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The method of isomonodromy deformations and connection formulas for the second Painlevé transcendent. (English. Russian original) Zbl 0681.34053
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. 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 $a\sp 2=e\sp{\pi \alpha\sp 2}- 1, sign a=2(1/2-\epsilon).$ 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.
Reviewer: A.M.Šermenev

34E99Asymptotic theory of ODE
34B30Special ODE (Mathieu, Hill, Bessel, etc.)
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