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On the Exact Gevrey Order of Formal Puiseux Series Solutions to the Third Painlevé Equation

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Abstract

In this paper, we study the third Painlevé equation with parameters γ = 0, αδ ≠  0. The Puiseux series formally satisfying this equation (after a certain change of variables) asymptotically approximate of Gevrey order one solutions to this equation in sectors with vertices at infinity. We present a family of values of the parameters δ = −β2/2 ≠ 0 such that these series are of exact Gevrey order one, and hence diverge. We prove the 1-summability of them and provide analytic functions which are approximated of Gevrey order one by these series in sectors with the vertices at infinity.

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Acknowledgements

We are grateful to the referees whose comments were very helpful.

Funding

The work of the first author was supported by grant RFBR-CNRS 16-51-150005-a.

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Correspondence to A. Parusnikova.

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Parusnikova, A., Vasilyev, A. On the Exact Gevrey Order of Formal Puiseux Series Solutions to the Third Painlevé Equation. J Dyn Control Syst 25, 681–690 (2019). https://doi.org/10.1007/s10883-019-09449-2

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  • DOI: https://doi.org/10.1007/s10883-019-09449-2

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