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<p>This work investigates the application of the Kumar–Malik method to construct and analyze soliton solutions for the nonlinear Estevez–Mansfield–Clarkson equation, a significant model for describing shallow water wave dynamics. The concentration is on obtaining the exact solutions that not only enhance the mathematical insight of wave dynamics but also have applications. The Kumar–Malik method, grounded in the formulation of a first-order differential equation, is demonstrated to be an efficient and systematic technique for solving nonlinear partial differential equations. This study presents the first application of this method for obtaining soliton solutions in this specific context. Changing to a moving coordinate frame reduces the governing equation to a single nonlinear ordinary differential equation. This method demonstrates considerable efficiency with respect to the analysis of nonlinear partial differential equations. By using this method, the Jacobi elliptic function solutions, hyperbolic function solutions, trigonometric function solutions, and exponential function solutions to the Estevez–Mansfield–Clarkson equation have been obtained. The extensive range of derived solutions enables a detailed analysis of their inherent wave properties. Three-dimensional plots, density distributions, and two-dimensional graphs are utilized to consistently depict behaviors such as the periodic, dark, singular, singular kink and the singular bell-shaped. Importantly, these soliton solutions provide a fresh and important perspective into the dynamics of the Estevez–Mansfield–Clarkson equation, which in turn provides a deeper understanding of wave coupling and propagation characteristics. An analysis of these solutions may help to understand the model’s behaviors in different environments. These solutions illuminate complex problems, such as the dispersion of configurations in liquid droplets and wave behavior in shallow water. The results provide a useful framework for future investigations of the system in question. This work offers a basis for further exploration and visualization of additional dynamical characteristics inherent in the physical processes.</p>

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Uncovering nonlinear dynamics in shallow water: an analytic approach to the \((1+1)\)-dimensional Estevez–Mansfield–Clarkson equation

  • 期刊论文

  • 发表日期：

    2025年7月20日

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  • Ghulam Hussain Tipu,
  • Waqas Ali Faridi,
  • Fengping Yao,
  • Mubariz Garayev

摘要

This work investigates the application of the Kumar–Malik method to construct and analyze soliton solutions for the nonlinear Estevez–Mansfield–Clarkson equation, a significant model for describing shallow water wave dynamics. The concentration is on obtaining the exact solutions that not only enhance the mathematical insight of wave dynamics but also have applications. The Kumar–Malik method, grounded in the formulation of a first-order differential equation, is demonstrated to be an efficient and systematic technique for solving nonlinear partial differential equations. This study presents the first application of this method for obtaining soliton solutions in this specific context. Changing to a moving coordinate frame reduces the governing equation to a single nonlinear ordinary differential equation. This method demonstrates considerable efficiency with respect to the analysis of nonlinear partial differential equations. By using this method, the Jacobi elliptic function solutions, hyperbolic function solutions, trigonometric function solutions, and exponential function solutions to the Estevez–Mansfield–Clarkson equation have been obtained. The extensive range of derived solutions enables a detailed analysis of their inherent wave properties. Three-dimensional plots, density distributions, and two-dimensional graphs are utilized to consistently depict behaviors such as the periodic, dark, singular, singular kink and the singular bell-shaped. Importantly, these soliton solutions provide a fresh and important perspective into the dynamics of the Estevez–Mansfield–Clarkson equation, which in turn provides a deeper understanding of wave coupling and propagation characteristics. An analysis of these solutions may help to understand the model’s behaviors in different environments. These solutions illuminate complex problems, such as the dispersion of configurations in liquid droplets and wave behavior in shallow water. The results provide a useful framework for future investigations of the system in question. This work offers a basis for further exploration and visualization of additional dynamical characteristics inherent in the physical processes.