Issues of Multi-Dimensional Integrable Systems

Abstract

This chapter discusses the issues of multi-dimensional integrable systems. Enormous list of examples are accumulated for many years of intensive studies from both the physical and mathematical sides. Most of this list is occupied by the soliton equations, which are so named because of their origin in soliton phenomena. Mathematically, soliton equations describe nonlinear waves propagating in one-dimensional space like a canal. Even the Kadomtsev–Petviashvili (KP) equation is considered as such though physically it has been introduced as a two-space-dimensional generalisation of the Korteweg-de Vries (KdV) equation. A few examples of multi-dimensional integrable systems have been discovered from a distinct point of view, that is, twistor theory. The equations of motion of self-dual connections, self-dual metrics, and their extensions to dimensions greater than four provide such examples.