Bilinear Bäcklund Transformations, as well as N-Soliton, Breather, Fission/Fusion and Hybrid Solutions for a (3+1)-Dimensional Integrable Wave Equation in a Fluid

Fluid mechanics has been applied in aviation, navigation, water conservancy, civil engineering, petroleum industry, meteorology, environmental protection and so on. In this paper, we investigate a (3+1)-dimensional integrable wave equation in a fluid: (i) According to some exchange formulae and the Hirota method, we obtain four sets of the bilinear Bäcklund transformations. (ii) We construct the N-soliton solutions, and then build up the higher-order breather, fusion/fission and certain hybrid solutions. The hybrid solutions are composed of the breathers/fusions/fissions and solitons, as well as mixed by the fusions/fissions and breathers. (iii) We graphically discuss the space-time structure of the fusion/fission wave. The interactions between a breather/fusion/fission wave and a soliton, and the interaction between a breather and a fusion/fission wave are also analyzed graphically.