Asymptotic approach for the rigid condition of appearance of the oscillations in the solution of the Painleve-2 equation

Abstract

The asymptotic solution for the Painleve-2 equation with small parameter is considered. The solution has algebraic behavior before point $t_{\ast }$ and fast oscillating behavior after the point $t_{\ast }$. In the transition layer the behavior of the asymptotic solution is more complicated. The leading term of the asymptotics satisfies the Painleve-1 equation and some elliptic equation with constant coefficients, where the solution of the Painleve-1 equation has poles. The uniform smooth asymptotics are constructed in the interval, containing the critical point $t_{\ast }$.

arXiv commentLatex, 18 pages

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