Dunajski generalization of the second heavenly equation: dressing method and the hierarchy

TL;DR

This paper develops a dressing method and describes the hierarchy for the Dunajski generalization of the second heavenly equation, advancing solution techniques and understanding of its integrable structure.

Contribution

It introduces a dressing scheme for the Dunajski equation and characterizes its hierarchy via a Lax-Sato form, highlighting volume form conservation.

Findings

Constructed solutions using implicit functions

Presented the Lax-Sato form of the hierarchy

Identified volume form conservation in the hierarchy

Abstract

Dunajski generalization of the second heavenly equation is studied. A dressing scheme applicable to Dunajski equation is developed, an 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 characterized by conservation of three-dimensional volume form, in which a spectral variable is taken into account.