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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References
Balser W. A different characterization of multi-summable power series. Analysis 1992;12:57–65.
Braaksma B. Multisummability of formal power series solutions of nonlinear meromorphic differential equations. Ann Iinst Fourier 1992;42(3):517–40.
Costin O. Exponential asymptotics, transseries, and generalized Borel summation for analytic, nonlinear, rank-one systems of ordinary differential equations. Int Math Res Not 1995;1995:377–417.
Costin O. On Borel summation and stokes phenomena for rank-1 nonlinear systems of ordinary differential equations. Duke Math J 1998;93:289–344.
Costin O. Topological construction of transseries and introduction to generalized Borel summability. Analyzable functions and applications, Contemp, Math., vol 373, Amer Math Soc, Providence, RI; 2005. p. 137–75.
Gromak VI, Lukashevich NA. 1990. Analytical properties of solutions to Painlevé equations. Minsk: Universitetskoe (Russian).
Lin Y, Dai D, Tibboel P. Existence and uniqueness of tronquée solutions of the third and fourth Painlevé equations. Nonlinearity 2014;27(2, 02):171–86.
Lukashevich NA. On a theory of Painlevé third equation. (Russian) Differencial’nye Uravneniya 1967;3(11):1913–23.
Malgrange B. Sur le théorème de Maillet. Asymptot Anal 1989;2:1–4.
Mitschi C, Sauzin D. Divergent series, summability and resurgence I monodromy and resurgence. Lect Notes Math 2016;2153:298.
Parusnikova AV, Vasilyev AV. On divergence of Puiseux series asymptotic expansions of solutions to the third Painleé equation. arXiv:1702.05758.
Bruno AD, Parusnikova AV. 2012. Expansions and asymptotic forms of solutions to the fifth Painlevé equation near infinity. Preprint of KIAM, No 61. Moscow, pp 32, in Russian.
Parusnikova AV. On Gevrey orders of formal power series solutions to the third and fifth Painlevé equations near infinity. Opuscula Math 2014;34(3):591–9. arXiv:1310.5345.
Parusnikova AV. 2013. Gevrey orders of solutions to the fourth Painleveé equation near infinity in a general case. Preprint of KIAM, No 97. Moscow, pp 12, in Russian.
Vasilyev AV, Parusnikova AV. 2017. On various approaches to asymptotics of solutions to the third Painlevé equation in a neighborhood of infinity. Itogi Nauki i Techniki. Sovremennaya matematika i ee prilojeniya. Tematicheskie obzory 139, 70-78 (Russian). To be translated in “Journal of Mathematical Sciences”.
Xia X. Tronquée solutions of the third and fourth Painlevé equations. arXiv:1803.11230v4.
Ramis J-P. 1993. Séries Divergentes et Théories Asymptotiques. Bulletin Sociéte Mathématique de France Panoramas et Synthéses, vol 121.
Acknowledgements
We are grateful to the referees whose comments were very helpful.
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The work of the first author was supported by grant RFBR-CNRS 16-51-150005-a.
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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