Symmetry Reductions of the Lax Pair of the Four-Dimensional Euclidean Self-Dual Yang-Mills Equations

TL;DR

This paper investigates symmetry reductions of the Lax pair associated with four-dimensional Euclidean self-dual Yang-Mills equations, deriving simplified systems and explicit Lax pairs for related lower-dimensional equations.

Contribution

It provides a systematic symmetry reduction of the Lax pair for self-dual Yang-Mills equations and constructs explicit Lax pairs for reduced systems including Nahm's equations.

Findings

Derived reduced systems of differential equations from symmetry considerations.

Constructed explicit Lax pairs for modified Nahm's equations.

Identified reductions leading to integrable lower-dimensional PDEs.

Abstract

The reduction by symmetry of the linear system of the self-dual Yang-Mills equations in four-dimensions under representatives of the conjugacy classes of subgroups of the connected part to the identity of the corresponding Euclidean group under itself is carried out. Only subgroups leading to systems of differential equations nonequivalent to conditions of zero curvature without parameter, or to systems of uncoupled first order linear O.D.E.'s are considered. Lax pairs for a modified form of the Nahm's equations as well as for systems of partial differential equations in two and three dimensions are written out.