Singular asymptotics for solutions of the inhomogeneous Painlevé II equation

Abstract

We consider a family of solutions to the Painlevé II equation $$ u''(x)=2u^3(x)+xu(x)-α\qquad \textrm{with } \a \in \mathbb{R} \cut \{0\}, $$ 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.

34 pages, 13 figures. Accepted by Nonlinearity on 28th Mar 2019