Some reductions of the self-dual Yang–Mills equations to integrable systems in 2+1 dimensions

Abstract

A reduction of the self-dual Yang–Mills (SDYM) equations is studied by imposing two space–time symmetries and by requiring that the connection one-form belongs to a Lie algebra of formal matrix-valued differential operators in an auxiliary variable. In this article, the scalar case and the canonical cases for 2×2 matrices are examined. In the scalar case, it is shown that the field equations can be reduced to the forced Burgers equation. In the matrix case, several well-known 2+1 integrable equations are obtained. Also examined are certain transformation properties between the solutions of some of these 2+1 equations.