Geodesic
Asymptotic expansions for partial solutions of the sixth Painlev\'e equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 128 (2001) no. 2, pp. 193-204

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A formalism for an averaging method for the Painlevé equations, in particular, the sixth equation, is developed. The problem is to describe the asymptotic behavior of the sixth Painlevé transcendental in the case where the module of the independent variable tends to infinity. The corresponding expansions contain an elliptic function (ansatz) in the principal term. The parameters of this function depend on the variable because of the modulation equation. The elliptic ansatz and the modulation equation for the sixth Painlevé equation are obtained in their explicit form. A partial solution of the modulation equation leading to a previously unknown asymptotic expansion for the partial solution of the sixth Painlevé equation is obtained. 
@article{TMF_2001_128_2_a3,
     author = {V. L. Vereshchagin},
     title = {Asymptotic expansions for partial solutions of the sixth {Painlev\'e} equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {193--204},
     publisher = {mathdoc},
     volume = {128},
     number = {2},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2001_128_2_a3/}
}
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V. L. Vereshchagin. Asymptotic expansions for partial solutions of the sixth Painlev\'e equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 128 (2001) no. 2, pp. 193-204. http://geodesic.mathdoc.fr/item/TMF_2001_128_2_a3/