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Zbl 0677.34052
Kapaev, A.A.
Asymptotics of solutions of the Painlevé equation of the first kind. (English. Russian original)
[J] Differ. Equations 24, No.10, 1107-1115 (1988); translation from Differ. Uravn. 24, No.10, 1684-1695 (1988). ISSN 0012-2661

The Painlevé equation $y\sb{xx}=6y\sp 2+x$ 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.
[I.Dorfman]

MSC 2000:: *34E05 Asymptotic expansions (ODE)
34E20 Asymptotic singular perturbations, methods (ODE)
34L99 Ordinary differential operators

Keywords: inverse scattering method; Painlevé equation; isomonodromic deformations; linear differential equations with rational coefficients; inverse problems; monodromy; explicit formulas; asymptotics; weakly nonlinear; complex solutions; strongly nonlinear; real solutions; separatrix solutions

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