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Construction of fractional granular model and bright, dark, lump, breather types soliton solutions using Hirota bilinear method

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  • Biswas, Swapan
  • Ghosh, Uttam
  • Raut, Santanu

Abstract

The present article designs the granular metamaterials considering the granular structures of discrete particles which are different from elastic metamaterials consisting of continuous media. In granular metamaterials, the wave propagates through contact with neighboring particles. To identify the propagating properties of wave quantities in the rough granular medium the fractional granular equation is formulated directly in a pre-compressed spherical chain adopting Hertz law and long wave approximation theory. Using phase and group velocities, Caputo fractional derivatives are used to illustrate normal and anomalous dispersion wave dependence. To demonstrate in depth the dynamical behavior of the wave profile, various types of complex solutions like multi-shock, multi-solitons, lump, and breather solutions of the one-dimensional time fractional granular equations are explored employing Hirota’s bilinear approach. Finally, the more complicated hybrid solutions such as kink with the lump, soliton with the lump, etc. are exhibited from numerical understanding. The numerical graphs and figures demonstrate the crucial role of the order of derivative (roughness parameter) in the formation of different types of soliton solutions.

Suggested Citation

  • Biswas, Swapan & Ghosh, Uttam & Raut, Santanu, 2023. "Construction of fractional granular model and bright, dark, lump, breather types soliton solutions using Hirota bilinear method," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    • Handle: RePEc:eee:chsofr:v:172:y:2023:i:c:s0960077923004216
    DOI: 10.1016/j.chaos.2023.113520
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    References listed on IDEAS

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    1. Hereman, Willy & Nuseir, Ameina, 1997. "Symbolic methods to construct exact solutions of nonlinear partial differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 43(1), pages 13-27.
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    1. Yuan, Rui-rui & Shi, Ying & Zhao, Song-lin & Wang, Wen-zhuo, 2024. "The mKdV equation under the Gaussian white noise and Wiener process: Darboux transformation and stochastic soliton solutions," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    2. Adel Elmandouh & Aqilah Aljuaidan & Mamdouh Elbrolosy, 2024. "The Integrability and Modification to an Auxiliary Function Method for Solving the Strain Wave Equation of a Flexible Rod with a Finite Deformation," Mathematics, MDPI, vol. 12(3), pages 1-15, January.
    3. Shao, Hangbing & Bilige, Sudao, 2026. "Dynamical asymptotic analysis to a (3+1)-dimensional B-type Kadomtsev–Petviashvili equation: The superposition formulas of rational solutions and interaction solutions under the bilinear vector method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 240(C), pages 938-952.
    4. Ghosh, Uttam & Roy, Subrata & Biswas, Swapan & Raut, Santanu, 2024. "A non-autonomous fractional granular model: Multi-shock, Breather, Periodic, Hybrid solutions and Soliton interactions," Chaos, Solitons & Fractals, Elsevier, vol. 187(C).
    5. Cao, Na & Li, XuHai & Yin, XiaoJun, 2025. "The control model of Rossby waves and dynamic characteristics in stratified fluids," Chaos, Solitons & Fractals, Elsevier, vol. 201(P1).
    6. Xu, Shuwei & He, Jingsong, 2025. "Various rational solutions generated from the higher order Kaup–Newell type equation," Chaos, Solitons & Fractals, Elsevier, vol. 201(P2).
    7. Mandal, Uttam Kumar & Malik, Sandeep & Kumar, Sachin & Zhang, Yi & Das, Amiya, 2024. "Integrability aspects, rational type solutions and invariant solutions of an extended (3+1)-dimensional B-type Kadomtsev–Petviashvili equation," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).

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