Soliton structures for the (3 + 1)-dimensional Painlevé integrable equation in fluid mediums

The (3 + 1)-dimensional Painlevé integrable equation are a class of nonlinear differential equations with special properties, which play an important role in nonlinear science and are of great significance in solving various practical problems, such as many important models in fields such as quantum mechanics, statistical physics, nonlinear optics, and celestial mechanics. In this work, we utilize the Hirota bilinear form and Mathematica software to formally obtain the interaction solution among lump wave, solitary wave and periodic wave, which has not yet appeared in other literature. Additionally, using the