Hybrid waves for a (2 + 1)-dimensional extended shallow water wave equation

Abstract

Shallow water waves are studied for the applications in hydraulic engineering and environmental engineering. In this paper, a (2 + 1)-dimensional extended shallow water wave equation is investigated. Hybrid solutions consisting of H-soliton, M-breather, and J-lump solutions have been constructed via the modified Pfaffian technique, where H, M, and J are the positive integers. One-breather solutions with a real function ϕ(y) are derived, where y is the scaled space variable; we notice that ϕ(y) influences the shapes of the background planes, and the one-breather solutions are localized along the curve (k1R+k2R)x+(k1Rk1I2+k1R2+k2Rk2I2+k2R2)ϕ(y)+w1t+Ω1R+Ω2R=0, while periodic along the curve (k1I+k2I)x−(k1Ik1I2+k1R2+k2Ik2I2+k2R2)ϕ(y)+w2t+Ω1I+Ω2I=0, where k1R, k1I, k2R, k2I, w1, w2, Ω1R, Ω1I, Ω2R, and Ω2I are the real constants. Discussions on the hybrid waves consisting of one breather and one soliton indicate that the one breather is not affected by one soliton after interaction. One-lump solutions with ϕ(y) are obtained with the condition k1R2<k1I2; we notice that the one lump consists of two low valleys and one high peak, and the amplitude and velocity keep invariant during its propagation. Hybrid waves consisting of the one lump and one soliton imply that the shape of the one soliton becomes periodic when ϕ(y) is changed from a linear function to a periodic function. Constant coefficient α can affect the propagation direction and velocity of the one breather and one lump, respectively.