Studies on the second member of the second Painlev\'e hierarchy

TL;DR

This paper investigates the second member of the second Painlevé hierarchy, revealing its Hamiltonian structure, symmetries, and associated differential equations, thus advancing understanding of integrable systems in mathematical physics.

Contribution

It introduces a new polynomial Hamiltonian system for the second Painlevé hierarchy, analyzes its symmetries, and connects it to higher-order differential equations.

Findings

Hamiltonian system is a 1-parameter family of coupled Painlevé systems

System admits extended affine Weyl group symmetry of type A1^{(1)}

Derived a fifth-order ODE with symmetry and holomorphy conditions

Abstract

In this paper, we study the second member of the second Painlev\'e hierarchy $P_{II}^{(2)}$. We show that the birational transformations take this equation to the polynomial Hamiltonian system in dimension four, and this Hamiltonian system can be considered as a 1-parameter family of coupled Painlev\'e systems. This Hamiltonian is new. We also show that this system admits extended affine Weyl group symmetry of type $A_1^{(1)}$, and can be recovered by its holomorphy conditions. We also study a fifth-order ordinary differential equation satisfied by this Hamiltonian. After we transform this equation into a system of the first-order ordinary differential equations of polynomial type in dimension five by birational transformations, we give its symmetry and holomorphy conditions.