Introduction to KP Theory and KP Solitons

Abstract

We begin with a study of the Burgers equation, which is the simplest equation combining both nonlinear propagation effects and diffusive effects, and can be used to describe a weak shock phenomena in gas dynamics (see e.g. [131]). The Burgers equation can be linearized by a nonlinear transformation, known as the Cole-Hopf transformation. The linearization then shows that the Burger equation has an infinite number of symmetries, and the set of those symmetries defines the Burgers hierarchy. The linearization enables us to construct several exact solutions such as multi-shock solutions. It turns out that the set of those exact solutions forms a subclass of the solutions of the KP equation, and multi-shock solutions give examples of the resonant interactions in the KP solutions. We then extend the Cole-Hopf transformation to construct a multi-component Burgers hierarchy, and introduce the \(\tau \)-function, which generates a large class of exact solutions of the KP equation, referred to as KP solitons. Based on the study of this multi-component Burgers hierarchy, we explain its connection to the Sato theory, which provides a mathematical foundation of the KP hierarchy in terms of an infinite dimensional Grassmann variety called the Sato Grassmannian [112, 113]. In this book, we consider a finite dimensional version of the Sato Grassmannian and construct each KP soliton from a point of this Grassmannian.