Painlevé analysis, Bäcklund transformations and traveling-wave solutions for a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation in a fluid

Fluids are seen in such disciplines as atmospheric science, oceanography and astrophysics. In this paper, we investigate a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation in a fluid. On the basis of the Painlevé analysis, we find that the equation is Painlevé integrable under a certain constraint. Through the truncated Painlevé expansion, we give an auto-Bäcklund transformation. By virtue of the Hirota method, we derive a bilinear auto-Bäcklund transformation. Via the polynomial-expansion method, traveling-wave solutions are obtained. We observe that the amplitude of a traveling wave remains invariant during the propagation. We graphically demonstrate that the amplitude of the traveling-wave is affected by the coefficients corresponding to the dispersion and nonlinearity effects, while other coefficients have no influence on the traveling-wave amplitude, which represent the perturbed effects and disturbed wave velocity effects.