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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.