Interactions among different types of nonlinear waves described by the Kadomtsev–Petviashvili equation

In nonlinear science, the interactions among solitons are well studied because the multiple soliton solutions can be obtained by various effective methods. However, it is very difficult to study interactions among different types of nonlinear waves such as the solitons (or solitary waves), the cnoidal periodic waves and Painlevé waves. In this paper, taking the Kadomtsev–Petviashvili (KP) equation as an illustration model, a new method is established to find interactions among different types of nonlinear waves. The nonlocal symmetries related to the Darboux transformation (DT) of the KP equation is localized after embedding the original system to an enlarged one. Then the DT is used to find the corresponding group invariant solutions. It is shown that the essential and unique role of the DT is to add an additional soliton on a Boussinesq-type wave or a KdV-type wave, which are two basic reductions of the KP equation.