**4. Remarks**

Based on the cosmological principle [5] that the universe is both homogeneous and isotropic, it is justifiable to use constant system parameters for the NLS equation, especially in dealing with solitons, where pulse energy is confined to a narrow spectral width. It is well known in physics, for example, that the dispersion coefficient varies with wavelength. But, when solitons are used in optical communication, a single group velocity dispersion coefficient is always used. It should be noted that the cosmological principle does not deny the existence of local deviations from the averages. Therefore, the propagation theory as described here will predict the averaged redshift while local conditions, such as peculiar velocity, gravity, or concentration, will produce deviations in observed data, for example, redshift. Although the deviations could be positive or negative, they may not cancel out each other due to the cosmic scale involved. Deviations from the Hubble's law, which can sometimes be quite large [6], have been observed by cosmologists in the redshiftdistance relationship.

Proponents of the Big Bang theory have considered the cosmic microwave background radiation (CMBR) as the second major piece of scientific evidence. The argument is that the short wavelengths gamma rays at the beginning of the Big Bang had been stretched, due to space expansion, to microwaves. All those CMBR have been trapped in the cosmos ever since that time. With the propagation theory, such an explanation is not needed. Those CMBR are simply electromagnetic waves from far away sources that have been broadened through the distance traveled.

Propagation theory is not intended to explain the origin of the universe. It is simply meant to illustrate that the linear relationship existed between the change of wavelength and distance traveled as found in the numerical solutions of the NLS equation. In addition, just like any other measuring instrument, calibration could be used to change the readings to a particular set of units. There is no new principle involved in this approach.

With numerical methods, the LCP method has been used because of fewer equations are solved. Any other numerical method could be used providing that the system is a fixed length dispersion managed map together with an iteration loop based on Eq. (9).

It should be noted that both the solutions for dark and bright solitons could be used to give the calibrated redshift-distance relationship and **Figure 6** is applicable to *z* values outside the plot.
