Solving the $$\mathbf{(3+1) }$$ ( 3 + 1 ) -dimensional KP–Boussinesq and BKP–Boussinesq equations by the simplified Hirota’s method

Abstract

We study two (3 $$+$$ 1)-dimensional generalized equations, namely the Kadomtsev–Petviashvili–Boussinesq equation and the B-type Kadomtsev–Petviashvili–Boussinesq equation. We use the simplified Hirota’s method to conduct this study and to find the general phase shift of these equations. We obtain one- and two-soliton solutions, for each equation, with the coefficients of the three spatial variables are left as free parameters. However, we also develop special conditions on the coefficients of the spatial variables guarantee the existence of three-soliton solutions for each of these two equation.