Tronqu\'ee solutions of the Painlev\'e equation \P1

Abstract

We analyze the one parameter family of tronqu\'ee solutions of the Painlev\'e equation \P1 in the pole-free sectors together with the region of the first array of poles. We find a convergent expansion for these solutions, containing one free parameter multiplying exponentially small corrections to the Borel summed power series. We link the position of the poles in the first array to the free parameter, and find the asymptotic expansion of the pole positions in this first array (in inverse powers of the independent variable). We show that the tritronqu\'ees are given by the condition that the parameter be zero. We show how this analysis in conjunction with the asymptotic study of the pole sector of the tritronqu\'ee in \cite{inprep} leads to a closed form expression for the Stokes multiplier directly from the Painlev\'e property, not relying on isomonodromic or related type of results.