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The 0-instanton solution of Painlevé I is a sequence (u n,0 ) of complex numbers which appears universally in many enumerative problems in algebraic geometry, graph theory, matrix models and 2-dimensional quantum gravity. The asymptotics of the 0-instanton (u n,0 ) for large n were obtained by the third author using the Riemann-Hilbert approach. For k = 0, 1, 2, . . . , the k-instanton solution of Painlevé I is a doubly-indexed sequence (u n,k ) of complex numbers that satisfies an explicit quadratic non-linear recursion relation. The goal of the paper is three-fold: (a) to compute the asymptotics of the 1-instanton sequence (u n,1 ) to all orders in 1/n by using the Riemann-Hilbert method, (b) to present formulas for the asymptotics of (u n,k ) for fixed k and to all orders in 1/n using resurgent analysis, and (c) to confirm numerically the predictions of resurgent analysis. We point out that the instanton solutions display a new type of Stokes behavior, induced from the tritronquée Painlevé transcendents, and which we call the induced Stokes phenomenon. The asymptotics of the 2-instanton and beyond exhibits new phenomena not seen in 0 and 1-instantons, and their enumerative context is at present unknown.

Abstract. Using the Riemann-Hilbert approach, the Ψ-function corresponding to the solution of the first Painlevé equation yxx = 6y 2 + x with the asymptotic behavior y ∼ ± −x/6 as |x| → ∞ is constructed. The exponentially small jump in the dominant solution and the coefficient asymptotics in the power-like expansion to the latter are found.

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Using the Riemann-Hilbert approach, we study the quasi-linear Stokes phenomenon for the second Painlevé equation yxx = 2y 3 + xy − α. The precise description of the exponentially small jump in the dominant solution approaching α/x as |x| → ∞ is given. For the asymptotic power expansion of the dominant solution, the coefficient asymptotics is found.

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