Kadomtsev-Petviashvili reduction and rational solutions of the generalized (2+1)-dimensional Boussinesq equation

In this work, we study rational solutions of the generalized (2 + 1)-dimensional Boussinesq equation which includes the classical (1 + 1)-dimensional Boussinesq equation, the classical (2 + 1)-dimensional Boussinesq equation, the classical (1 + 1)-dimensional Benjamin-Ono equation and the (2 + 1)-dimensional Benjamin-Ono equation. By fixing the reduction condition, the bilinear form of Kadomtsev-Petviashvili equation is reduced to the corresponding bilinear form of the generalized (2 + 1)-dimensional Boussinesq equation under three different variable transformation. As a result, three families of the N th-order rational solutions of the generalized (2 + 1)-dimensional Boussinesq equation are successfully derived by means of Kadomtsev-Petviashvili hierarchy reduction technique. These rational solutions can generate the rogue waves and lumps since they can be localized in spatial variables and spatial-temporal variables. Their rich dynamic behaviors are graphically exhibited. It is found that this equation admits bright- and dark-type rogue waves depending on the sign of coefficient alpha. Especially, various rogue wave patterns, such as triangle and pentagon, are established in the framework of the classical (1 + 1)-dimensional Benjamin-Ono equation.