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arXiv · 1611.05285

Connection formulas for the Ablowitz-Segur solutions of the inhomogeneous Painlevé II equation

Dan Dai·Weiying Hu
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Abstract

We consider the second Painlevé equation $$ u"(x)=2u^3(x)+xu(x)-α, $$ where $α$ is a nonzero constant. Using the Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems, we rigorously prove the asymptotics as $x \to \pm \infty$ for both the real and purely imaginary Ablowitz-Segur solutions, as well as the corresponding connection formulas. We also show that the real Ablowitz-Segur solutions have no real poles when $α\in (-1/2, 1/2)$.

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Dan Dai, Weiying Hu. 2017-05-12. Connection formulas for the Ablowitz-Segur solutions of the inhomogeneous Painlevé II equation. https://doi.org/10.1088/1361-6544%2Faa72c1

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