Rational and semi-rational solutions for the (3 + 1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation

Abstract

Nonlinear waves are seen in nature, such as the water waves and plasma waves. Investigated in this paper is a (3 + 1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation. Based on the bilinear method, we get the rational solutions, which are different from the published ones, semi-rational solutions and breather-type kink soliton solutions. Through the rational solutions, we observe two types of waves: the lump waves and line rogue waves. The semi-rational solutions depict two types of interactions: (1) The fusion or fission between the lump wave and soliton; (2) The interaction between the line rogue wave and soliton. During the interaction between the line rogue wave and soliton, the line rogue wave evolves with three different shapes: the bright rogue waves, bright–dark rogue waves and dark rogue waves. Via the breather-type kink soliton solutions, we observe the breather-soliton mixture.