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Zbl 0647.34029
Sulejmanov, B.I.
The relation between asymptotic properties of solutions of the second Painlevé equation in different directions towards infinity. (English. Russian original)
[J] Differ. Equations 23, 569-576 (1987); translation from Differ. Uravn. 23, No.5, 834-842 (1987). ISSN 0012-2661

It is known that the second Painlevé equation (*) $q''=xq(x)+2q\sp 3(x),$ for $0<a<1$, has bounded solutions, with the asymptotic $$ q(x)\sim aAi(x)\sim (ax\sp{-1/4}\exp (- \frac{2}{3}x\sp{3/2}))/2\sqrt{\pi}, $$ for $x\to +\infty$, and for $x\to -\infty$ the following relations occur: $$ q'(x)=-(-x)\sp{1/4}\rho (x)[\cos \theta (x)+O(\vert x\vert\sp{-3/2}\ln \vert x\vert], $$ where $\rho\sp 2=\alpha\sp 2+O(\vert x\vert\sp{-3/2}\ln \vert x\vert),\rho$ (x)$\ge 0$, $\theta =(2/3)\vert x\vert\sp{3/2}-(3/4)\alpha\sp 2\ln \vert x\vert +\gamma +O(\vert x\vert\sp{-3/2}\ln \vert x\vert).$ The author proves the following relations between $\alpha$, $\gamma$ and $a:\alpha\sp 2=-\pi\sp{-1}\ln (1-a\sp 2),\gamma =\pi /4-\arg \{\Gamma (1- i\alpha\sp 2/2)\}-(3\alpha\sp 2\ln 2)/2.$ To this end a process of ``integration'' of (*) based on the study of some auxiliary linear problems is used.
[P.Talpalaru]

MSC 2000:: *34C11 Qualitative theory of solutions of ODE: Growth, etc.

Keywords: Painlevé equation; bounded solutions

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