# Singular asymptotics for solutions of the inhomogeneous Painlev\'e II equation

## TL;DR

This paper rigorously derives the singular asymptotics of a family of solutions to the Painlevé II equation with poles on the negative real axis, extending known asymptotic results to all real parameters and providing connection formulas.

## Contribution

It introduces a rigorous derivation of singular asymptotics for Painlevé II solutions with poles, extending asymptotic results to all real parameters and deriving connection formulas.

## Key findings

- Derived singular asymptotics for solutions as x → -∞
- Extended asymptotic results for x → +∞ to all real α
- Obtained connection formulas between asymptotic regimes

## Abstract

We consider a family of solutions to the Painlev\'e II equation $$ u''(x)=2u^3(x)+xu(x)-\alpha \qquad \textrm{with } \a \in \mathbb{R} \cut \{0\}, $$ which have infinitely many poles on $(-\infty, 0)$. Using Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems, we rigorously derive their singular asymptotics as $x \to -\infty$. In the meantime, we extend the existing asymptotic results when $x\to +\infty$ from $\a-\frac{1}{2} \notin \mathbb{Z}$ to any real $\a$. The connection formulas are also obtained.