Novel Fractal Soliton Solutions of a (3+1)-Dimensional Benjamin–Bona–Mahony Equation on a Cantor Set

• M. M. Alqarni ^[1] ; Emad E. Mahmoud ^[2] ; M.A. Aljohani ^[3] ; Shabir Ahmad ^[4]
  1. [1] King Khalid University

     King Khalid University

     Arabia Saudí

     
  2. [2] Taif University

     Taif University

     Arabia Saudí

     
  3. [3] Taibah University

     Taibah University

     Arabia Saudí

     
  4. [4] University of Campania “Luigi Vanvitelli”

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• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 2, 2025
• Idioma: inglés
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• Resumen

  • The Benjamin–Bona–Mahony (BBM) equation is used to model unidirectional wave propagation in dispersive media, which describes wave behavior in physical systems where dispersion plays a significant role. This paper considers the extended version of the BBM equation in (3+1)-dimension. It has been noted that the considered equation was not studied when the fractal calculus is utilized. Here, in this work, the fractal form of the considered equation is taken into the consideration using the He’s fractal operator on semi-domain. The variational direct method, which integrates variational theory and the Ritz-like technique, is used in conjunction with the fractal based two scale transformation to develop exact solutions in semi-domain for considered equation. The obtained solutions are displayed via 3D simulations. The use of and other values introduces fractal element into the solution, contributing to complex dynamics observed in plot. This complexity is the reason of fractal effects on the solution and showcases how different parameters’ values can can result in varied wave behaviors of the proposed model. The results obtained for the proposed model has potential applications in fluid dynamics and oceanography. It can model waves in the coastal environments having fractal-like characteristics.

• Referencias bibliográficas
  • 1. Negro, S.: Integrable structures in quantum field theory. J. Phys. A: Math. Theor. 49(32), 323006 (2016)
  • 2. Møller, F.S., Perfetto, G., Doyon, B., Schmiedmayer, J.: Euler-scale dynamical correlations in integrable systems with fluid motion. SciPost...
  • 3. Fokas, A.S.: Integrable nonlinear evolution partial differential equations in 4+ 2 and 3+ 1 dimensions. Phys. Rev. Lett. 96(19),...
  • 4. Wazwaz, A.M.: Higher dimensional integrable Vakhnenko–Parkes equation: multiple soliton solutions. Int. J. Num. Methods Heat Fluid Flow...
  • 5. Yoku¸s, A., Duran, S., Kaya, D.: An expansion method for generating travelling wave solutions for the (2+ 1)-dimensional Bogoyavlensky–Konopelchenko...
  • 6. Duran, S., Yokus, A., Kilinc, G.: A study on solitary wave solutions for the Zoomeron equation supported by two-dimensional dynamics. Phys....
  • 7. Wang, K.J., Li, S.: Complexiton, complex multiple kink soliton and the rational wave solutions to the generalized (3+ 1)-dimensional...
  • 8. Zhou, X., Ilhan, O.A., Manafian, J., Singh, G., Tuguz, N.S.: N-lump and interaction solutions of localized waves to the (2+ 1)-dimensional...
  • 9. Zhang, M., Xie, X., Manafian, J., Ilhan, O.A., Singh, G.: Characteristics of the new multiple rogue wave solutions to the fractional generalized...
  • 10. Gu, Y., Malmir, S., Manafian, J., Ilhan, O.A., Alizadeh, A.A., Othman, A.J.: Variety interaction between k-lump and k-kink solutions for...
  • 11. Alquran, M., Ali, M., Al-Khaled, K., Grossman, G.: Simulations of fractional time-derivative against proportional time-delay for solving...
  • 12. Al-Deiakeh, R., Alquran, M., Ali, M., Yusuf, A., Momani, S.: On group of Lie symmetry analysis, explicit series solutions and conservation...
  • 13. Wei, C.F.: New solitary wave solutions for the fractional Jaulent–Miodek hierarchy model. Fractals 31(05), 2350060 (2023)
  • 14. Wang, K.L.: Novel solitary wave and periodic solutions for the nonlinear Kaup–Newell equation in optical fibers. Opt. Quantum Electron....
  • 15. Wang, K.L.: Novel investigation of fractional long-and short-wave interaction system. Fractals 32(01), 2450023 (2024)
  • 16. Wang, K.L.: New mathematical approaches to nonlinear coupled Davey–Stewartson Fokas system arising in optical fibers. Math. Methods Appl....
  • 17. Eslami, M., Matinfar, M., Asghari, Y., Rezazadeh, H.: Soliton solutions to the conformable timefractional generalized Benjamin–Bona–Mahony...
  • 18. Eslami, M., Rezazadeh, H.: The first integral method for Wu–Zhang system with conformable timefractional derivative. Calcolo 53, 475–485...
  • 19. Mirzazadeh, M., Hashemi, M.S., Akbulu, A., Ur Rehman, H., Iqbal, I., Eslami, M.: Dynamics of optical solitons in the extended (3+...
  • 20. Asghari, Y., Eslami, M., Matinfar, M., Rezazadeh, H.: Novel soliton solution of discrete nonlinear Schrödinger system in nonlinear optical...
  • 21. Khater, M.M.: Wave propagation analysis in the modified nonlinear time fractional harry dym equation: insights from khater ii method and...
  • 22. Khater, M.M.: Computational method for obtaining solitary wave solutions of the (2+ 1)-dimensional AKNS equation and their physical...
  • 23. Lin, Y., Khater, M.M.: Plenty of accurate, explicit solitary unidirectional wave solutions of the nonlinear Gilson–Pickering model. Int....
  • 24. Khater, M.M.: Dynamic insights into nonlinear evolution: Analytical exploration of a modified widthBurgers equation. Chaos Solitons Fractals...
  • 25. Khater, M.M.: Unraveling dynamics: analytical insights into liquid–gas interactions. Chaos Solitons Fractals 184, 114977 (2024)
  • 26. Khater, M.M., Alfalqi, S.H.: Solitary wave solutions for the (1+ 1)-dimensional nonlinear Kakutani– Matsuuchi model of internal gravity...
  • 27. Chen, H., Shahi, A., Singh, G., Manafian, J., Eslami, B., Alkader, N.A.: Behavior of analytical schemes with non-paraxial pulse propagation...
  • 28. Gaber, A.A., Bekir, A.: Integrability, similarity reductions and new classes of exact solutions for (3+ 1)-D potential Yu–Toda–Sasa–Fukuyama...
  • 29. Badshah, F., Tariq, K.U., Bekir, A., Tufail, R.N., Ilyas, H.: Lump, periodic, travelling, semi-analytical solutions and stability analysis...
  • 30. Xie, G., Fu, B., Li, H., Du, W., Zhong, Y., Wang, L., Geng, H., Zhang, J., Si, L.: A gradient-enhanced physics-informed neural networks...
  • 31. Benjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R Soc. Lond....
  • 32. Ghanbari, B., Baleanu, D., Al Qurashi, M.: New exact solutions of the generalized Benjamin–Bona– Mahony equation. Symmetry 11(1), 20 (2018)
  • 33. Yokus, A., Sulaiman, T.A., Bulut, H.: On the analytical and numerical solutions of the Benjamin– Bona–Mahony equation. Opt. Quantum Electron....
  • 34. Al-deiakeh, R., Alquran,M., Ali,M., Qureshi, S.,Momani, S.,Malkawi, A.: Lie symmetry, convergence analysis, explicit solutions, and conservation...
  • 35. Wazwaz, A.M.: Exact soliton and kink solutions for new (3+ 1)-dimensional nonlinear modified equations of wave propagation. Open Eng....
  • 36. Jaradat, I., Alquran, M.: A variety of physical structures to the generalized equal-width equation derived from Wazwaz–Benjamin–Bona–Mahony...
  • 37. Rezazadeh, H., Inc, M., Baleanu, D.: New solitary wave solutions for variants of (3+ 1)-dimensional Wazwaz–Benjamin–Bona–Mahony equations....
  • 38. Abbas, N., Bibi, F., Hussain, A., Ibrahim, T.F., Dawood, A.A., Birkea, F., Hassan, A.M.: Optimal system, invariant solutions and dynamics...
  • 39. Bekir, A., Shehata, M.S., Zahran, E.H.: New perception of the exact solutions of the 3D-fractional Wazwaz–Benjamin–Bona–Mahony (3D-FWBBM)...
  • 40. Yang, X.J.: An insight on the fractal power law flow: from a Hausdorff vector calculus perspective. Fractals 30(03), 2250054 (2022)
  • 41. Martin-Vergara, F., Rus, F., Villatoro, F.R.: Fractal structure of the soliton scattering for the graphene superlattice equation. Chaos...
  • 42. Wang, K.: A new fractal model for the soliton motion in a microgravity space. Int. J. Num. Methods Heat Fluid Flow 31(1), 442–451 (2021)
  • 43. Yang, X.J., Machado, J.T., Cattani, C., Gao, F.: On a fractal LC-electric circuit modeled by local fractional calculus. Commun. Nonlinear...
  • 44. Wang, K.J., Liu, J.H.: On the zero state-response of the J-order RC circuit within the local fractional calculus. COMPEL-Int. J. Comput....
  • 45. Wang, K.J.: The fractal active low-pass filter within the local fractional derivative on the Cantor set. COMPEL-Int. J. Comput. Math....
  • 46. Jiang, H., Li, S.M., Wang, W.G.: Moderate deviations for parameter estimation in the fractional Ornstein–Uhlenbeck processes with periodic...
  • 47. Wang, K.J., Liu, J.H., Shi, F.: On the semi-domain soliton solutions for the fractal (3+ 1)-dimensional generalized Kadomtsev–Petviashvili–Boussinesq...
  • 48. Wang, K.J., Si, J., Wang, G.D., Shi, F.: A new fractal modified Benjamin–Bona–Mahony equation: its generalized variational principle and...
  • 49. Wang, K.J., Li, S.: Study on the local fractional (3+ 1)-dimensional modified Zakharov–Kuznetsov equation by a simple approach. Fractals...
  • 50. Xu, P., Huang, H., Liu, H.: Semi-domain solutions to the fractal (3+ 1)-dimensional Jimbo–Miwa equation. Fractals 32, 1–8 (2024)
  • 51. He, J.-H.: Fractal calculus and its geometrical explanation. Results Phys. 10, 272–276 (2018)
  • 52. He, J.-H., Ain, Q.-T.: New promises and future challenges of fractal calculus: from two-scale thermodynamics to fractal variational principle....