Analysis of wave phenomena in shallow water with rational solutions of the (1  <b>+</b>  1)-dimensional Korteweg–de-Vries–Calogero–Bogoyavlenski–Schiff equation

Understanding the behavior of shallow-water waves plays an important role in many areas of science and technology, such as oceanography and optical telecommunications. This study presents an analytical exploration of the (1+1)-dimensional Korteweg–de-Vries–Calogero–Bogoyavlensky–Schiff equation, through which a variety of rational solutions are obtained. Using mathematical methods, several wave structures are identified and categorized, like lump interactions, rogue waves, periodic kink configurations, and breather-like structures. The classification of such waves elucidates and collates their significance in the physical world and further constructs a cohesive framework for the understanding of transitions in integrable systems. Special attention is given to complex nonlinear interactions including kink-crossing phenomena and the development of M-shaped wave profiles. The results provide a detailed interpretation of the evolution of nonlinear wave motion in shallow-water waves and enhance the understanding of integrable systems.