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 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 \(\alpha=\tfrac{1}{2}+\mathrm{i}p\), \(p\in\mathbb{R}\setminus\{0\}\), one of the increasing tritronquée solutions has no poles or zeros whatsoever along the bisecting axis.