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 well-known Whitham equations describing the slow modulation of multiphase wavetrains of the (defocusing) nonlinear Schrödinger (NLS) equation. The model equation is of hyperbolic type, and NRZ pulse with certain initial 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. We also discuss the effect of third order dispersion by using an integrable hierarchy of the NLS equation, and we give a condition to avoid a shock formation.

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