The Whitham Equations for Optical Communications: Mathematical Theory of NRZ

We present a model of optical communication system for high-bit-rate data transmission in the nonreturn-to-zero (NRZ) format over transoceanic distance. The system operates in a small group velocity dispersion regime, and the model equation is given by the Whitham equations describing the slow modulation of multi-phase wavetrains of the (defocusing) nonlinear Schr\"odinger (NLS) equation. The model equation is of hyperbolic type, and certain initial NRZ pulse with phase modulation develops a shock. We then show how one can obtain a global solution by choosing an appropriate Riemann surface on which the Whitham equation is defined. The present analysis may be interpreted as an alternative to the method of inverse scattering transformation for the NLS solitons. We also discuss wavelength-division-multiplexing (WDM) in the NRZ format by using the Whitham equation for a coupled NLS equation, and show that there exists a hydro-dynamic-type instability between channels.