All asymptotic expansions of solutions to the sixth Painlevé equation

A. D. Bruno, I. V. Goryuchkina


Abstract: We obtain all asymptotic expansions of solutions to the sixth Painlevé equation near all three its singular points $x=0$, $x=1$ and $x=\infty$ for all values of its four complex parameters. They form all together 111 families and include expansions of four types: power, power-logarithmic, complicated and exotic. In the expansions, the independent variable can have complex power exponents. First by methods of the power geometry, we obtain all such expansions near the singular point $x=0$ with the order of the first term less than one. These expansions are called basic. They form 19 families. All other asymptotic expansions of the solutions near three singular points of the equation can be computed from basic by means of symmetries of the equation. Most of these expansions are new. We give examples and comparisons with known results.