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The Boutroux ansatz for the second Painlevé equation in the complex domain. (Russian) Zbl 0719.34014
The author obtains an asymptotic representation of the general solution of the second Painlevé equation in a sector of the complex z-plane. In order to solve the problem the author uses the isomonodromic deformation method developed in a previous work [A. R. Its and the author, Isomonodromic deformation method in the theory of Painlevé equations. Lect. Notes in Math., 1191. Berlin (1986; Zbl 0592.34001)] for the real line. The main member of the presentation is an elliptic function, while in the real case it is either a trigonometric or a hyperbolic function.

MSC:
34M55Painlevé and other special equations; classification, hierarchies
34E05Asymptotic expansions (ODE)
30D05Functional equations in the complex domain, iteration and composition of analytic functions
34E20Asymptotic singular perturbations, turning point theory, WKB methods (ODE)