The WKB Riemann-Hilbert Problem for the Painlevé I System

Abstract

In this paper, we solve the direct monodromy problem for the Borel-summed $0$-parameter solution of Painlevé~I. Using the auxiliary time equation, we first construct formal WKB wave matrices whose amplitudes are regular at the double turning point. Rigorous Borel summation then yields exact WKB matrices in each Stokes region. Uniform estimates on moving action strips provide a common exact wave matrix in a full neighbourhood of the double point, proving that all four incident jumps are identities for consistent column normalisations. We formulate the resulting normalised Riemann--Hilbert problem and prove its existence and uniqueness on the Borel domains of the construction. Finally, after Painlevé scaling, the five Borel-summed solutions coincide with the five rotated tritronquée solutions of Painlevé~I.