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The relation between asymptotic properties of solutions of the second Painlevé equation in different directions towards infinity. (English) Zbl 0647.34029

It is known that the second Painlevé equation (*) q '' =xq(x)+2q 3 (x), for 0<a<1, has bounded solutions, with the asymptotic

q(x)aAi(x)(ax -1/4 exp(-2 3x 3/2 ))/2π,

for x+, and for x- the following relations occur:

q ' (x)=-(-x) 1/4 ρ(x)[cosθ(x)+O(|x| -3/2 ln|x|],

where ρ 2 =α 2 +O(|x| -3/2 ln|x|),ρ (x)0, θ=(2/3)|x| 3/2 -(3/4)α 2 ln|x|+γ+O(|x| -3/2 ln|x|)· The author proves the following relations between α, γ and a:α 2 =-π -1 ln(1-a 2 ),γ=π/4-arg{Γ(1-iα 2 /2)}-(3α 2 ln2)/2· To this end a process of “integration” of (*) based on the study of some auxiliary linear problems is used.

Reviewer: P.Talpalaru
MSC:
34C11Qualitative theory of solutions of ODE: growth, boundedness