Multi-component Painlev\'{e} ODE's and related non-autonomous KdV stationary hierarchies

TL;DR

This paper develops a method to derive non-autonomous Painlevé hierarchies from autonomous Stäckel systems and reconstructs related non-autonomous KdV stationary hierarchies, linking integrable ODEs and PDEs.

Contribution

It introduces a novel approach to deform autonomous Stäckel systems into Painlevé hierarchies and reconstructs associated non-autonomous KdV stationary hierarchies.

Findings

Established a connection between autonomous Stäckel systems and Painlevé hierarchies.

Reconstructed non-autonomous KdV stationary hierarchies from Painlevé systems.

Provided a framework for deriving non-autonomous integrable hierarchies.

Abstract

First, starting from two hierarchies of autonomous St\"{a}ckel ODE's, we reconstruct the hierarchy of KdV stationary systems. Next, we deform considered autonomous St\"{a}ckel systems to non-autonomous Painlev\'{e} hierarchies of ODE's. Finally, we reconstruct the related non-autonomous KdV stationary hierarchies from respective Painlev\'{e} systems.