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This paper has been submitted for publication in SIGMA. To download a preprint of this paper, just click here.

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1513054. 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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On the Increasing Tritronquée Solutions of the Painlevé-II Equation

Peter D. Miller

Department of Mathematics, University of Michigan

Abstract:

The increasing tritronquée solutions of the Painlevé-II equation with parameter \(\alpha\) exhibit square-root asymptotics in sectors of opening angle \(4\pi/3\) and have recently appeared in applications where it is necessary to understand the behavior of these solutions for complex values of \(\alpha\). Here these solutions are investigated from the point of view of a Riemann-Hilbert representation related to the Lax pair of Jimbo and Miwa, which naturally arises in the analysis of rogue waves of infinite order. We show that for generic complex \(\alpha\), all such solutions are asymptotically pole-free along the bisecting ray of the sector of opening angle \(2\pi/3\) that contains the poles far from the origin. This allows the definition of a total integral of the solution along the axis containing the bisecting ray, in which certain algebraic terms are subtracted at infinity and the poles are dealt with in the principal-value sense. We compute the value of this integral for all such solutions. We also prove that if the Painlevé-II parameter \(\alpha\) is of the form \(\alpha=\tfrac{1}{2}+\mathrm{i}p\), \(p\in\mathbb{R}\setminus\{0\}\), one of the increasing tritronquée solutions has no poles or zeros whatsoever along the bisecting axis.