Exact soliton and interaction solutions of the Estevez-Mansfield-Clarkson equation

• Used Hirota bilinear method to study the Estevez-Mansfield Clarkson (EMC) equation. • Derived M-shaped, rogue, kink, breather, lump, and cross kink wave solutions. • Visualized results with 3D, contour, and density plots using Mathematica. • Explained soliton dynamics and travelling wave deformation in dispersive media. • Applications in nonlinear fibre optics and telecommunications. • Demonstrated the effectiveness of Hirota method for nonlinear models. In this work, we employed the Hirota bilinear method to investigate solitary wave solutions to the Estévez–Mansfield–Clarkson equation, a model used in the study of shape formation in liquid droplets, mathematical physics, shallow water waves, optics, etc. Although several analytical techniques have been applied to this equation in earlier studies, a comprehensive use of the Hirota bilinear method for extracting a wide range of nonlinear traveling wave structures has not been previously reported. These solutions include M-shaped wave solutions with one and two kinks, M-shaped waves interacting with rogue and kink waves, multiple breather and lump waves, and periodic lump and cross kink wave solutions. Many of these interaction structures, especially the interactions of the various M-profile waves, multi-wave structures, and periodic cross-kink patterns, are new for the EMC equation and extend the existing family of known solutions. In addition to the Hirota bilinear framework, we applied the new generalized Kudryashov approach to deduce further exact solutions of the EMC equation. This additional method broadens the solution space and yields several new soliton structures, including dark, mixed, and exponential solitons. With these solutions and carefully chosen parameter values, we depict various 3D graphs and their corresponding contour and density plots using Mathematica. These graphical representations allow us to verify the behavior of the constructed solutions, analyze changes in amplitude and geometry, and highlight how parameter variations influence wave evolution and interaction patterns. The soliton phenomenon is explained by the obtained solutions and the physical structures, which also replicate the dynamic characteristics of the front of the traveling wave deformation produced in the dispersive medium. This shows the strength, compatibility, and potential use of the Hirota bilinear method in future research to find unique solutions for different types of nonlinear model that arise in engineering and physical sciences. These solutions are of great importance in the fields of nonlinear fiber optics and telecommunications, contributing to our understanding of the fundamental physical model.