On the geometry of the first and second Painlev\'e equations

Abstract

In this paper we \emph{explicitly} compute the transformation that maps the generic second order differential equation $y''= f(x, y, y')$ to the Painlev\'e first equation $y''=6y^2+x$ (resp. the Painlev\'e second equation ${y''=2 y^{3}+yx+ \alpha}$). This change of coordinates, which is function of $f$ and its partial derivatives, does not exist for every $f$; it is necessary that the function $f$ satisfies certain conditions that define the equivalence class of the considered Painlev\'e equation. In this work we won't consider these conditions and the existence issue is solved \emph{on line} as follows: If the input equation is known then it suffices to specialize the change of coordinates on this equation and test by simple substitution if the equivalence holds. The other innovation of this work lies in the exploitation of discrete symmetries for solving the equivalence problem.