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

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

1999-02-05

arxiv.org/abs/solv-int/9902007v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Latex, 18 pages

Scientific paper

The asymptotic solution for the Painleve-2 equation with small parameter is considered. The solution has algebraic behavior before point $t_*$ and fast oscillating behavior after the point $t_*$. 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 $t_*$.

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