  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.
