Solitary wave solutions of the delayed KP–BBM equation

In this paper, a type of shallow water wave model referred to as the Kadomtsev–Petviashvili–Benjamin–Bona–Mahony (KP–BBM) equation is studied. Initially, the unperturbed form of the KP–BBM equation is examined. By employing geometric singular perturbation (GSP) theory, particularly the invariant manifold theory, methods from dynamical systems, and Melnikov analysis, the persistence of solitary wave solutions is established for the delayed KP–BBM equation. Furthermore, the equation is analyzed under various types of nonlinear terms. Finally, the analytical results are validated through numerical simulations.