The Sigma Form for the PII Hierarchy

TL;DR

This paper explores the sigma form of the second Painlevé hierarchy, utilizing Hamiltonian structures and Lenard operators to deepen understanding of its mathematical properties.

Contribution

It introduces a sigma form for the second Painlevé hierarchy based on Hamiltonian and Lenard operator techniques, advancing the theoretical framework.

Findings

Derived the sigma form for the second Painlevé hierarchy

Identified key properties of the Hamiltonian structure

Enhanced understanding of the hierarchy's mathematical structure

Abstract

In this paper we study the so-called sigma form of the second Painlev\'e hierarchy. To obtain this form, we use some properties of the Hamiltonian structure of the second Painlev\'e hierarchy and of the Lenard operator.