The state transition mechanism of nonlinear waves with external force control in the fluid or plasma

In this paper, the state transition mechanism in the variable-coefficient (3+1)-dimensional Kadomtsev–Petviashvili equation with external force control is reported, which could be beneficial for providing potential theoretical insights in fluid mechanics or plasma. Utilizing the Hirota bilinear method, the non-autonomous two-soliton solution can be derived. Subsequently, the transformed waves under various external forces are modulated based on the state transition condition. The graphical representation indicates that the external force affects the modulation plane of the transformed waves. In conclusion, the discovery that wave solutions can still occur in non-autonomous systems through the addition of external forces is more conducive to providing practical guidance for applications.