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. The initial spectral data of the Zakharov Shabhat (ZS) operator, a particular linear operator having the solution to NLS as the potential, are calculated from the initial data of the NLS; they evolve in a simple way as a result of the integrability of the problem, and produce the solution through the inverse spectral transformation.
In collaboration with A. Tovbis, we have developed
a one parameter family of initial data for 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 breaking 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.
