On symmetries of the non-stationary PIIn hierarchy and their applications

TL;DR

This paper explores the symmetries of the non-stationary second Painlevé hierarchy, constructs associated affine Weyl groups, and links rational solutions to special polynomials with determinant representations.

Contribution

It introduces a detailed symmetry analysis of the hierarchy, constructs affine Weyl groups, and connects rational solutions with Yablonskii-Vorobiev-type polynomials and tau-functions.

Findings

Constructed affine Weyl group and its extension for the hierarchy

Linked rational solutions to Yablonskii-Vorobiev-type polynomials

Derived determinant representation of tau-functions in Jacobi-Trudi form

Abstract

In the current paper we study auto-B\"acklund transformations of the non-stationary second Painlev\'e hierarchy $\text{P}_\text{II}^{(n)}$ depending on $n$ parameters: a parameter $\alpha_n$ and times $t_1, \dots, t_{n-1}$. Using generators $s^{(n)}$ and $r^{(n)}$ of these symmetries, we have constructed an affine Weyl group $W^{(n)}$ and its extension $\tilde{W}^{(n)}$ associated with the $n$-th member considered hierarchy. We determined $\text{P}_\text{II}^{(n)}$ rational solutions via Yablonskii-Vorobiev-type polynomials $u_m^{(n)} (z)$. We brought out a correlation between Yablonskii-Vorobiev-type polynomials and polynomial $\tau$-functions $\tau_m^{(n)} (z)$ and found their determinant representation in the Jacobi-Trudi form.