Tritronquée Solutions of Perturbed First Painlevé Equations

Abstract

We consider solutions of the class of ODEs y″ = 6y2 − xμ, which contains the first Painleve equation (PI) for μ = 1. It is well known that PI has a unique real solution (called a tritronquee solution) asymptotic to \( - \sqrt {x/6} \) and decaying monotonically on the positive real line. We prove the existence and uniqueness of a corresponding solution for each real nonnegative μ ≠ 1.