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On symmetries of the nonstationary \(\mathrm{P}_\mathrm{II}^{(n)}\) hierarchy and their applications

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Abstract

We study auto-Bäcklund transformations of the nonstationary second Painlevé hierarchy \( \mathrm{P}_\mathrm{II}^{(n)} \) 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 construct an affine Weyl group \(W^{(n)}\) and its extension \(\widetilde{W}^{(n)}\) associated with the \(n\)th member of the hierarchy. We determine rational solutions of \( \mathrm{P}_\mathrm{II}^{(n)} \) in terms of Yablonskii–Vorobiev-type polynomials \(u_m^{(n)}(z)\). We show that Yablonskii–Vorobiev-type polynomials are related to the polynomial \(\tau\)-function \(\tau_m^{(n)}(z)\) and find their determinant representation in the Jacobi–Trudi form.

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Notes

  1. The difference–differential equation in this theorem was also found in [15], where polynomials \(u_m^{(n)}\) were called the generalized Yablonskii–Vorobiev polynomials. The author of this paper obtained this result independently of [15] and gives the most complete proof of this remarkable fact.

  2. Many thanks go to Marta Mazzocco, who drew the author’s attention to this fact.

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Acknowledgments

The author is grateful to her scientific advisors Vladimir Poberezhnyi and Vladimir Rubtsov, who introduced her to this amazing area of mathematical physics, constantly supported her in the course of research, and developed her interest in the theory of Painlevé equations. The author expresses her special gratitude to Vladimir Rubtsov for setting the scientific problem and his continuous attention to this work. The author is also grateful to Marta Mazzocco, who drew the author’s attention to a different way of constructing affine Weyl group generators, and to Ilia Gaiur for the fruitful discussions. The author thanks the referee for the careful reading of the paper and helpful remarks, which, in particular, improved the proof of Theorem 5.

Funding

This paper is a part of the author’s PhD studies at the HSE University and has been carrying out at the Faculty of Mathematics. The author thanks this faculty for giving her such an opportunity. This paper was partially supported by the Russian Foundation for Basic Research grant No. 18-01-00461_a and the International Laboratory of Cluster Geometry, HSE, under the RF Government grant No. 075-15-2021-608.

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  1. National Research University “Higher School of Economics”, Moscow, Russia

    I. A. Bobrova

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 213, pp. 65–94 https://doi.org/10.4213/tmf10173.

Appendix Plots of the roots of Yablonskii–Vorobiev-type polynomials

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Bobrova, I.A. On symmetries of the nonstationary \(\mathrm{P}_\mathrm{II}^{(n)}\) hierarchy and their applications. Theor Math Phys 213, 1369–1394 (2022). https://doi.org/10.1134/S0040577922100063

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