On the Increasing Tritronquée Solutions of the Painlevé-II Equation

Abstract

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

References in corpus (3)

• Total integrals of global solutions to Painleve II
• Rational Solutions of the Painlevé-III Equation
• Rational Solutions of the Painlevé-III Equation: Large Parameter Asymptotics

Cited by in corpus (5)

• Extreme Superposition: Rogue Waves of Infinite Order and the Painlevé-III Hierarchy
• Far-Field Asymptotics for Multiple-Pole Solitons in the Large-Order Limit
• Open Problems for Painlevé Equations
• Trans-Series Asymptotics of Solutions to the Degenerate Painlevé III Equation: A Case Study
• On integrals of the tronquée solutions and the associated Hamiltonians for the Painlevé II equation