Dynamics of nonlinear dark waves and multi-dark wave interactions for a new extended (3+1)-dimensional Kadomtsev–Petviashvili equation

The Kadomtsev–Petviashvili (KP) equation is widely used to model nonlinear wave phenomena occurring in plasmas and fluids, such as water waves with weak nonlinear restoring forces, and shock waves propagation phenomena in plasma. In this paper, we investigate a new extended (3 + 1)-dimensional KP equation. Based on the Hirota bilinear method, the N-soliton solutions for the equation are derived. By introducing a new constraint to the N-soliton solutions, namely considering different dispersion coefficients to zero, the fission and fusion phenomena in multi solitons have been discovered. Subsequently, the M-lump wave and rogue wave solutions were derived using the long wave limit, and systematically analyzed the dynamics of the lump waves through numerical simulations. Additionally, based on the method mentioned earlier, combined with the complex conjugate construction method, we investigated some novel interaction phenomena such as lump waves with fusion (fission) waves, fusion (fission) waves with breather waves, line rogue waves with lump waves, and so on. These results may provide some explanation for the fission or fusion phenomena occurring in nonlinear models, as well as the complex nonlinear superposition between waves in fluid systems.