All expansions of solutions to the sixth Painlevé equation near its nonsingular point. (English. Russian original) Zbl 1181.34097
Dokl. Math. 79, No. 3, 397-402 (2009); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 426, No. 5, 586-591 (2009).
For the sixth Painlevé equation for all the values of its four complex parameters \(a, b, c\), and \(d\) near its nonsingular point \(x=x_0\neq 0,1,\infty\), we find all the asymptotic expansions of solutions of four types (power, powerlogarithmic, complicated, and exotic) and exponential asymptotic forms. Altogether, they form 17 families and all of them are of the power type. The other three types and exponential asymptotic forms are lacking, as must be for the Painlevé equation. Eight of these 17 families are new. The remaining nine families are known.
MSC:
| 34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |
References:
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