Breather and hybrid solutions for a generalized (3 + 1)-dimensional B-type Kadomtsev–Petviashvili equation for the water waves

Abstract

Water waves are one of the most common phenomena in nature, the study of which helps in designing the related industries. In this paper, a generalized ( $$3+1$$ )-dimensional B-type Kadomtsev–Petviashvili equation for the water waves is investigated. Gramian solutions are constructed via the Kadomtsev–Petviashvili hierarchy reduction. Based on the Gramian solutions, we construct the breathers. We graphically analyze the breather solutions and find that the breathers can be reduced to the homoclinic orbits. For the higher-order breather solutions, we obtain the mixed solutions consisting of the breathers and homoclinic orbits. According to the long-wave limit method, rational solutions are constructed. We look at two types of the rational solutions, i.e., the lump and line rogue wave solutions, and give the condition for the lumps being reduced to the line rogue waves. Taking another set of the parameters for the Gramian solutions, we also derive the kinky breather solutions which can be reduced to the kink solitons. For the higher-order kinky breather solutions, we obtain the mixed solutions consisting of the breathers and kink solitons. Combining the breather and rational solutions, we construct two kinds of the hybrid solutions composed of the breathers, lumps, line rogue waves and kink solitons. Characteristics of those hybrid solutions are graphically analyzed and the conditions for the generation of those hybrid solutions are given.