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Asymptotics of solutions of the Painlevé equation of the first kind. (English) Zbl 0677.34052
The Painlevé equation ${y}_{xx}=6{y}^{2}+x$ is considered as an equation for isomonodromic deformations of the associated system of linear differential equations with rational coefficients. The direct and the inverse problems of monodromy are investigated. This allows to give explicit formulas for the asymptotics of any solution of the Painlevé equation satisfying appropriate restrictions. The results presented refer to weakly nonlinear complex solutions, strongly nonlinear real solutions and also separatrix solutions.
Reviewer: I.Dorfman
##### MSC:
 34E05 Asymptotic expansions (ODE) 34E20 Asymptotic singular perturbations, turning point theory, WKB methods (ODE) 34L99 Ordinary differential operators