Tritronquée Painlevé-II asymptotics for the focusing nonlinear Schrödinger equation on a modulationally unstable background

We study the long-time asymptotics of the focusing nonlinear Schr\"odinger equation with nonzero boundary conditions in the transition region. Biondini and Mantzavinos showed that, away from the transition curves, the $(x,t)$-plane decomposes into two constant-amplitude plane-wave regions and a central region described by slowly modulated elliptic oscillations. However, their asymptotic formulae are not uniform near the boundaries separating these regions. The purpose of this paper is to resolve this transition problem. Using a double-scaling nonlinear steepest-descent analysis of the associated Riemann--Hilbert problem, we show that the leading term in the transition region is still a plane wave, while the first nontrivial correction is of order $t^{-1/3}$. The coefficient of this correction is expressed in terms of a distinguished tritronqu\'ee solution of an inhomogeneous Painlev\'e-II equation. This Painlev\'e-II tritronqu\'ee structure is also known to appear in the asymptotic analysis of rogue waves of infinite order.