Rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schrödinger hydrodynamics

Abstract The rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schrödinger (NLS) hydrodynamics is a very interesting problem with many challenges. To date, the full analysis of this problem remains lacking. In this work, the long-time asymptotics for the defocusing NLS 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 estimations 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.