Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 125, 38 pages      arXiv:1804.03173      https://doi.org/10.3842/SIGMA.2018.125
Contribution to the Special Issue on Painlevé Equations and Applications in Memory of Andrei Kapaev

On the Increasing Tritronquée Solutions of the Painlevé-II Equation

Peter D. Miller
Department of Mathematics, University of Michigan, East Hall, 530 Church St., Ann Arbor, MI 48109, USA

Received April 11, 2018, in final form November 12, 2018; Published online November 15, 2018

Abstract
The increasing tritronquée solutions of the Painlevé-II equation with parameter $\alpha$ exhibit square-root asymptotics in the maximally-large sector $|\arg(x)|$<$\tfrac{2}{3}\pi$ 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 complementary sector $|\arg(-x)|$<$\tfrac{1}{3}\pi$ 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 $\alpha=\pm\tfrac{1}{2}+{\rm i} p$, $p\in\mathbb{R}\setminus\{0\}$, one of the increasing tritronquée solutions has no poles or zeros whatsoever along the bisecting axis.

Key words: Painlevé-II equation; tronquée solutions.

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