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.

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

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• Geometric Aspects of Painlevé Equations
• Determinant Structure of the Rational Solutions for the Painlevé II Equation
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