On the Riemann–Hilbert approach to asymptotics of tronquée solutions of Painlevé I

Abstract

Abstract In this paper, we revisit large variable asymptotic expansions of tronquée solutions of the Painlevé I equation, obtained via the Riemann–Hilbert approach and the method of steepest descent. The explicit construction of an extra local parametrix around the recessive stationary point of the phase function, in terms of complementary error functions, makes it possible to give detailed information about exponential-type contributions beyond the standard Poincaré expansions for tronquée and tritronquée solutions.