Abstract
We examine asymptotic expansions of the third Painlevé transcendents for αδ ≠ = 0 and γ = 0 in the neighborhood of infinity in a sector of aperture <2π by the method of dominant balance). We compare intermediate results with results obtained by methods of three-dimensional power geometry. We find possible asymptotics in terms of elliptic functions, construct a power series, which represents an asymptotic expansion of the solution to the third Painlevé equation in a certain sector, estimate the aperture of this sector, and obtain a recurrent relation for the coefficients of the series.
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 139, Differential Equations. Mathematical Physics, 2017.
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Vasilyev, A.V., Parusnikova, A.V. On Various Approaches to Asymptotics of Solutions to the Third Painlevé Equation in a Neighborhood of Infinity. J Math Sci 241, 318–326 (2019). https://doi.org/10.1007/s10958-019-04426-3
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DOI: https://doi.org/10.1007/s10958-019-04426-3