Doubrov-Ferapontov general heavenly equation and the hyper-K\"ahler hierarchy

TL;DR

This paper explores the Doubrov-Ferapontov general heavenly equation, linking it to hyper-K"ahler geometry and hierarchies, and develops a $ar{ ext{d}}$-dressing method for solutions.

Contribution

It provides a new differential form description of the general heavenly equation and connects it to hyper-K"ahler structures and hierarchies, including multidimensional generalizations.

Findings

Describes the heavenly equation via a closed differential Pl"ucker two-form.

Establishes the equation as a generator in the hyper-K"ahler hierarchy.

Develops a $ar{ ext{d}}$-dressing scheme for solutions.

Abstract

We give a description of recently introduced Doubrov-Ferapontov general heavenly equation in terms of closed differential Pl\"ucker two-form, rationally depending on the spectral parameter. We demonstrate that general heavenly equation is an important generating equation in the context of Takasaki hyper-K\"ahler hierarchy, and it is also directly connected to hyper-K\"ahler geometry through the Gindikin construction. We develop a $\bar{\partial}$-dressing scheme and introduce a formula for the potential satisfying the general heavenly equation. Multidimensional generalization is also outlined.