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

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.