Construction of coupled Harry Dym hierarchy and its solutions from St\"ackel systems

TL;DR

This paper develops a method to construct the coupled Harry Dym hierarchy from classical St"ackel systems, providing new solutions and insights into the hierarchy's structure and solutions.

Contribution

It introduces a novel approach to derive the coupled Harry Dym hierarchy from St"ackel systems and constructs explicit solutions from these systems.

Findings

Constructed the coupled Harry Dym hierarchy from St"ackel systems.

Derived both nonlocal and differential parts of the hierarchy.

Generated explicit solutions from St"ackel system solutions.

Abstract

In this paper we show how to construct the coupled (multicomponent) Harry Dym (cHD) hierarchy from classical St\"ackel separable systems. Both nonlocal and purely differential parts of hierarchies are obtained. We also construct various classes of solutions of cHD hierarchy from solutions of corresponding St\"ackel systems.