On a class of multidimensional integrable hierarchies and their reductions

TL;DR

This paper introduces a new class of multidimensional integrable hierarchies linked to vector field commutation, providing their representations and a dressing scheme, with examples including the Manakov-Santini and Dunajski systems.

Contribution

It presents a novel class of integrable hierarchies, their formulations, and a dressing method based on nonlinear vector Riemann problems, expanding understanding of multidimensional integrable systems.

Findings

Hierarchies connected with Manakov-Santini and Dunajski systems are analyzed.

A dressing scheme using nonlinear vector Riemann problems is developed.

The hierarchies are represented via generating equations and Lax-Sato form.

Abstract

A class of multidimensional integrable hierarchies connected with commutation of general (unreduced) (N+1)-dimensional vector fields containing derivative over spectral variable is considered. They are represented in the form of generating equation, as well as in the Lax-Sato form. A dressing scheme based on nonlinear vector Riemann problem is presented for this class. The hierarchies connected with Manakov-Santini equation and Dunajski system are considered as illustrative examples.