On the dressing method for Dunajski anti-self-duality equation

TL;DR

This paper develops a dressing method for the Dunajski anti-self-duality equation, introduces its hierarchy, and explores reductions, including waterbag-type, providing new tools for constructing solutions in integrable systems.

Contribution

A novel dressing scheme for the Dunajski equation and its hierarchy, with analysis of reductions and solution construction methods.

Findings

Developed a dressing scheme for Dunajski equation

Presented the hierarchy and Lax-Sato form of the equation

Studied reductions including waterbag-type

Abstract

A dressing scheme applicable to Dunajski equation is developed. Simple example of constructing solutions in terms of implicit functions is considered. Dunajski equation hierarchy is described, its Lax-Sato form is presented. Dunajsky equation hierarchy is characterised by conservation of three-dimensional volume form, in which a spectral variable is taken into account. Some reductions of Dunajsky equation hierarchy, including waterbag-type reduction, are studied.