Geodesic
On various approaches to asymptotics of solutions to the third Painlev\'e equation in a neighborhood of infinity
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential equations. Mathematical physics, Tome 139 (2017), pp. 70-78.

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We examine asymptotic expansions of the third Painlevé transcendents for $\alpha \delta \ne 0$ and $\gamma=0$ in a neighborhood of infinity in a sector of aperture ${}2 \pi$ 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 a 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.
Keywords: Painlevé equations, Newton polygon, asymptotic expansion, Gevrey order. 
@article{INTO_2017_139_a6,
     author = {A. V. Vasilyev and A. V. Parusnikova},
     title = {On various approaches to asymptotics of solutions to the third {Painlev\'e} equation in a neighborhood of infinity},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {70--78},
     publisher = {mathdoc},
     volume = {139},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2017_139_a6/}
}
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A. V. Vasilyev; A. V. Parusnikova. On various approaches to asymptotics of solutions to the third Painlev\'e equation in a neighborhood of infinity. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential equations. Mathematical physics, Tome 139 (2017), pp. 70-78. http://geodesic.mathdoc.fr/item/INTO_2017_139_a6/