Bilinear form, auto-Bäcklund transformations and kink solutions of a \((3+1)\)-dimensional variable-coefficient Kadomtsev-Petviashvili-like equation in a fluid

Fluid mechanics has been linked to a wide range of disciplines, such as atmospheric science, oceanography and astrophysics. In this paper, we focus our attention on a \((3+1)\) $(3+1)$ -dimensional variable-coefficient Kadomtsev-Petviashvili-like equation in a fluid. Through the Hirota method, we derive a bilinear form. We obtain an auto-Bäcklund transformation based on the truncated Painlev \(\acute{\textrm{e}}\) $\acute{\text{e}}$ expansion and a bilinear Bäcklund transformation based on the bilinear form. With the variable coefficients \(\alpha (t)\) $\alpha (t)$ , \(\beta (t)\) $\beta (t)$ , \(\gamma (y,t)\) $\gamma (y,t)$ , \(\delta (t)\) $\delta (t)$ and \(\mu (t)\) $\mu (t)$ taken as certain constraints, one- and two-kink solutions are shown. Based on the one-kink solutions, we take \(\gamma (y,t)\) $\gamma (y,t)$ as the linear and trigonometric functions of y, and then give the ring-type and periodic-type one-kink waves, where t and y are the independent variables. According to the two-kink solutions, we obtain the parabolic-type, linear-type and periodic-type kink waves.