The lump, lumpoff and rouge wave solutions of a (3+1)-dimensional generalized shallow water wave equation

Abstract

We consider the (3[Formula: see text]+[Formula: see text]1)-dimensional generalized shallow water wave (GSWW) equation. By virtue of the binary Bell polynomials theory, we obtain the bilinear form of the equation. Then its lump wave solutions, a kind of rational solution localized in all directions of the space, are derived by employing its bilinear form at the special situation for [Formula: see text]. Furthermore, it is worth noting that the lump wave solutions can interact with single-stripe soliton waves and double-stripe solution waves to generate lumpoff waves and a kind of predictable rouge waves, respectively. Especially, it is interesting that we can predict when and where the peculiar rouge waves will occur. Moreover, in order to understand the dynamics and propagation of the lump waves and the interaction solution, some graphic analyses are exhibited by selecting special parameters. The results of this work can be used to understand the propagation behavior of these solutions of the GSWW equation, which is of great significance for ocean engineering.