Dynamics of localized waves and interactions in a (2+1)-dimensional equation from combined bilinear forms of Kadomtsev–Petviashvili and extended shallow water wave equations

This work investigates new exact solutions within a unique (2+1)-dimensional problem by combining the Hirota bilinear forms of the Kadomtsev–Petviashvili and the Shallow Water Wave equations. We obtain resonant Y-type and X-type soliton, breather, and lump waves by introducing novel limitations on the $N$-soliton. Additionally, it generates various types of solutions (bulk-soliton, fissionable soliton-lump, breather-lump, and soliton-breather-lump) based on velocity and module resonance conditions. Furthermore, we investigate the lump wave’s paths interacting with the waves prior to and following collision using the long-wave limit approach. We show that the lump wave either avoids collision with other waves or maintains a persistent state of collision with them by applying additional limitations. Specifically, we provide a series of figures depicting all solutions, comprehensively capturing their dynamical behavior and interactions.