A special rainbow function

In order to characterize the common feature of the general Lorentz violation models that the local speed of light is variable at ultrahigh energy scale, we introduced a parameter n to characterize the variation of the speed of light between inertial systems. And in order not to violate some fundamental principles and experiments’ results, some constraints were imposed on n. Then a coordinate transformation with the parameter n, which meets the symmetry of inertial systems, was naturally obtained. But just to satisfy the symmetry of inertial systems, the expression for n cannot be determined. Inspired by the idea of DSR model, we then discussed a specific expression for n that makes the particle's energy have a limit rather than be infinite derived from the Lorentz model. We found that the modified dispersion relation corresponding to the specific expression for n is deeply associated with the general formula of dispersion relation from the DSR. The motivation of introducing such a parameter n was also discussed. Finally, we discussed the possible LIV effects from astrophysical observations in GRB events and photon annihilation events. Using the data from GRB 190114C we investigated the special dispersion relation obtained in this paper and found that it appears to fit these data better than the two models corresponding to the first or second order approximation of the general formula of dispersion relation. And similar to the other Lorentz violation models corresponding to the subluminal case, the modified dispersion relation obtained in this paper can also result in increasing the energy threshold of photon annihilation reaction, which is expected to be verified by the future energetic photons events.