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<p>The (3+1)-dimensional integrable nonlinear partial differential equations play a critical role in modeling complex phenomena across fields such as ocean engineering, quantum mechanics, nonlinear optics, and statistical physics. This study investigates the soliton dynamics of a (3+1)-dimensional nonlinear evolution equation governing wave propagation in fluid media. By applying a systematic wave transformation, the equation is reduced to a nonlinear ordinary differential equation. Three analytical methods–the Kumar-Malik method, the modified Sardar sub-equation approach, and the extended Arnous approach–are employed to derive a diverse set of soliton solutions, including mixed, dark, bright-dark, and composite profiles. Additionally, multistability analysis reveals the coexistence of multiple stable wave states under specific parametric conditions. Numerical simulations and graphical illustrations are provided to explore the dynamical behavior and stability of the solutions. The findings contribute to the theoretical understanding of nonlinear wave interactions and offer practical implications for simulating real-world wave phenomena in fluid dynamics. This study highlights the effectiveness of modern analytical methods in exploring higher-dimensional nonlinear systems.</p>

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Investigating higher-dimensional nonlinear evolution equation: dynamics of waves and multistability in fluid mediums

  • 期刊论文

  • 发表日期：

    2025年9月30日

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  • J. Muhammad,
  • U. Younas,
  • Karim K. Ahmed

摘要

The (3+1)-dimensional integrable nonlinear partial differential equations play a critical role in modeling complex phenomena across fields such as ocean engineering, quantum mechanics, nonlinear optics, and statistical physics. This study investigates the soliton dynamics of a (3+1)-dimensional nonlinear evolution equation governing wave propagation in fluid media. By applying a systematic wave transformation, the equation is reduced to a nonlinear ordinary differential equation. Three analytical methods–the Kumar-Malik method, the modified Sardar sub-equation approach, and the extended Arnous approach–are employed to derive a diverse set of soliton solutions, including mixed, dark, bright-dark, and composite profiles. Additionally, multistability analysis reveals the coexistence of multiple stable wave states under specific parametric conditions. Numerical simulations and graphical illustrations are provided to explore the dynamical behavior and stability of the solutions. The findings contribute to the theoretical understanding of nonlinear wave interactions and offer practical implications for simulating real-world wave phenomena in fluid dynamics. This study highlights the effectiveness of modern analytical methods in exploring higher-dimensional nonlinear systems.