Travelling-wave, Mixed-lump-kink and Mixed-rogue-wave-kink Solutions for an Extended (3+1)-dimensional Shallow Water Wave Equation in Oceanography and Atmospheric Science

Shallow water waves are seen in oceanography, atmospheric science, and other fields. In this paper, we investigate an extended (3+1)-dimensional shallow water wave equation. We get the travelling-wave solutions via the polynomial-expansion method. Applying the Hirota method and symbolic computation, we derive some mixed-lump-kink and mixed-rogue-wave-kink solutions. Based on the mixed-lump-kink solutions, we graphically show the interaction between a lump and a kink soliton, and find two different cases: (1) the lump merges into the kink soliton; (2) the lump separates from the kink soliton. Based on the mixed-rogue-wave-kink solutions, we graphically analyze the interaction between the rogue wave and two-kink solitons, and find that the rogue wave emerges from the one kink soliton and merges into the other kink soliton.