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Transition from circular to spiral waves and from Mexican hat to upside-down Mexican hat-solutions: The cases of local and nonlocal λ−ω reaction-diffusion-convection fractal systems with variable coefficients

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  • El-Nabulsi, Rami Ahmad

Abstract

Nonlinear partial differential equations admitting traveling wave solutions play an important role in the description and analysis of real-life physical processes and nonlinear phenomena. In this study, we prove that the excitable λ−ωreaction-diffusion-convection system introduced by Kopell and Howard can exhibit, in fractal dimensions, a large variety of spatial patterns. We have considered two independent models: a local reaction-diffusion-convection model characterized by variable coefficients that are subject to particular power laws and a nonlocal reaction-diffusion model characterized by symmetric kernels and a variable diffusion coefficient. Each model is characterized by a number of motivating properties and features. In the 1st model, the amplitude is governed by a 2nd-order differential equation, whereas in the 2nd-model, the amplitude is governed by a 4th-order differential equation, which is, under some conditions, comparable to the Swift-Hohenberg equation with variable coefficients that arise in the study of pattern formation, which belongs to the family of extended Fisher-Kolmogorov stationary equations used to study pattern-forming systems in biological and chemical systems. We report the emergence of superstructures that are suppressed for fractal dimensions much less than unity. These superstructures include superspiral waves characterized by a circular symmetry detected in various oscillatory media and the emergence of reflection of waves that take place in non-uniform reaction-diffusion systems, besides the emergence of micro-spiral waves that emerge at the cellular level. A transition from spiral waves to perfectly rotating waves is observed, besides a transition from Mexican hat shaped solutions to upside-down Mexican hat shaped solutions. The domain size has a very strong impact on the rotational frequency of spiral and circular waves. These new phenomena associated with configuration patterns through a reaction-diffusion-convection system with different scales and characterized by variable coefficients can be applied for modeling a wide class of reaction-diffusion-convection problems. Supplementary properties have been obtained and discussed accordingly.

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  • El-Nabulsi, Rami Ahmad, 2024. "Transition from circular to spiral waves and from Mexican hat to upside-down Mexican hat-solutions: The cases of local and nonlocal λ−ω reaction-diffusion-convection fractal systems with variable coef," Chaos, Solitons & Fractals, Elsevier, vol. 189(P2).
  • Handle: RePEc:eee:chsofr:v:189:y:2024:i:p2:s096007792401289x
    DOI: 10.1016/j.chaos.2024.115737
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    References listed on IDEAS

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    1. Polyanin, Andrei D., 2019. "Functional separable solutions of nonlinear reaction–diffusion equations with variable coefficients," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 282-292.
    2. Tang, Jun & Luo, Jin-Ming & Ma, Jun & Yi, Ming & Yang, Xian-Qing, 2013. "Spiral waves in systems with fractal heterogeneity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(22), pages 5764-5771.
    3. Heng Cheng & Zebin Xing & Yan Liu, 2023. "The Improved Element-Free Galerkin Method for 3D Steady Convection-Diffusion-Reaction Problems with Variable Coefficients," Mathematics, MDPI, vol. 11(3), pages 1-19, February.
    4. Qureshi, Sania & Aziz, Shaheen, 2020. "Fractional modeling for a chemical kinetic reaction in a batch reactor via nonlocal operator with power law kernel," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 542(C).
    5. Alanoud Almutairi & Ali Hasan Ali & Omar Bazighifan & Loredana Florentina Iambor, 2023. "Oscillatory Properties of Fourth-Order Advanced Differential Equations," Mathematics, MDPI, vol. 11(6), pages 1-11, March.
    6. Ma, Jun & Wang, Chun-Ni & Li, Yan-Long & Li, Shi-Rong, 2007. "Suppression of spiral waves in light-sensitive media using chaotic signal modulated scheme," Chaos, Solitons & Fractals, Elsevier, vol. 33(3), pages 965-970.
    7. Karthikeyan Rajagopal & Shirin Panahi & Mo Chen & Sajad Jafari & Bocheng Bao, 2021. "Suppressing Spiral Wave Turbulence In A Simple Fractional-Order Discrete Neuron Map Using Impulse Triggering," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(08), pages 1-10, December.
    8. Atangana, Abdon, 2017. "Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 396-406.
    9. Sameerah Jamal, 2020. "Imaging Noise Suppression: Fourth-Order Partial Differential Equations and Travelling Wave Solutions," Mathematics, MDPI, vol. 8(11), pages 1-10, November.
    10. Rami Ahmad El-Nabulsi & Waranont Anukool, 2023. "A generalized nonlinear cubic-quartic Schrodinger equation and its implications in quantum wire," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 96(5), pages 1-8, May.
    11. Bo Yan & Shaobo He & Shaojie Wang, 2020. "Multistability and Formation of Spiral Waves in a Fractional-Order Memristor-Based Hyperchaotic Lü System with No Equilibrium Points," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-12, June.
    12. El-Nabulsi, Rami Ahmad & Anukool, Waranont, 2023. "A family of nonlinear Schrodinger equations and their solitons solutions," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    13. Etémé, A.S. & Tabi, C.B. & Mohamadou, A. & Kofané, T.C., 2019. "Elimination of spiral waves in a two-dimensional Hindmarsh–Rose neural network under long-range interaction effect and frequency excitation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 533(C).
    14. Atangana, Abdon & Khan, Muhammad Altaf, 2019. "Validity of fractal derivative to capturing chaotic attractors," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 50-59.
    15. Taher S. Hassan & Qingkai Kong & Rami Ahmad El-Nabulsi & Waranont Anukool, 2022. "New Hille Type and Ohriska Type Criteria for Nonlinear Third-Order Dynamic Equations," Mathematics, MDPI, vol. 10(21), pages 1-12, November.
    16. Zhou, Jiaying & Ye, Yong & Arenas, Alex & Gómez, Sergio & Zhao, Yi, 2023. "Pattern formation and bifurcation analysis of delay induced fractional-order epidemic spreading on networks," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    17. Ma, Jun & Jia, Ya & Yi, Ming & Tang, Jun & Xia, Ya-Feng, 2009. "Suppression of spiral wave and turbulence by using amplitude restriction of variable in a local square area," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1331-1339.
    18. Clemens Bachmair & Eckehard Schöll, 2014. "Nonlocal control of pulse propagation in excitable media," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 87(11), pages 1-10, November.
    19. Ji-Huan He & Yusry O. El-Dib, 2021. "A Tutorial Introduction To The Two-Scale Fractal Calculus And Its Application To The Fractal Zhiber–Shabat Oscillator," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(08), pages 1-9, December.
    20. Wang, Zhen & Rostami, Zahra & Jafari, Sajad & Alsaadi, Fawaz E. & Slavinec, Mitja & Perc, Matjaž, 2019. "Suppression of spiral wave turbulence by means of periodic plane waves in two-layer excitable media," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 229-233.
    21. Osama Moaaz & Belgees Qaraad & Rami Ahmad El-Nabulsi & Omar Bazighifan, 2020. "New Results for Kneser Solutions of Third-Order Nonlinear Neutral Differential Equations," Mathematics, MDPI, vol. 8(5), pages 1-12, May.
    22. Meron, Ehud, 2012. "Pattern-formation approach to modelling spatially extended ecosystems," Ecological Modelling, Elsevier, vol. 234(C), pages 70-82.
    23. Hail S. Alrashdi & Osama Moaaz & Khaled Alqawasmi & Mohammad Kanan & Mohammed Zakarya & Elmetwally M. Elabbasy, 2024. "Asymptotic and Oscillatory Properties of Third-Order Differential Equations with Multiple Delays in the Noncanonical Case," Mathematics, MDPI, vol. 12(8), pages 1-15, April.
    24. Osama Moaaz & Rami Ahmad El-Nabulsi & Ali Muhib & Sayed K. Elagan & Mohammed Zakarya, 2021. "New Improved Results for Oscillation of Fourth-Order Neutral Differential Equations," Mathematics, MDPI, vol. 9(19), pages 1-12, September.
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