"<div><div>This paper presents an explicit integration scheme that incorporates septic-order triangular elements. Cubic arcs are utilized to approximate the curved edges of these elements. This methodology is particularly effective for discretizing curved domains, and its primary objective is the approximation of equations involving differential operators. A 36-node septic-order triangular element with a curved boundary, which consists of one curved edge and two straight edges, is introduced in this study. This element serves as the foundation for the isoparametric coordinate transformation discussed herein. A standard triangle in the local coordinate system is mapped onto the curved triangular element in the global coordinate system by means of a unique point transformation. The curved triangular element is replaced by septic arcs, and the coordinates located on the curved edge are embedded within the parameters that define these arc equations. Each arc consistently represents a distinct cubic arc due to the relationships involved in the parameter calculations. Consequently, the overall numerical approximation is highly accurate. For higher-order curved triangular elements, the finite element method, in conjunction with numerical integration that utilizes curved boundary point transformations (applicable to both the exterior and interior of each curved triangular element), will act as a robust subparametric coordinate transformation and, as a result, a formidable numerical technique. The efficacy of this method is demonstrated through the resolution of three boundary value problems. Numerical outcomes affirm that the proposed technique significantly surpasses existing methods in the approximation of boundary value problems.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"17 ","pages":"Article 101332"},"PeriodicalIF":0.0,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791937","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-03-01Epub Date: 2025-12-12DOI: 10.1016/j.padiff.2025.101325
Dian K. Palagachev
The non-homogeneous conormal derivative problems for nonlinear, second-order divergence form elliptic equations with singular data appear naturally in mathematical modeling of real phenomena involving problems of image recovery, the thermistor problem, or studies of non-Newtonian fluids.
We prove suitable estimates for certain surface integrals, related to non-homogeneous conormal derivative problems, which lead to essential boundedness of the weak solutions under quite general hypotheses on the data.
{"title":"On certain surface integrals related to the conormal derivative problem","authors":"Dian K. Palagachev","doi":"10.1016/j.padiff.2025.101325","DOIUrl":"10.1016/j.padiff.2025.101325","url":null,"abstract":"<div><div>The non-homogeneous conormal derivative problems for nonlinear, second-order divergence form elliptic equations with singular data appear naturally in mathematical modeling of real phenomena involving problems of image recovery, the thermistor problem, or studies of non-Newtonian fluids.</div><div>We prove suitable estimates for certain surface integrals, related to non-homogeneous conormal derivative problems, which lead to essential boundedness of the weak solutions under quite general hypotheses on the data.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"17 ","pages":"Article 101325"},"PeriodicalIF":0.0,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145760656","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-01Epub Date: 2025-09-19DOI: 10.1016/j.padiff.2025.101299
Mehari Fentahun Endalew, Xiaoming Zhang
Hybrid nanofluids have emerged as a promising medium for enhancing heat transfer, with power-law hybrid nanofluids (PLHNF) exhibiting superior thermal conductivity compared to conventional power-law nanofluids (PLNF). Despite these advantages, their transport behavior under complex flow conditions — particularly in ionized Darcy–Forchheimer regimes influenced by slip effects and non-classical heat conduction — remains largely unexplored. This study addresses this gap by developing a comprehensive theoretical framework for PLHNF flow over a stretching surface, incorporating magnetic field inclination, Navier slip, and a modified Fourier’s law of heat conduction. The governing nonlinear system is transformed via similarity techniques and solved numerically using MATLAB’s bvp4c solver, with validation against established benchmarks. The findings reveal that PLHNF not only sustain higher thermal transport but also exhibit distinctive flow responses: velocity slip significantly suppresses both axial and radial components, while inclined magnetic fields enhance axial transport but reduce radial motion. The superior thermal conductivity of PLHNF amplifies these effects, yielding higher surface heat transfer rates compared to PLNF. By elucidating the coupled influence of magnetic, slip, and non-Fourier heat conduction effects, this work extends the theoretical foundation of non-Newtonian hybrid nanofluids and highlights their potential for high-efficiency thermal management systems.
{"title":"Analysis of Navier slip effects in ionized power-law hybrid nanofluid flow through a Darcy–Forchheimer porous medium with modified Fourier heat transfer","authors":"Mehari Fentahun Endalew, Xiaoming Zhang","doi":"10.1016/j.padiff.2025.101299","DOIUrl":"10.1016/j.padiff.2025.101299","url":null,"abstract":"<div><div>Hybrid nanofluids have emerged as a promising medium for enhancing heat transfer, with power-law hybrid nanofluids (PLHNF) exhibiting superior thermal conductivity compared to conventional power-law nanofluids (PLNF). Despite these advantages, their transport behavior under complex flow conditions — particularly in ionized Darcy–Forchheimer regimes influenced by slip effects and non-classical heat conduction — remains largely unexplored. This study addresses this gap by developing a comprehensive theoretical framework for PLHNF flow over a stretching surface, incorporating magnetic field inclination, Navier slip, and a modified Fourier’s law of heat conduction. The governing nonlinear system is transformed via similarity techniques and solved numerically using MATLAB’s bvp4c solver, with validation against established benchmarks. The findings reveal that PLHNF not only sustain higher thermal transport but also exhibit distinctive flow responses: velocity slip significantly suppresses both axial and radial components, while inclined magnetic fields enhance axial transport but reduce radial motion. The superior thermal conductivity of PLHNF amplifies these effects, yielding higher surface heat transfer rates compared to PLNF. By elucidating the coupled influence of magnetic, slip, and non-Fourier heat conduction effects, this work extends the theoretical foundation of non-Newtonian hybrid nanofluids and highlights their potential for high-efficiency thermal management systems.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"16 ","pages":"Article 101299"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145119553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-01Epub Date: 2025-09-27DOI: 10.1016/j.padiff.2025.101305
Shaher Momani , Rabha W. Ibrahim
This work investigates soliton solutions of nonlinear wave equations modeling light propagation in optical metamaterials with nonlocal nonlinear responses, incorporating external optical potentials. The residual power series method (RPSM) is employed to construct enhanced analytical solutions, capturing both dispersive and memory effects effectively. In addition, this study investigates the propagation of solitons in optical metamaterials with nonlocal responses using -fractional calculus. This calculus is based on the generalization of the quantum gamma function (). By employing -fractional derivatives in the form of the -Mittag-Leffler function, we explore the dynamics of soliton fields in these materials. The model considers key parameters such as the fractional order , the generalized parameters and , and the initial weight parameter . The flexibility of these parameters allows for a more accurate description of optical metamaterials, capturing both classical soliton behavior and more complex nonlocal and memory effects. We compare fractional models with classical models and demonstrate the advantages of using fractional calculus to model memory effects and nonlocal interactions. Numerical simulations, including the residual series method, reveal the enhanced accuracy and insights provided by the fractional approach in optical metamaterials. The study provides a detailed framework for understanding soliton propagation in advanced optical materials, paving the way for the design of next-generation optical devices.
{"title":"Soliton propagation in optical metamaterials with nonlocal responses: A fractional calculus approach using (q,τ)-Mittag-Leffler functions","authors":"Shaher Momani , Rabha W. Ibrahim","doi":"10.1016/j.padiff.2025.101305","DOIUrl":"10.1016/j.padiff.2025.101305","url":null,"abstract":"<div><div>This work investigates soliton solutions of nonlinear wave equations modeling light propagation in optical metamaterials with nonlocal nonlinear responses, incorporating external optical potentials. The residual power series method (RPSM) is employed to construct enhanced analytical solutions, capturing both dispersive and memory effects effectively. In addition, this study investigates the propagation of solitons in optical metamaterials with nonlocal responses using <span><math><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>τ</mi><mo>)</mo></mrow></math></span>-fractional calculus. This calculus is based on the generalization of the quantum gamma function (<span><math><mrow><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>τ</mi><mo>)</mo></mrow><mo>−</mo><mi>Γ</mi><mrow><mo>(</mo><mo>.</mo><mo>)</mo></mrow></mrow></math></span>). By employing <span><math><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>τ</mi><mo>)</mo></mrow></math></span>-fractional derivatives in the form of the <span><math><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>τ</mi><mo>)</mo></mrow></math></span>-Mittag-Leffler function, we explore the dynamics of soliton fields in these materials. The model considers key parameters such as the fractional order <span><math><mi>α</mi></math></span>, the generalized parameters <span><math><mi>q</mi></math></span> and <span><math><mi>τ</mi></math></span>, and the initial weight parameter <span><math><mi>β</mi></math></span>. The flexibility of these parameters allows for a more accurate description of optical metamaterials, capturing both classical soliton behavior and more complex nonlocal and memory effects. We compare fractional models with classical models and demonstrate the advantages of using fractional calculus to model memory effects and nonlocal interactions. Numerical simulations, including the residual series method, reveal the enhanced accuracy and insights provided by the fractional approach in optical metamaterials. The study provides a detailed framework for understanding soliton propagation in advanced optical materials, paving the way for the design of next-generation optical devices.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"16 ","pages":"Article 101305"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145221956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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