## Information:

This material is based upon work supported by the National Science Foundation under Grant Nos. DMS-0807653 and DMS-1206131. Any opinions, findings and conclusions or recomendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF).

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## Large-Degree Asymptotics of Rational Painlevé-II Functions. Noncritical behavior.

Robert J. Buckingham and Peter D. Miller

Department of Mathematical Sciences, University of Cincinnati
Department of Mathematics, University of Michigan, Ann Arbor

### Abstract:

Rational solutions of the inhomogeneous Painlevé-II equation and of a related coupled Painlevé-II system have recently arisen in studies of fluid vortices and of the sine-Gordon equation. For the sine-Gordon application in particular it is of interest to understand the large-degree asymptotic behavior of the rational Painlevé-II functions. We explicitly compute the leading-order large-degree asymptotics of these two families of rational functions valid in the whole complex plane with the exception of a neighborhood of a certain piecewise-smooth closed curve. We obtain rigorous error bounds by using the Deift-Zhou nonlinear steepest-descent method for Riemann-Hilbert problems.

Left: $$m=2$$. Right: $$m=9$$.