Expansions of solutions to the sixth painleve equation in cases $a=0$ and $b=0$

A. D. Bruno, I. V. Goryuchkina


Abstract: We consider the sixth Painleve equation for $a=0$. For the case we obtain six new expansions of solutions for $x\to0$ and $x\to\infty$. They are different from expansions known for $a\ne0$ and found early (Doklady Mathematics, 2004, v. 69, no. 2). Among six new expansions, four are power expansions, and two are complicated ones, i.e., they have nonpower asymptotics. Except that, we obtain new expansions of solutions for $b=0$. We use for this symmetry of the equation. All found here expansions are new.