Fractal formation and chaotic soliton phenomena in nonlinear conformable Heisenberg ferromagnetic spin chain equation
The present study constructs and investigates solitonic phenomena in the complex structured (3+1)-dimensional conformable Heisenberg ferromagnetic spin chain equation (CHFSCE). This model explains the behavior of ferromagnetic spin chains and is an extension of the integer-order Heisenberg equation...
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De Gruyter
2025-06-01
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| Series: | Open Physics |
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| Online Access: | https://doi.org/10.1515/phys-2025-0166 |
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| author | Ali Rashid Imtiaz Rabia Junjua Moin-ud-Din Awwad Fuad A. Ismail Emad A. A. Hendy Ahmed S. |
| author_facet | Ali Rashid Imtiaz Rabia Junjua Moin-ud-Din Awwad Fuad A. Ismail Emad A. A. Hendy Ahmed S. |
| author_sort | Ali Rashid |
| collection | DOAJ |
| description | The present study constructs and investigates solitonic phenomena in the complex structured (3+1)-dimensional conformable Heisenberg ferromagnetic spin chain equation (CHFSCE). This model explains the behavior of ferromagnetic spin chains and is an extension of the integer-order Heisenberg equation in nonlinear physics that controls the magnetization of the ferromagnetic solid. We present a new array of soliton solutions in the form exponential, rational, hyperbolic, and rational-hyperbolic functions, using the Riccati modified extended simple equation method (RMESEM). The proposed anstaz uses a complex structured wave transformation to convert CHFSCE into a more manageable nonlinear ordinary differential equation (NODE) and constraint conditions. The resulting NODE is assumed to have a close form solution that further converts it into a system of nonlinear algebraic equations via substitution in order to identify fresh plethora of optical soliton solutions. Moreover, the fundamental characteristics and theory of the employed conformable derivative, specifically, the chain rule, have been described. We demonstrate the existence of quasi-periodic solitons, including smooth-, multiple-, and periodic-periodic solitons, in the context of CHFSCE using a range of 3D, 2D, and contour visual representations. The obtained quasi-periodic solitons prominently result in the development of fractal structures while their squared norms result in the development of hump, peakon, and parabolic solitons. Additionally, we investigate bifurcating and chaotic behavior, detecting its existence in the perturbed dynamical system and finding advantageous results that suggest quasi-periodicity and fractal behavior in the model. According to the results we obtained, the suggested strategy is a powerful method for detecting novel soliton phenomena in such types of nonlinear settings. |
| format | Article |
| id | doaj-art-b7111d66eb3b412d978b79d5a9caef82 |
| institution | DOAJ |
| issn | 2391-5471 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Open Physics |
| spelling | doaj-art-b7111d66eb3b412d978b79d5a9caef822025-08-20T03:20:13ZengDe GruyterOpen Physics2391-54712025-06-012313351110.1515/phys-2025-0166Fractal formation and chaotic soliton phenomena in nonlinear conformable Heisenberg ferromagnetic spin chain equationAli Rashid0Imtiaz Rabia1Junjua Moin-ud-Din2Awwad Fuad A.3Ismail Emad A. A.4Hendy Ahmed S.5School of Information and Mathematics, Yangtze University, Jingzhou, Hubei, 434023, ChinaDepartment of Mathematics, University of Engineering and Technology, Lahore 39161, PakistanSchool of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, Zhejiang, P.R. ChinaDepartment of Quantitative Analysis, College of Business Administration, King Saud University, P.O. Box 71115, Riyadh 11587, Saudi ArabiaDepartment of Quantitative Analysis, College of Business Administration, King Saud University, P.O. Box 71115, Riyadh 11587, Saudi ArabiaDepartment of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., Yekaterinburg 620002, RussiaThe present study constructs and investigates solitonic phenomena in the complex structured (3+1)-dimensional conformable Heisenberg ferromagnetic spin chain equation (CHFSCE). This model explains the behavior of ferromagnetic spin chains and is an extension of the integer-order Heisenberg equation in nonlinear physics that controls the magnetization of the ferromagnetic solid. We present a new array of soliton solutions in the form exponential, rational, hyperbolic, and rational-hyperbolic functions, using the Riccati modified extended simple equation method (RMESEM). The proposed anstaz uses a complex structured wave transformation to convert CHFSCE into a more manageable nonlinear ordinary differential equation (NODE) and constraint conditions. The resulting NODE is assumed to have a close form solution that further converts it into a system of nonlinear algebraic equations via substitution in order to identify fresh plethora of optical soliton solutions. Moreover, the fundamental characteristics and theory of the employed conformable derivative, specifically, the chain rule, have been described. We demonstrate the existence of quasi-periodic solitons, including smooth-, multiple-, and periodic-periodic solitons, in the context of CHFSCE using a range of 3D, 2D, and contour visual representations. The obtained quasi-periodic solitons prominently result in the development of fractal structures while their squared norms result in the development of hump, peakon, and parabolic solitons. Additionally, we investigate bifurcating and chaotic behavior, detecting its existence in the perturbed dynamical system and finding advantageous results that suggest quasi-periodicity and fractal behavior in the model. According to the results we obtained, the suggested strategy is a powerful method for detecting novel soliton phenomena in such types of nonlinear settings.https://doi.org/10.1515/phys-2025-0166nonlinear fractional partial differential equationsconformable fractional derivativesriccati modified extended simple equation methodquasi-periodic solitons |
| spellingShingle | Ali Rashid Imtiaz Rabia Junjua Moin-ud-Din Awwad Fuad A. Ismail Emad A. A. Hendy Ahmed S. Fractal formation and chaotic soliton phenomena in nonlinear conformable Heisenberg ferromagnetic spin chain equation Open Physics nonlinear fractional partial differential equations conformable fractional derivatives riccati modified extended simple equation method quasi-periodic solitons |
| title | Fractal formation and chaotic soliton phenomena in nonlinear conformable Heisenberg ferromagnetic spin chain equation |
| title_full | Fractal formation and chaotic soliton phenomena in nonlinear conformable Heisenberg ferromagnetic spin chain equation |
| title_fullStr | Fractal formation and chaotic soliton phenomena in nonlinear conformable Heisenberg ferromagnetic spin chain equation |
| title_full_unstemmed | Fractal formation and chaotic soliton phenomena in nonlinear conformable Heisenberg ferromagnetic spin chain equation |
| title_short | Fractal formation and chaotic soliton phenomena in nonlinear conformable Heisenberg ferromagnetic spin chain equation |
| title_sort | fractal formation and chaotic soliton phenomena in nonlinear conformable heisenberg ferromagnetic spin chain equation |
| topic | nonlinear fractional partial differential equations conformable fractional derivatives riccati modified extended simple equation method quasi-periodic solitons |
| url | https://doi.org/10.1515/phys-2025-0166 |
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