Construction of M-shaped solitons for a modified regularized long-wave equation via Hirota's bilinear method

This study examines the effects of various M-shaped water wave shapes on coastal environments for the modified regularized long-wave equation (MRLWE). This work explores the complex dynamics of sediment transport, erosion, and coastal stability influenced by different wave structures using the Hirota bilinear transformation as a basic analytical tool. By providing insightful information about how these wave patterns impact coastal stability, it seeks to broaden our knowledge of dynamic coastlines. As we explore the intricate interactions between water waves and beaches, the knowledge gained from this research could help direct sustainable coastal management and preservation initiatives. For convenience, a range of M-shaped wave structures are depicted, demonstrating the adaptability of the Hirota bilinear transformation approach in recognizing novel wave patterns. Overall, this work contributes to a better understanding of the dynamics of the coastal environment, highlights the wide range of applications for mathematical models in science and engineering, and helps to develop more sensible and practical coastal management and conservation strategies for the protection of coastal areas against changing water wave patterns. Finally, as far as the authors could verify, this is the first work in the literature in which M-shaped soliton solutions are derived for the MRLWE using any method.