Singular normal form for the Painlevé equation P1

Abstract

We show that there exists a rational change of coordinates of Painlevé's P1 equation $y''=6y^2+x$ and of the elliptic equation $y''=6y^2$ after which these two equations become analytically equivalent in a region in the complex phase space where $y$ and $y'$ are unbounded. The region of equivalence comprises all singularities of solutions of P1 (i.e. outside the region of equivalence, solutions are analytic). The Painlevé property of P1 (that the only movable singularities are poles) follows as a corollary. Conversely, we argue that the Painlevé property is crucial in reducing P1, in a singular regime, to an equation integrable by quadratures.