Painlevé analysis and analytic solutions of a generalized (3+1)-dimension variable-coefficient nonlinear evolution equation in fluid mechanics and plasma physics

In this paper, a generalized (3+1)-dimension variable-coefficient nonlinear evolution equation is investigated, which serves as a model for describing nonlinear wave behaviors in shallow water, ion-acoustic wave fluid mechanics and plasma physics. The Painlevé integrability is tested by the Weiss, Tabor and Carnevale (WTC) method with the simplified form of Krustal. The bilinear form of the equation is derived through the application of the Hirota bilinear method. Building on the bilinear equation, a broad range of analytical solutions are then obtained, including X-shaped and Y-shaped soliton solutions, lump solution, breather solution, and interaction solutions. In addition, another type of soliton solution, periodic solution, and ratio of trigonometric functions are derived.