Interaction of high-order breather, periodic wave, lump, rational soliton solutions and mixed solutions for reductions of the (4+1)-dimensional Fokas equation

The interaction of high-order breather, periodic-wave, lump, rational soliton solutions and mixed solutions for reductions of the (4+1)-dimensional Fokas equation are investigated by means of the Kadomtsev-Petviashvili (KP) hierarchy reduction method. Through analyzing the structural characteristics of periodic wave solutions, we find that evolution of the breather is decided by two characteristic lines. Interestingly, growingdecaying amplitude periodic wave and amplitude-invariant periodic wave are given through some conditions posed on the parameters. Some fascinating nonlinear wave patterns composed of high-order breathers and high-order periodic waves are shown. Furthermore, taking the long wave limit on the periodic-wave solutions, the semi-rational solutions composed of lumps, moving solitons, breathers, and periodic waves are obtained. Some novel dynamical processes are graphically analyzed. Additionally, we provide a new method to derive periodic-wave and semi-rational solutions for the (3+1)-dimensional KP equation by reducing the solutions of the (4+1)-dimensional Fokas equation. The presented results might help to understand the dynamic behaviors of nonlinear waves in the fluid fields and may provide some new perspectives for studying nonlinear wave solutions of high dimensional integrable systems.