Two-dimensional reductions of the self-dual Yang–Mills equations in self-dual spaces

Abstract

Many completely integrable equations have been shown to be reductions of the Yang–Mills equations in a flat space. In this paper two-dimensional symmetry reductions of the self-dual Yang–Mills equations, together with the corresponding linear equations, in an arbitrary self-dual space are classified. The reduced equations are given in different forms. Some new equations are found, in particular one which generalizes the sine-Gordon equation and the Liouville equation.