A family of solutions of a nonlinear ordinary differential equation and its application to Painlevé equations (III), (V) and (VI)
We deal with an equation of the form \[ (E)\quad x(xu')'=F_ 0(x,e^{- u},xe\quad u)+F_ 1(x,e^{-u},xe\quad u)(xu')+F_ 2(x,e^{- u\quad},xe\quad u)(xu')^ 2\quad ('=d/dx). \] Here \(F_ j(x,\xi,\eta)'s\) are holomorphic and bounded in the domain \(| x| <r_ 0\), \(| \xi | <r_ 1\), \(| \eta | <r_ 1\) \((r_ 0,r_ 1>0)\) and satisfy \(F_ j(0,0,0)=0\). In the neighbourhood of \(x=0\), we construct a two-parameter family of solutions of equation (E), and using the result, we give some analytic representations of solutions of Painlevé equations (III), (V), and (VI) near the singular point of the regular type. In some special cases, our representations of solutions of Painlevé equations are valid in the domain where the solution have infinitely many movable poles and zeroes.
scientific article; zbMATH DE number 4040318