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arXiv · 1908.05950

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

Weiying Hu
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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.

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Weiying Hu. 2019-08-16. Singular asymptotics for solutions of the inhomogeneous Painlev\'e II equation. https://doi.org/10.1088/1361-6544/ab142a

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