Soliton interactions, soliton bifurcations and molecules, breather molecules, breather-to-soliton transitions, and conservation laws for a nonlinear (3+1)-dimensional shallow water wave equation

In this study, we employ the bilinear method to construct Nth-order solutions for a nonlinear (3+1)-dimensional shallow water equation. Subsequently, we delve into a comprehensive exploration of novel dynamical features within the equation, including soliton interactions, soliton bifurcations, soliton molecules, breather and soliton interactions, breather molecules, and breather-to-soliton transitions. Analytical expressions for the constraint conditions required to generate soliton bifurcations and soliton molecules are presented. Our study reveals that the newly obtained waves exhibit distinctive characteristics: they can form multi-layered step-like flat blocks, propagate with stable energies and shapes, thereby reflecting an intrinsic balance between nonlinearities and dispersions. We also verify that this equation has an infinite number of conservation laws. These intriguing findings contribute to a deeper understanding of the dynamic behaviors exhibited by solitons, breathers, and their hybrid forms within the realm of shallow water waves characterized by nonlinear motions.