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Article

On the exact Gevrey order of formal Puiseux series solutions to the third Painlevé equation

Journal of Dynamical and Control Systems. 2019. Vol. 25. No. 4. P. 681-690.
Parusnikova A., Vasilyev A. V.

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 = −β^2/2 ≠ 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.