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

Abstract

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.