Diversity of the soliton solutions and sensitivity analysis for a fractional stochastic dynamical system: Applications to certain ferromagnetic materials

Badr Saad T. Alkahtani; Badr Saad T. Alkahtani

doi:10.3934/math.2025649

AIMS Mathematics

2025, Volume 10, Issue 6: 14434-14458. doi: 10.3934/math.2025649

Previous Article Next Article

Research article Special Issues

Diversity of the soliton solutions and sensitivity analysis for a fractional stochastic dynamical system: Applications to certain ferromagnetic materials

Badr Saad T. Alkahtani ^,

Department of Mathematics, College of Science, King Saud University, Riyadh 11989, Saudi Arabia

Received: 12 December 2024 Revised: 29 January 2025 Accepted: 13 February 2025 Published: 23 June 2025
MSC : 35C08, 60H15

In this article, we methodically investigate the fractional stochastic Kraenkel–Manna–Merle system (KMMS), which explains how a magnetic field propagates in a ferromagnet with zero conductivity and may shed light on a number of intriguing scientific occurrences. A suitable wave transformation is used to convert the governing equation into an ordinary differential equation (ODE). We thoroughly evaluate the innovative soliton solutions in the forms of dark, bright–dark, dark–bright, periodic, singular, hyperbolic, mixed trigonometric, and rational forms using the improved $ \mathscr{F} $-expansion approach and the new extended direct algebraic method (NEDAM). Furthermore, a sensitivity analysis is carried out to investigate the impact of different factors on the behavior of the system. In order to shed light on the model's physical behavior, the study displays graphical plots of the chosen solutions using the selected methodologies. These techniques offer a strong foundation for resolving nonlinear fractional differential equations, which are crucial for simulating intricate ferromagnetistic physical processes. The resulting solutions demonstrate the fractional stochastic KMMS's complex structures and dynamic behavior.
- fractional stochastic KMMS,
- stochastic soliton solution,
- NEDAM,
- improved $ \mathscr{F} $-expansion scheme,
- sensitivity analysis
Citation: Badr Saad T. Alkahtani. Diversity of the soliton solutions and sensitivity analysis for a fractional stochastic dynamical system: Applications to certain ferromagnetic materials[J]. AIMS Mathematics, 2025, 10(6): 14434-14458. doi: 10.3934/math.2025649

Related Papers:

Abstract

In this article, we methodically investigate the fractional stochastic Kraenkel–Manna–Merle system (KMMS), which explains how a magnetic field propagates in a ferromagnet with zero conductivity and may shed light on a number of intriguing scientific occurrences. A suitable wave transformation is used to convert the governing equation into an ordinary differential equation (ODE). We thoroughly evaluate the innovative soliton solutions in the forms of dark, bright–dark, dark–bright, periodic, singular, hyperbolic, mixed trigonometric, and rational forms using the improved $ \mathscr{F} $-expansion approach and the new extended direct algebraic method (NEDAM). Furthermore, a sensitivity analysis is carried out to investigate the impact of different factors on the behavior of the system. In order to shed light on the model's physical behavior, the study displays graphical plots of the chosen solutions using the selected methodologies. These techniques offer a strong foundation for resolving nonlinear fractional differential equations, which are crucial for simulating intricate ferromagnetistic physical processes. The resulting solutions demonstrate the fractional stochastic KMMS's complex structures and dynamic behavior.

References

[1]	J. Liu, Z. Li, Bifurcation analysis and soliton solutions to the Kuralay equation via dynamic system analysis method and complete discrimination system method, Qual. Theory Dyn. Syst., 23 (2024), 126. https://doi.org/10.1007/s12346-024-00990-5 doi: 10.1007/s12346-024-00990-5
[2]	M. A. El-Shorbagy, S. Akram, M. U. Rahman, Propagation of solitary wave solutions to (4+1)-dimensional Davey-Stewartson-Kadomtsev-Petviashvili equation arise in mathematical physics and stability analysis, Partial Differ. Equ. Appl. Math., 10 (2024), 100669. https://doi.org/10.1016/j.padiff.2024.100669 doi: 10.1016/j.padiff.2024.100669
[3]	M. U. Rahman, M. Sun, S. Boulaaras, D. Baleanu, Bifurcations, chaotic behavior, sensitivity analysis, and various soliton solutions for the extended nonlinear Schrödinger equation, Bound. Value Probl., 2024 (2024), 15. https://doi.org/10.1186/s13661-024-01825-7 doi: 10.1186/s13661-024-01825-7
[4]	I. Alazman, B. S. T. Alkahtani, M. U. Rahman, M. N. Mishra, Nonlinear complex dynamical analysis and solitary waves for the (3+1)-D nonlinear extended quantum Zakharov-Kuznetsov equation, Results Phys., 58 (2024), 107432. https://doi.org/10.1016/j.rinp.2024.107432 doi: 10.1016/j.rinp.2024.107432
[5]	W. X. Qiu, Z. Z. Si, D. S. Mou, C. Q. Dai, J. T. Li, W. Liu, Data-driven vector degenerate and nondegenerate solitons of coupled nonlocal nonlinear Schrödinger equation via improved PINN algorithm, Nonlinear Dyn., 113 (2025), 4063–4076. https://doi.org/10.1007/s11071-024-09648-y doi: 10.1007/s11071-024-09648-y
[6]	Y. Q. Chen, Y. H. Tang, J. Manafian, H. Rezazadeh, M. S. Osman, Dark wave, rogue wave and perturbation solutions of Ivancevic option pricing model, Nonlinear Dyn., 105 (2021), 2539–2548. https://doi.org/10.1007/s11071-021-06642-6 doi: 10.1007/s11071-021-06642-6
[7]	B. Li, Y. Zhang, X. Li, Z. Eskandari, Q. He, Bifurcation analysis and complex dynamics of a Kopel triopoly model, J. Comput. Appl. Math., 426 (2023), 115089. https://doi.org/10.1016/j.cam.2023.115089 doi: 10.1016/j.cam.2023.115089
[8]	Z. Eskandari, Z. Avazzadeh, R. Khoshsiar Ghaziani, B. Li, Dynamics and bifurcations of a discrete‐time Lotka-Volterra model using nonstandard finite difference discretization method, Math. Meth. Appl. Sci., 48 (2025), 7197–7212. https://doi.org/10.1002/mma.8859 doi: 10.1002/mma.8859
[9]	X. Zhu, P. Xia, Q. He, Z. Ni, L. Ni, Ensemble classifier design based on perturbation binary salp swarm algorithm for classification, Comput. Model. Eng. Sci., 135 (2022), 653–671. https://doi.org/10.32604/cmes.2022.022985 doi: 10.32604/cmes.2022.022985
[10]	Q. He, P. Xia, C. Hu, B. Li, Public information, actual intervention, and inflation expectations, Transform. Bus. Econ., 21 (2022), 644–666.
[11]	R. Luo, Rafiullah, H. Emadifar, M. U. Rahman, Bifurcations, chaotic dynamics, sensitivity analysis and some novel optical solitons of the perturbed non-linear Schrödinger equation with Kerr law non-linearity, Results Phys., 54 (2023), 107133. https://doi.org/10.1016/j.rinp.2023.107133 doi: 10.1016/j.rinp.2023.107133
[12]	G. H. Tipu, W. A. Faridi, Z. Myrzakulova, R. Myrzakulov, S. A. AlQahtani, N. F. AlQahtani, et al., On optical soliton wave solutions of non-linear Kairat-X equation via new extended direct algebraic method, Opt. Quant. Electron., 56 (2024), 655. https://doi.org/10.1007/s11082-024-06369-9 doi: 10.1007/s11082-024-06369-9
[13]	A. M. Talafha, A. Jhangeer, S. S. Kazmi, Dynamical analysis of (4+1)-dimensional Davey Srewartson Kadomtsev Petviashvili equation by employing Lie symmetry approach, Ain Shams Eng. J., 14 (2023), 102537. https://doi.org/10.1016/j.asej.2023.102537 doi: 10.1016/j.asej.2023.102537
[14]	A. M. Alqahtani, S. Akram, J. Ahmad, K. A. Aldwoah, M. U. Rahman, Stochastic wave solutions of fractional Radhakrishnan-Kundu-Lakshmanan equation arising in optical fibers with their sensitivity analysis, J. Opt., 2024. https://doi.org/10.1007/s12596-024-01850-w doi: 10.1007/s12596-024-01850-w
[15]	A. Farooq, M. I. Khan, K. S. Nisar, N. A. Shah, A detailed analysis of the improved modified Korteweg-de Vries equation via the Jacobi elliptic function expansion method and the application of truncated M-fractional derivatives, Results Phys., 59 (2024), 107604. https://doi.org/10.1016/j.rinp.2024.107604 doi: 10.1016/j.rinp.2024.107604
[16]	H. Yépez-Martínez, M. S. Hashemi, A. S. Alshomrani, M. Inc, Analytical solutions for nonlinear systems using Nucci's reduction approach and generalized projective Riccati equations, AIMS Mathematics, 8 (2023), 16655–16690. https://doi.org/10.3934/math.2023852 doi: 10.3934/math.2023852
[17]	S. Khajir, Z. Karimzadeh, M. Khoubnasabjafari, V. Jouyban-Gharamaleki, E. Rahimpour, A. Jouyban, A Rayleigh light scattering technique based on $\beta$-cyclodextrin modified gold nanoparticles for phenytoin determination in exhaled breath condensate, J. Pharm. Biomed. Anal., 223 (2023), 115141. https://doi.org/10.1016/j.jpba.2022.115141 doi: 10.1016/j.jpba.2022.115141
[18]	T. Yin, Z. Xing, J. Pang, Modified Hirota bilinear method to (3+1)-D variable coefficients generalized shallow water wave equation, Nonlinear Dyn., 111 (2023), 9741–9752. https://doi.org/10.1007/s11071-023-08356-3 doi: 10.1007/s11071-023-08356-3
[19]	S. Naowarat, S. Saifullah, S. Ahmad, M. De la Sen, Periodic, singular and dark solitons of a generalized geophysical KdV equation by using the tanh-coth method, Symmetry, 15 (2023), 135. https://doi.org/10.3390/sym15010135 doi: 10.3390/sym15010135
[20]	S. Tomar, M. Singh, K. Vajravelu, H. Ramos, Simplifying the variational iteration method: A new approach to obtain the Lagrange multiplier, Math. Comput. Simulat., 204 (2023), 640–644. https://doi.org/10.1016/j.matcom.2022.09.003 doi: 10.1016/j.matcom.2022.09.003
[21]	J. Ahmad, S. Akram, K. Noor, M. Nadeem, A. Bucur, Y. Alsayaad, Soliton solutions of fractional extended nonlinear Schrödinger equation arising in plasma physics and nonlinear optical fiber, Sci. Rep., 13 (2023), 10877. https://doi.org/10.1038/s41598-023-37757-y doi: 10.1038/s41598-023-37757-y
[22]	M. T. Islam, T. Rahman, M. Inc, M. A. Akbar, Comprehensive soliton solutions of fractional stochastic Kraenkel-Manna-Merle equations in ferromagnetic materials, Opt. Quant. Electron., 56 (2024), 927. https://doi.org/10.1007/s11082-024-06471-y doi: 10.1007/s11082-024-06471-y
[23]	F. M. Al-Askar, The solitary wave solutions of the stochastic Heisenberg ferromagnetic spin chain equation using two different analytical methods, Front. Phys., 11 (2023), 1267673. https://doi.org/10.3389/fphy.2023.1267673 doi: 10.3389/fphy.2023.1267673
[24]	M. A. E. Abdelrahman, M. A. Sohaly, Y. F. Alharbi, The new structure of stochastic solutions for the Heisenberg ferromagnetic spin chain equation, Opt. Quant. Electron., 55 (2023), 694. https://doi.org/10.1007/s11082-023-04923-5 doi: 10.1007/s11082-023-04923-5
[25]	W. W. Mohammed, M. El-Morshedy, C. Cesarano, F. M. Al-Askar, Soliton solutions of fractional stochastic Kraenkel-Manna-Merle equations in ferromagnetic materials, Fractal Fract., 7 (2023), 328. https://doi.org/10.3390/fractalfract7040328 doi: 10.3390/fractalfract7040328
[26]	S. Zhao, Z. Li, The analysis of traveling wave solutions and dynamical behavior for the stochastic coupled Maccari's system via Brownian motion, Ain Shams Eng. J., 15 (2024), 103037. https://doi.org/10.1016/j.asej.2024.103037 doi: 10.1016/j.asej.2024.103037
[27]	Z. Li, J. Lyu, E. Hussain, Bifurcation, chaotic behaviors and solitary wave solutions for the fractional Twin-Core couplers with Kerr law non-linearity, Sci. Rep., 14 (2024), 22616. https://doi.org/10.1038/s41598-024-74044-w doi: 10.1038/s41598-024-74044-w
[28]	B. Q. Li, Y. L. Ma, Loop-like periodic waves and solitons to the Kraenkel-Manna-Merle system in ferrites, J. Electromagn. Waves Appl., 32 (2018), 1275–1286. https://doi.org/10.1080/09205071.2018.1431156 doi: 10.1080/09205071.2018.1431156
[29]	B. Q. Li, Y. L. Ma, Rich soliton structures for the Kraenkel-Manna-Merle (KMM) system in ferromagnetic materials, J. Supercond. Nov. Magn., 31 (2018), 1773–1778. https://doi.org/10.1007/s10948-017-4406-9 doi: 10.1007/s10948-017-4406-9
[30]	H. T. Tchokouansi, V. K. Kuetche, T. C. Kofane, On the propagation of solitons in ferrites: The inverse scattering approach, Chaos Soliton Fract., 86 (2016), 64–74. https://doi.org/10.1016/j.chaos.2016.02.032 doi: 10.1016/j.chaos.2016.02.032
[31]	F. T. Nguepjouo, V. K. Kuetche, T. C. Kofane, Soliton interactions between multivalued localized waveguide channels within ferrites, Phys. Rev. E, 89 (2014), 063201. https://doi.org/10.1103/PhysRevE.89.063201 doi: 10.1103/PhysRevE.89.063201
[32]	S. Saini, R. Kumar, Deeksha, R. Arora, K. Kumar, Symmetry analysis and wave solutions of the fisher equation using conformal fractional derivatives, J. Appl. Math., 2023 (2023), 1633450. https://doi.org/10.1155/2023/1633450 doi: 10.1155/2023/1633450
[33]	A. Hussain, H. Ali, F. Zaman, N. Abbas, New closed form solutions of some nonlinear pseudo-parabolic models via a new extended direct algebraic method, Int. J. Math. Comput. Eng., 2 (2023), 35–58. https://doi.org/10.2478/IJMCE-2024-0004 doi: 10.2478/IJMCE-2024-0004
[34]	M. A. El-Shorbagy, S. Akram, M. U. Rahman, H. A. Nabwey, Analysis of bifurcation, chaotic structures, lump and $M-W$-shape soliton solutions to (2+1) complex modified Korteweg-de-Vries system, AIMS Mathematics, 9 (2024), 16116–16145. https://doi.org/10.3934/math.2024780 doi: 10.3934/math.2024780

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)