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We elaborate a systematic way to obtain higher order contributions in thenonlinear steepest descent method for Riemann-Hilbert problem associated withhomogeneous Painleve II equation. The problem is reformulated as a matrixfactorization problem on two circles and can be solved perturbatively reducingit to finite systems of algebraic linear equations. The method is applied tofind explicitly long-time asymptotic behaviour for tau function of Painleve IIequation.

Long-time asymptotic series for the Painleve II equation: Riemann-Hilbert approach