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Симметрии нестационарной иерархии PIIn и их приложения
Теоретическая и математическая физика. 2022. Т. 213. № 1. С. 65–94.
Bobrova I.
We study auto-Bäcklund transformations of the nonstationary second Painlevé hierarchy $\rm{P}_{\rm{II}}^{(n)}$ depending on n parameters: a parameter $\alpha_n$ and times $t_1, …, t_{n−1}$. Using generators $s^{(n)}$ and $r^{(n)}$ of these symmetries, we construct an affine Weyl group $W^{(n)}$ and its extension $\tilde{W}^{(n)}$ associated with the nth member of the hierarchy. We determine rational solutions of $\rm{P}_{\rm{II}}^{(n)}$ in terms of Yablonskii–Vorobiev-type polynomials $u_m^{(n)} (z)$. We show that Yablonskii–Vorobiev-type polynomials are related to the polynomial τ-function $\tau_m^{(n)} (z)$ and find their determinant representation in the Jacobi–Trudi form.
Keywords: уравнения ПенлевеPainlevé equationsBäcklund transformationsaffine Weyl groupsаффинные группы ВейляYablonskii–Vorobiev polynomialspolynomial τ-functionsJacobi–Trudi determinantsпреобразования Беклундаполиномы Яблонского–Воробьеваполиномиальные τ-функциидетерминанты Якоби–Труди
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