This project is concerned with several aspects for a class of cubic nonlinear integrable systems with peakons and blow-up solutions. We mainly investigate the initial value problem and the initial boundary value problem for the systems. The local well-posedness of the systems are established, and the blow-up phenomena as well as the global existence for strong solutions to the systems are studied. The existence and the uniqueness of global weak solutions for the systems are proved, and the orbitally stability of peakon solutions are discussed. Furthermore, the numerical simulations of interactions of the solitons to the systems are considered. Researches on the above issues for the equations will help us deeply understand the important physical phenomena of solitons and wave-breaking in water wave from the view of mathematics, so the project is very important for both mathematical theories and physical applications.