Six-dimensional heavenly equation. Dressing scheme and the hierarchy

TL;DR

This paper develops a dressing scheme and hierarchy for the six-dimensional heavenly equation, providing a method to construct solutions via wave functions and generating relations within a dispersionless integrable hierarchy.

Contribution

It introduces a novel dressing scheme and hierarchy reduction for the six-dimensional heavenly equation, enabling explicit solution construction.

Findings

Developed a dressing scheme for the reduced hierarchy.

Constructed solutions using implicit functions.

Characterized the reduction via wave functions and generating relations.

Abstract

We consider six-dimensional heavenly equation as a reduction in the framework of general six-dimensional linearly degenerate dispersionless hierarchy. We characterise the reduction in terms of wave functions, introduce generating relation, Lax-Sato equations and develop the dressing scheme for the reduced hierarchy. Using the dressing scheme, we construct a class of solutions for six-dimensional heavenly equation in terms of implicit functions.