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Application of uniform asymptotics to the second Painlevé transcendent. (English) Zbl 0912.34007
Connection problems of Painlevé differential equations of the form d 2 ϕ/dη 2 =-ξ 2 F(η,ξ)ϕ are studied. These problems involve finding uniform approximations to solutions to this equation when the independent variable passes towards infinity along different directions in the complex plane. By the method used the need to match solutions is avoided. The treatment depends on the locations of the zeros of the function F in the limit. If they are isolated a uniform approximation to solutions can be derived in terms of Airy functions of suitable argument. If two of the zeros of F coalesce as |ξ| then an approximation can be derived in terms of parabolic cylinder functions.
Reviewer: V.Burjan (Praha)

34A25Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.)
34A45Theoretical approximation of solutions of ODE
34M55Painlevé and other special equations; classification, hierarchies
33C10Bessel and Airy functions, cylinder functions, 0 F 1
34M40Stokes phenomena and connection problems (ODE in the complex domain)