Families of semi-rational solutions to the Kadomtsev-Petviashvili I equation

In this work we derive families of explicit breather solutions of any order to the Kadomtsev-Petviashvili equation (KPI) and the Boussinesq equation. We employ the Hirota bilinear method combined with the KP hierarchy reduction method to determine these solutions. By taking a long wave limit of breather solutions, two types of semi-rational solutions to the KPI equation are constructed via using the determinant expression. The first type of semi-rational solutions only consists of breathers of arbitrary order and lumps of arbitrary order in the (x, y)-plane. The second type of semi-rational solutions comprises of solitons of arbitrary order, breathers of arbitrary order and lumps of arbitrary order in the (x, y)-plane.