Gram-type, three-breather and hybrid solutions for a (3＋1)-dimensional generalized variable-coefficient shallow water wave equation

In this paper, a (3＋1)-dimensional generalized variable-coefficient shallow water wave equation is studied, which characterizes the flow below a pressure surface in a fluid. Through the Kadomtsev–Petviashvili hierarchy reduction, we construct some Gram-type solutions, three-breather solutions and three kinds of the hybrid solutions. Via the asymptotic analysis, we choose certain parameters to make the three connected breathers parallel to the x axis and reduce the first-order breather to the first-order homoclinic orbit on the x–z plane. Interaction among the three breathers is displayed and we observe that the one breather splits into the two breathers on the x–y plane, where x, y, z are the scaled spatial coordinates. For the three hybrid solutions, we reduce the V-shaped soliton to the one kink soliton on the y–z plane and show that the one breather converts into the one kink soliton on the y–z plane. Asymptotic analysis indicates that the interaction between the V-shaped soliton and kink soliton leads to the increase/decrease of the constant background of the V-shaped soliton as the constant background of the kink soliton decrease/increase. Furthermore, we present the two interactions among the V-shaped soliton and three breathers.