2026/06/28 by Haibing Zhang, Xianguo Geng, Kedong Wang · 1 voice
Physics and Astronomy · #nlin.SI
arxiv published 2026/06/28 · arxiv updated 2026/07/05
We study the long-time asymptotics of the focusing nonlinear Schrödinger equation with nonzero boundary conditions in the transition regions between the plane-wave and modulated elliptic-wave regimes. Biondini and Mantzavinos showed that, away from the transition curves \(x=± 4√(2) qo t\), the \((x,t)\)-half-plane decomposes, to leading order, into two 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 missing boundary layer. Using a double-scaling nonlinear steepest descent analysis of the associated Riemann--Hilbert problem, we show that the leading term in each 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 tritronquée solution of an inhomogeneous Painlevé-II equation. This Painlevé-II tritronquée structure is also known to appear in the asymptotic analysis of rogue waves of infinite order.