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Zbl 0681.34053
Its, A.R.; Kapaev, A.A.
The method of isomonodromy deformations and connection formulas for the second Painlevé transcendent. (English. Russian original)
[J] Math. USSR, Izv. 31, No.1, 193-207 (1988); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 51, No.4, 878-892 (1987). ISSN 0025-5726

The authors consider the equation $u\sb{xx}-ux-2u\sp 3=0$ (the special case of the Painleve equation of the second type). Using the method of monodromy preserving deformation they give the following description of asymptotic behavior of pure imaginary solution of the equation. As $x\to -\infty$ the asymptotic formula $$ u(x)=i\alpha (-x)\sp{-1/4} \sin \{(2/3)(-x)\sp{3/2}+(3/4)\alpha\sp 2 \ln (-x)+\phi)+O((-x)\sp{-1/4}) $$ holds, where $\alpha$,$\phi$ are the parameters of the solution u(x), $\alpha >0,0\le \phi <2\pi$. As $x\to +\infty$ two cases are possible. \par If the parameters $\alpha$,$\phi$ satisfy some conditions then the asymptotic formula $$ u(x)=(ia/(2\sqrt{\pi}))\kappa\sp{-1/4}e\sp{- 2/3\kappa\sp{3/2}}(1+O(1)) $$ holds where $a\sp 2=e\sp{\pi \alpha\sp 2}- 1, sign a=2(1/2-\epsilon).$ \par If this condition is violated then the asymptotic formula $$ u(x)=\pm i\sqrt{\kappa /2}\pm i(2x)\sp{-1/4}\rho \cos \{(2\sqrt{2}/3)x\sp{3/2}- (3/2)r\sp 2 \ln x+\theta \}+O(x\sp{-1/4}) $$ holds. Given is the explicit formula for calculating the parameters $\rho$,$\theta$ in terms of the parameters $\alpha$,$\phi$. The authors admit that some of these results are not new.
[A.M.Šermenev]

MSC 2000:: *34E99 Asymptotic theory of ODE
34B30 Special ODE

Keywords: Painleve equation of the second type; method of monodromy; asymptotic formula

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