Rogue waves for a variable-coefficient Kadomtsev–Petviashvili equation in fluid mechanics

Abstract

Under investigation in this paper is a variable-coefficient Kadomtsev–Petviashvili equation in fluid mechanics, which describes the shallow-water waves with the weak nonlinearity and dispersion. Employing the Kadomtsev–Petviashvili hierarchy reduction, we obtain the rogue-wave solutions in terms of the Gramian. Periodic, cubic- and s-shaped line rogue waves are presented with different forms of the dispersion coefficient. The second-order rogue waves and multi-rogue waves are also graphically discussed. It is observed that only parts of the second-order rogue wave approach the constant background, and the other parts move to the far distance with the undiminished amplitudes. The multi-rogue waves describe the interaction of several first-order rogue waves. We plot the interactions of two periodic, cubic- and s-shaped line rogue waves. The lump wave, which propagates stably in all directions, is also depicted.