Lumps and rouge waves for a $$(3+1)$$ ( 3 + 1 ) -dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluid mechanics

Abstract

In this paper, a $$(3+1)$$ -dimensional variable-coefficient Kadomtsev–Petviashvili equation, which describes the long water waves and small-amplitude surface waves with the weak nonlinearity, weak dispersion and weak perturbation in fluid mechanics, is investigated. Lump, lump–soliton and rouge–soliton solutions are obtained with the aid of symbolic computation. For the lump and soliton, amplitudes are related to the nonlinearity coefficient and dispersion coefficient, while velocities are related to the perturbation coefficients. Fusion and fission phenomena between the lump and soliton are observed, respectively. Graphic analysis shows that: (i) soliton’s amplitude becomes larger after the fusion interaction, and becomes smaller after the fission interaction; (ii) after the interaction, the soliton propagates along the opposite direction to before when any one of the perturbation coefficients is a time-dependent function. For the interactions between the rogue wave and two solitons, the rogue wave splits from one soliton and merges into the other one, and the two solitons exchange the amplitudes through the energy transfer by the rogue wave.