The NLS equation describes solitonic transmission in fiber optic communication and is generically encountered in propagation through nonlinear media. One of its most important aspects is its modulational instability: regular wavetrains are unstable to modulation and break up to more complicated structures.

The IVP for the NLS equation is solvable by the method of inverse scattering, which utilizes the spectral information of a linear operator, the Zakharov Shabhat (ZS) operator, associated with and written in terms of the solution. The initial spectral data of the ZS operator are calculated from the solution which is known at time zero and they evolve in a simple way. In the semicalssical limit,  the calculation of the spectral data at time zero is quite challenging. Even more challenging is the process of inverse scattering, i.e. recovering the unknown solution of NLS at a nonzero time from
the spectral data at that time.

In collaboration with A. Tovbis, we have developed an example in which the derivation of the spectral data is explicit.
Then,in collaboration with A. Tovbis and X. Zhou, we have obtained the following results:

1) We prove the existence and basic properties of the first break-curve (curve in space-time above which the character of the solution changes by the emergence of a new oscillatory phase) and show that for pure radiation no further breaks occur.

2) We construct the solution beyond the first break-time. 

3) We derive a rigorous estimate of the error. 

4) We derive rigorous asymptotics for the large time behavior of the system in the pure radiation case.