The inverse scattering approach linearizes the NLS initial value problem by associating it with a linear 2 by 2 first order ODE eigenvalue problem, the Zakharov Shabhat (ZS) equation. The scattering information of ZS constitutes the input to a Riemann-Hilbert problem (RHP) that is to nonlinear integrable systems what the Fourier integral represetation is to the solution of a linear PDE. Solving the RHP provides the solution to NLS together with the dependence of the ”eigenfunctions” of ZS in the spectral parameter for any point in space-time. The time evolution of the input is as simple as the evolution of the Fourier transform in a linear PDE. We will outline the asymptotic methods that lead to the following results:

1) Proof of existence and basic properties of the first breaking curve in space-time above which a phase transition occurs and show that for pure radiation no further breaks occur.

2) Post-break structure of the solution. 3) Rigorous error estimate. 4) Rigorous asymptotics for the large time behavior of the system in the pure radiation case.