Rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schr\"odinger hydrodynamics
Abstract
The rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schr\"{o}dinger hydrodynamics is a very interesting problem with many challenges. To date, the full analysis of this problem remains open. In this work, the long-time asymptotics for the defocusing nonlinear Schr\"{o}dinger equation with general step-like initial data is investigated by the Whitham modulation theory and Riemann-Hilbert formulation. The Whitham modulation theory shows that there are six cases for the initial discontinuity problem according to the orders of the Riemann invariants. The leading-order terms and the corresponding error estimates for each region of the six cases are formulated by the Deift-Zhou nonlinear steepest descent method for oscillatory Riemann-Hilbert problems. It is demonstrated that the long-time asymptotic solutions match very well with the results from Whitham modulation theory and the numerical simulations.
Keywords
Cite
@article{arxiv.2305.12968,
title = {Rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schr\"odinger hydrodynamics},
author = {Deng-Shan Wang and Peng Yan},
journal= {arXiv preprint arXiv:2305.12968},
year = {2026}
}