Dynamic mechanism of nonlinear waves for the (3+1)-dimensional generalized variable-coefficient shallow water wave equation

Under investigation in this paper is a (3+1)-dimensional generalized variable-coefficient shallow water wave equation, which can be used to describe the flow below a pressure surface in oceanography and atmospheric science. Employing the Kadomtsev-Petviashvili hierarchy reduction, we obtain the breather and lump solutions in terms of Grammian. We investigate the generation mechanism and conversion of the breathers, lumps and rogue waves. We find that the breather is produced by the superposition of three parts: The soliton part, the periodic wave part and the background part. The angle between the soliton part and the periodic wave part affects the shape of the breather. Considering the influences of the variable coefficients, we observe the breathers propagating on the periodic backgrounds, with double peaks and the breathers propagating periodic with time, respectively. Taking the long-wave limits, we get the rational solutions which describe the lumps. We find that the characteristic lines keep unchanged on the x − y plane, which means that the lump is similar to a part of the breather. Linear rogue waves only appear on the y − z plane.