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Zbl 0912.34007
Bassom, Andrew P.; Clarkson, Peter A.; Law, C.K.; McLeod, J.Bryce
Application of uniform asymptotics to the second Painlevé transcendent. (English)
[J] Arch. Ration. Mech. Anal. 143, No.3, 241-271 (1998). ISSN 0003-9527; ISSN 1432-0673/e

Connection problems of Painlevé differential equations of the form $d^2\phi/d\eta^2= -\xi^2F(\eta,\xi)\phi$ 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 $|\xi|\to \infty$ then an approximation can be derived in terms of parabolic cylinder functions.
[V.Burjan (Praha)]

MSC 2000:: *34A25 Analytical theory of ODE
34A45 Theoretical approximation of solutions of ODE
34M55 Painlevé and other special equations
33C10 Cylinder functions, etc.
34M40 Stokes phenomena and connection problems

Keywords: transformations; Painlevé differential equations; uniform approximations to solutions; Airy functions; parabolic cylinder functions

Cited in: Zbl 1182.33034 Zbl 1090.34072 Zbl 1068.34085

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