Dark breather waves, dark lump waves and lump wave–soliton interactions for a $(3+1)$-dimensional generalized Kadomtsev–Petviashvili equation in a fluid

Abstract

Fluids are seen in a wide range of disciplines, including mechanical, civil, chemical and biomedical engineering, geophysics, astrophysics and biology. In this paper, we investigate a $(3+1)$-dimensional generalized Kadomtsev–Petviashvili equation for the long water waves and small-amplitude surface waves with the weak nonlinearity, weak dispersion and weak perturbation in a fluid. Breather-wave, lump-wave and lump wave–soliton solutions are derived under certain conditions via the Hirota method. With $h_1{h}_2^{-1}<0$, where $h_1$ and $h_2$ represent the coefficients of dispersion and nonlinearity, respectively, we obtain the dark breather wave and lump wave. We observe the effects of $h_1$, $h_2$, $h_4$, $h_6$ and $h_8$ on the dark breather wave and lump wave, where $h_6$ is the perturbed effect, $h_4$ and $h_8$ stand for the disturbed wave velocity effects corresponding to the $y$ and $z$ coordinates: $h_1$ and $h_2$ influence the amplitude of the dark breather wave; $h_1$, $h_4$ and $h_8$ influence the distance between the adjacent valleys of the dark breather wave; $h_1$, $h_4$, $h_6$ and $h_8$ influence the location of the dark breather wave; $h_2$, $h_4$, $h_6$ and $h_8$ influence the amplitude of the dark lump wave; $h_1$, $h_4$ and $h_8$ influence the width of the dark lump wave; $h_4$, $h_6$ and $h_8$ influence the location of the dark lump wave. When $h_1{h}_2^{-1}>0$, we present the fusion between a bright lump wave and one bright soliton as well as fission of one bright soliton. We also observe the fusion between a dark lump wave and one dark soliton as well as fission of one dark soliton with $h_1{h}_2^{-1}>0$.