The Coalescence Limit of the Second Painlev\'E Equation

Abstract

In this paper, we study a well known asymptotic limit in which the second Painlev\'e equation (P_II) becomes the first Painlev\'e equation (P_I). The limit preserves the Painlev\'e property (i.e. that all movable singularities of all solutions are poles). Indeed it has been commonly accepted that the movable simple poles of opposite residue of the generic solution of P_{II} must coalesce in the limit to become movable double poles of the solutions of P_I, even though the limit naively carried out on the Laurent expansion of any solution of P_{II} makes no sense. Here we show rigorously that a coalescence of poles occurs. Moreover we show that locally all analytic solutions of P_I arise as limits of solutions of P_{II}.