Self-dual Yang–Mills hierarchy: A nonlinear integrable dynamical system in higher dimensions

Abstract

A self-dual Yang–Mills (SDYM) hierarchy is presented that has some properties similar to those of the celebrated Kadomtsev–Petviashvili (KP) hierarchy having one spacial variable. The SDYM hierarchy is an infinite system of GL(∞)-invariant compatible SDYM equations depending on an infinite number of spacial variables, on which an exponential operator acts as a time evolution operator. A nontrivial GL(∞) symmetry of the SDYM hierarchy called the Bruhat transformation is found. The existence of the GL(∞) SDYM hierarchy supports the complete integrability of the single GL(∞)SDYM equation with a reduction condition, however, our observation suggests that the SDYM equation without the condition may not be completely integrable in sharp contrast with the KP equation.