Bilinear form and solutions of a (3+1)-dimensional generalized nonlinear evolution equation for the shallow-water waves

A (3+1)-dimensional generalized nonlinear evolution equation for the shallow-water waves is investigated. Bilinear form is derived and semi-rational solutions are constructed via the Kadomtsev–Petviashvili hierarchy reduction. Interactions between the lumps and solitons are analyzed. For the first-order semi-rational solutions, we observe that (1) the lump and the soliton fuse into the soliton; (2) the lump arises from the soliton and then separates from the soliton; (3) the first-order semi-rational solutions on the y−z plane possess a line profile and the wave shape changes with t varying. For the multi-semi-rational solutions, we find that on the x−y plane, the two lumps fuse into the two solitons, and on the x−z and y−z planes, the two lumps emerge on and then split from the two solitons. For the higher-order semi-rational solutions, we observe three kinds of interaction phenomena: (1) The two lumps fuse into the soliton; (2) The two lumps arise from the soliton and then separate from the soliton; (3) The two lumps which propagate towards each other fuse into one lump, and then that one splits into two other lumps. Influences of the coefficients in the original equation on the semi-rational solutions are also revealed.