Long-time limit studies of an obstruction in the g-function mechanism for semiclassical focusing NLS

TL;DR

This paper studies the long-time behavior of obstructions in the Riemann-Hilbert analysis of the semiclassical focusing NLS, revealing asymptotic boundary curves and their properties supported by numerical evidence.

Contribution

It provides the first detailed analysis of the obstruction curve in the long-time limit for the focusing NLS with specific initial conditions, including asymptotic behavior and numerical validation.

Findings

Obstruction curve approaches vertical asymptotes at x=±ln 2 as t→∞

Long-time asymptotics of the obstruction boundary are derived

Numerical results support the asymptotic analysis

Abstract

We consider the long-time properties of the an obstruction in the Riemann-Hilbert approach to one dimensional focusing Nonlinear Schr\"odinger equation in the semiclassical limit for a one parameter family of initial conditions. For certain values of the parameter a large number of solitons in the system interfere with the $g$-function mechanism in the steepest descent to oscillatory Riemann-Hilbert problems. The obstruction prevents the Riemann-Hilbert analysis in a region in $(x,t)$ plane. We obtain the long time asymptotics of the boundary of the region (obstruction curve). As $t\to\infty$ the obstruction curve has a vertical asymptotes $x=\pm \ln 2$. The asymptotic analysis is supported with numerical results.