Overview of the Kadomtsev–Petviashvili-hierarchy reduction method for solitons

Abstract

The Kadomtsev–Petviashvili (KP) hierarchy reduction method is a prominent direct method for deriving explicit solutions to integrable equations. This method is based on Hirota’s bilinear formulation of integrable systems, as well as the observation that bilinear forms of integrable systems belong to the KP hierarchy or its extensions. Thus, solutions to the KP hierarchy or its extensions, under proper reductions, would yield solutions to the underlying integrable system. In this article, we give a brief overview of this KP-hierarchy reduction method for the derivation of solitons, dark solitons, lumps and rogue waves in well-known integrable equations, such as the KP equation, the Korteweg–de Vries equation, the Davey–Stewartson equations, and the nonlinear Schrödinger equation.