`Interpolating' differential reductions of multidimensional integrable hierarchies

TL;DR

This paper extends the method of differential reductions to multidimensional integrable hierarchies, with a focus on four-dimensional cases related to the heavenly equation, providing new characterizations and solution techniques.

Contribution

It generalizes the differential reduction scheme from the Manakov-Santini hierarchy to multidimensional hierarchies, including the four-dimensional case, with new characterizations via Lax-Sato equations and dressing methods.

Findings

Characterization of differential reductions in multidimensional hierarchies

Extension of reduction schemes to four-dimensional integrable systems

Development of a dressing method based on nonlinear Riemann-Hilbert problems

Abstract

We transfer the scheme of constructing differential reductions, developed recently for the case of the Manakov-Santini hierarchy, to the general multidimensional case. We consider in more detail the four-dimensional case, connected with the second heavenly equation and its generalization proposed by Dunajski. We give a characterization of differential reductions in terms of the Lax-Sato equations as well as in the framework of the dressing method based on nonlinear Riemann-Hilbert problem.