On the Increasing Tritronqu\'ee Solutions of the Painlev\'e-II Equation

TL;DR

This paper analyzes the asymptotic behavior and pole structure of increasing tritronquée solutions of the Painlevé-II equation using Riemann-Hilbert methods, revealing pole-free regions and computing associated integrals.

Contribution

It introduces a Riemann-Hilbert approach to study these solutions, establishing pole-free regions and calculating integrals for complex parameters, advancing understanding of their complex asymptotics.

Findings

Solutions are asymptotically pole-free along certain rays.

The total integral of solutions is explicitly computed.

For specific parameters, solutions have no poles or zeros along the bisecting axis.

Abstract

The increasing tritronqu\'ee solutions of the Painlev\'e-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 Painlev\'e-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 tritronqu\'ee solutions has no poles or zeros whatsoever along the bisecting axis.