Dark localized waves for the (3+1)-dimensional Wazwaz-Kaur-Boussinesq equation

Abstract Starting from the nonzero background, dark localized waves are systematacially researched in the (3+1)-dimensional Wazwaz-Kaur-Boussinesq equation by Hirota bilinear method. Firstly, the generalized bilinear form with a nonzero constant C is obtained and N-dark-soliton solutions are successfully given. Subsequently, dark resonant soliton solutions, encompassing Y-shaped and X-shaped solitons, are directly derived through choosing some appropriate resonant conditions in the received N-dark-soliton. By introducing the velocity resonance conditions to the aforementioned N-soliton, the general dark soliton molecules (SMs) are then constructed. Moreover, these SMs can be classified into three categories: pure dark SMs, dark SMs interacting with standard dark solitons, and hybrid solutions of different dark SMs. Then, the concrete expressions of M-dark-lump solutions are given by the complex conjugate and the long-wave limit methods, especially, one- and two-dark lumps are discussed in detail. Next, the interactions between different types of dark localized waves were investigated, specifically: Y-shaped solitons and solitons, lumps and solitons, and lumps interacting with breathers. Finally, the mathematical formation mechanisms and some related dynamics of these nonlinear localized waves are systematically elucidated through comprehensive analysis.