Integrable systems and reductions of the self-dual Yang–Mills equations

Abstract

Many integrable equations are known to be reductions of the self-dual Yang–Mills equations. This article discusses some of the well known reductions including the standard soliton equations, the classical Painlevé equations and integrable generalizations of the Darboux–Halphen system and Chazy equations. The Chazy equation, first derived in 1909, is shown to correspond to the equations studied independently by Ramanujan in 1916.