On Soliton Content of Self Dual Yang-Mills Equations

TL;DR

This paper develops a method to construct solutions to the Self Dual Yang-Mills equations using soliton hierarchies and Riemann-Hilbert factorization, linking integrable systems techniques to gauge theory solutions.

Contribution

It introduces a framework to pull back soliton hierarchies to the Self Dual Yang-Mills equations via Riemann-Hilbert problems, enabling new solution constructions.

Findings

Solutions include those from symmetry reductions and genuine 4D solutions.

The method recovers known solutions like breaking solitons.

Provides a new perspective on integrable structures within gauge theories.

Abstract

Exploiting the formulation of the Self Dual Yang-Mills equations as a Riemann-Hilbert factorization problem, we present a theory of pulling back soliton hierarchies to the Self Dual Yang-Mills equations. We show that for each map $ \C^4 \to \C^{\infty } $ satisfying a simple system of linear equations formulated below one can pull back the (generalized) Drinfeld-Sokolov hierarchies to the Self Dual Yang-Mills equations. This indicates that there is a class of solutions to the Self Dual Yang-Mills equations which can be constructed using the soliton techniques like the $\tau$ function method. In particular this class contains the solutions obtained via the symmetry reductions of the Self Dual Yang-Mills equations. It also contains genuine 4 dimensional solutions . The method can be used to study the symmetry reductions and as an example of that we get an equation exibiting breaking solitons, formulated by O. Bogoyavlenskii, as one of the $2 + 1 $ dimensional reductions of the Self Dual Yang-Mills equations.