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

1999/02/05 by O. M. Kiselëv, Kiselev, O. M.
Mathematics · Physics and Astronomy · #Differential Equations and Numerical Methods #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Nonlinear Photonic Systems #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.solv-int/9902007

openalex publication_date 1999/02/05 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

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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