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Background <p>Nonlinear wave equations in higher dimensions play a crucial role in describing complex behaviors in dispersive media such as optics, plasma, and fluid dynamics. However, deriving exact solutions for generalized multi-dimensional models remains difficult due to the strong coupling between nonlinearity and dispersion.</p> Methods <p>In this work, we apply the Improved Modified Extended Tanh Method (IMETM) to a generalized (3+1)-dimensional nonlinear dispersive equation. Using a traveling wave transformation, the original partial differential equation is reduced to an ordinary differential equation. A polynomial ansatz combined with the balancing principle is introduced, and the resulting nonlinear algebraic systems are solved with symbolic computation software.</p> Results <p>This procedure generates diverse classes of analytical solutions, including bright and dark solitons, singular solutions, exponential forms, and periodic structures. Graphical simulations are provided to illustrate their qualitative behaviors, showing stability and structural richness across different parameter settings. The results highlight the efficiency of IMETM in capturing a broad spectrum of nonlinear wave patterns.</p> Conclusions <p>The findings extend the available set of exact soliton solutions for high-dimensional nonlinear evolution equations and enhance understanding of dispersive wave propagation. The IMETM framework demonstrates both robustness and adaptability, making it a valuable tool for studying more general nonlinear systems, including those involving fractional-order terms, stochastic effects, or hybrid analytical–numerical approaches.</p>

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Exploration of soliton structures in (3+1)-dimensional generalized evolution nonlinear equation using an efficient integration scheme

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月16日

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  • Ibrahim Saber,
  • Hamdy Ahmed,
  • Niveen Badra,
  • Islam Samir

摘要

Background

Nonlinear wave equations in higher dimensions play a crucial role in describing complex behaviors in dispersive media such as optics, plasma, and fluid dynamics. However, deriving exact solutions for generalized multi-dimensional models remains difficult due to the strong coupling between nonlinearity and dispersion.

Methods

In this work, we apply the Improved Modified Extended Tanh Method (IMETM) to a generalized (3+1)-dimensional nonlinear dispersive equation. Using a traveling wave transformation, the original partial differential equation is reduced to an ordinary differential equation. A polynomial ansatz combined with the balancing principle is introduced, and the resulting nonlinear algebraic systems are solved with symbolic computation software.

Results

This procedure generates diverse classes of analytical solutions, including bright and dark solitons, singular solutions, exponential forms, and periodic structures. Graphical simulations are provided to illustrate their qualitative behaviors, showing stability and structural richness across different parameter settings. The results highlight the efficiency of IMETM in capturing a broad spectrum of nonlinear wave patterns.

Conclusions

The findings extend the available set of exact soliton solutions for high-dimensional nonlinear evolution equations and enhance understanding of dispersive wave propagation. The IMETM framework demonstrates both robustness and adaptability, making it a valuable tool for studying more general nonlinear systems, including those involving fractional-order terms, stochastic effects, or hybrid analytical–numerical approaches.