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 ${\alpha }_n$ 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.

Some typos were fixed. The proof of Theorem 1.1 has been rewritten

References in corpus (2)

• The Hamiltonian Structure of the Second Painleve Hierarchy
• A remark on the Hankel determinant formula for solutions of the Toda equation