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

Abstract

In the current paper we study auto-Bäcklund transformations of the non-stationary second Painlevé hierarchy $\text{P}_\text{II}^{(n)}$ depending on $n$ parameters: a parameter $α_n$ 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 $τ$-functions $τ_m^{(n)} (z)$ and found their determinant representation in the Jacobi-Trudi form.