Interchanging Space and Time in Nonlinear Optics Modeling and Dispersion Management Models

Interchanging the role of space and time is widely used in nonlinear optics for modeling the evolution of light pulses in glass fibers. A phenomenological model for the mathematical description of light pulses in glass fibers with a periodic structure in this set-up is the so-called dispersion management equation. It is the purpose of this paper to answer the question whether the dispersion management equation or other modulation equations are more than phenomenological models in this situation. Using Floquet theory we prove that in case of comparable wave lengths of the light and of the fiber periodicity the NLS equation and NLS like modulation equations with constant coefficients can be derived and justified through error estimates under the assumption that rather strong stability and non-resonance conditions hold. This is the first NLS approximation result documented for time-periodic dispersive systems. We explain that the failure of these conditions allows us to prove that these modulation equations in general make wrong predictions. The reasons for this failure and the behavior of the system for a fiber periodicity much larger than the wave length of light shows that interchanging the role of space and time for glass fibers with a periodic structure leads to unwanted phenomena.