Three types of localized waves and their interactions in the (3+1)-dimensional Wazwaz–Kaur–Boussinesq equation

The (3+1)-dimensional Wazwaz–Kaur–Boussinesq equation, which always elucidate that the propagation of gravity waves over the water surface, more specifically, the head-on collision of oblique wave profiles, is investigated by the Hirota bilinear method. First, the N-soliton solutions in the exponential form are given. By introducing some resonant conditions in the N-soliton solutions, we successfully derive the resonant soliton solutions. Then, these three kinds of resonant solutions, including Y-shaped solitons, X-shaped solitons, as well as interactions between Y-shaped and the common solitons, are discussed in detail. Furthermore, multi-breathers are generated through the complex conjugate reduction method, especially, parallel and crossed breathers are discussed. Combining both the long-wave limit and the complex conjugate reduction methods, the concrete expressions of arbitrary lump solutions are also given. Finally, different kinds of interactional solutions among resonant solitons, breathers, and lumps are systematically studied, and then the related dynamics are also demonstrated.