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 $$ -expansion method, we provide a rich set of exact solutions for the (3 + 1)-dimensional Painlevé integrable equation, which includes two functions with arbitrary values. This method is the first to be applied to the (3 + 1)-dimensional Painlevé integrable equation. By giving some 3D graphics and density maps, the dynamic properties are analyzed and demonstrated, which is beneficial for promoting understanding and application of the (3 + 1)-dimensional Painlevé integrable equation.