$\ast$-SDYM Fields and Heavenly Spaces. I. $\ast$-SDYM equations as an integrable system

TL;DR

This paper demonstrates that the self-dual Yang-Mills equations on heavenly spaces can be reduced to a single master equation, with associated conservation laws, twistor transform, and solutions to the Riemann-Hilbert problem, advancing integrable systems theory.

Contribution

It introduces a reduction of SDYM equations to a master equation on heavenly spaces and constructs related conservation laws and twistor solutions, revealing new integrable structures.

Findings

Reduction of SDYM equations to a master equation

Construction of two hierarchies of conservation laws

Explicit twistor transform and Riemann-Hilbert solutions

Abstract

It is shown that the self-dual Yang-Mills (SDYM) equations for the $\ast$-bracket Lie algebra on a heavenly space can be reduced to one equation (the \it master equation\rm). Two hierarchies of conservation laws for this equation are constructed. Then the twistor transform and a solution to the Riemann-Hilbert problem are given.