Hard loss of stability in Painlev\'e-2 equation

TL;DR

This paper analyzes a special asymptotic solution of the Painlevé-2 equation near a bifurcation point, revealing a transition from slow variation to rapid oscillations and providing a detailed asymptotic description.

Contribution

It introduces a detailed asymptotic analysis of the Painlevé-2 equation's solution near a bifurcation point, including the transitional layer and smooth asymptotic approximation.

Findings

Identification of the bifurcation point $t_*$ and solution behavior change

Development of a smooth asymptotic solution across the transition

Detailed description of the transitional layer and oscillatory regime

Abstract

A special asymptotic solution of the Painlev\'e-2 equation with small parameter is studied. This solution has a critical point $t_*$ corresponding to a bifurcation phenomenon. When $t<t_*$ the constructed solution varies slowly and when $t>t_*$ the solution oscillates very fast. We investigate the transitional layer in detail and obtain a smooth asymptotic solution, using a sequence of scaling and matching procedures.