On Boutroux's tritronquée solutions of the first Painlevé equation

Abstract

The triply truncated solutions of the first Painlevé equation were specified by Boutroux in his famous paper of 1913 as those having no poles (of large modulus) except in one sector of angle 2π/5. There are five such solutions and each of them can be obtained from any other one by applying a certain symmetry transformation. One of these solutions is real on the real axis. We found a characteristic property of this solution, different from the asymptotic description given by Boutroux. This allows us to estimate numerically the position of its real pole and zero closest to the origin. We also study properties of asymptotic series for truncated solutions.