Probing the diversity of soliton phenomena within conformable Estevez-Mansfield-Clarkson equation in shallow water

This study aims to employ the extended direct algebraic method (EDAM) to generate and evaluate soliton solutions to the nonlinear, space-time conformable Estevez Mansfield-Clarkson equation (CEMCE), which is utilized to simulate shallow water waves. The proposed method entails transforming nonlinear fractional partial differential equations (NFPDEs) into nonlinear ordinary differential equations (NODEs) under the assumption of a finite series solution by utilizing Riccati ordinary differential equations. Various mathematical structures/solutions for the current model are derived in the form of rational, exponential, trigonometric, and hyperbolic functions. The wide range of obtained solutions allows for a thorough analysis of their actual wave characteristics. The 3D and 2D graphs are used to illustrate that these behaviors consistently manifest as periodic, dark, and bright kink solitons. Notably, the produced soliton solutions offer new and critical insights into the intricate behaviors of the CEMCE by illuminating the basic mechanics of the wave's interaction and propagation. By analyzing these solutions, academics can better understand the model's behavior in various settings. These solutions shed light on complicated issues such as configuration dispersion in liquid drops and wave behavior in shallow water.