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 \((3+1)\)-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\(\acute{\textrm{e}}\) expansion and a bilinear Bäcklund transformation based on the bilinear form. With the variable coefficients \(\alpha (t)\), \(\beta (t)\), \(\gamma (y,t)\), \(\delta (t)\) and \(\mu (t)\) taken as certain constraints, one- and two-kink solutions are shown. Based on the one-kink solutions, we take \(\gamma (y,t)\) 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.