Dynamics of transformed nonlinear waves in the (3+1)-dimensional B-type Kadomtsev–Petviashvili equation I: Transitions mechanisms

Abstract

We explore the dynamical properties of transformed nonlinear waves (TNWs) for the (3+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation describing the propagation of waves in fluids. The breath-wave solution is first given by the Hirota bilinear method. Different from the (1+1)- or (2+1)-dimensional case, three types of conversion conditions are analytically derived in different spatial coordinates, by which the breath waves can be converted into diverse TNWs, including the M-shaped kink soliton, kink soliton with multi peaks, (quasi-) kink soliton, and (quasi-) periodic wave. In addition, an attractive dynamic mechanism of high-dimensional nonlinear waves is investigated, where the shape-changed evolution of these waves can be observed. Then the gradient relationship of the TNWs is illustrated in terms of the wave number ratio of superposition components. The formation mechanism of TNWs is further analyzed based on the analysis of nonlinear superposition and phase shift. Different from previous result, the wave component for the (3+1)-dimensional BKP shows the kink-shaped profile, instead of the bell-shaped one. The principle of the nonlinear superposition is further used to explicate the essence of oscillation, locality and shape-changed evolution of the TNWs. The lump wave is finally transformed into the line rogue wave (LRW) showing the short-lived property. This indicates that the LRWs could be incorporated into the framework of TNWs in some high-dimensional systems.