Singular Normal Form for the Painlev

We show that there exists a rational change of coordinates of Painlev e's P1 equation y 00 = 6y 2 +x and of the elliptic equation y 00 = 6y 2 after which these two equations become analytically equivalent in a region in the complex phase space where y and y 0 are unbounded. The region of equivalence comprises all singularities of solutions of P1 (i.e. outside the region of equivalence, solutions are analytic). The Painlev e property of P1 (that the only movable singularities are poles) follows as a corollary. Conversely, we argue that the Painlev e property is crucial in reducing P1, in a singular regime, to an equation integrable by quadratures. 1. Introduction The problem of determining which classes of nonlinear diierential equations can deene new transcendents (special functions having good properties), received a special attention in the last century, especially due to the emphasis on nding \explicit" solutions to diierential equations. Fuchs had the intuition that the appropriate condition these equations must satisfy is that their solutions have no movable branch points. This feature of an equation is now known as the Painlev e property and proved to be a very relevant characteristic, in a wide range of problems. Fuchs' study was pursued by Briot and Bouquet, and then by Painlev e 10] and Gambier who showed that there are no new transcendents coming from rst order equations, but there are six second order equations which deene new special functions. These equations (now denoted usually as P1 to P6) were discovered as a result of a purely theoretical quest, but they later arose naturally in many distinct physical applications (see, e.g., 4] and 7]). Linearization of second order Painlev e equations through the isomonodromic transformation method 2], 4], 8], 9] is one of the most important recent developments. To this date, higher order equations have not yet been classiied from the point of view of Painlev e integrability. Perhaps surprisingly, proving the Painlev e property of an equation turns out to be quite diicult (although if one assumes that singularities are described locally by convergent power-logarithmic series, then it is usually easy to check for the absence of movable branch points) and some of the classical proofs for Painlev e equations have been subsequently challenged. See also 5] and 6]. The Painlev e property, being shared by all solutions, must reveal a particular structure of the …