Lumps, breathers and interaction solution to the (3 + 1)-dimensional modified Korteweg de Vries- Zakharov-Kuznetsov (mKdV-ZK) model

This study employs the Hirota bilinear technique to derive exact analytical solutions of the (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsov (mKdV-ZK) model. The governing equation is first transformed into its bilinear form, which is essential for constructing analytical solutions. Using this approach, a variety of exact wave structures are systematically obtained, including lump solutions, multi-soliton waves involving two- to four-soliton interactions, and various types of breather solutions, such as the Ma breather, Kuznetsov-Ma breather (KMB), and generalized breathers (GB). In addition, two distinct interaction scenarios between lump and solitary waves are presented with clear physical interpretation. The complex dynamical behaviours, all obtained solutions are reported for the first time, are illustrated through graphical simulations in Maple, offering a clear visualization of wave evolution and collision dynamics. These results significantly enhance understanding of nonlinear wave interactions, propagation, and stability, with potential applications in fluid dynamics, plasma physics, and other nonlinear dispersive media.