  The equations of in-medium gluodynamics are proposed. Their classical lowest
order solution is explicitly shown for a color charge moving with constant
speed. For nuclear permittivity larger than 1 it describes emission of
Cherenkov gluons resembling results of classical electrodynamics. The values of
the real and imaginary parts of the nuclear permittivity are obtained from the
fits to experimental data on the double-humped structure around the away-side
jet obtained at RHIC. The dispersion of the nuclear permittivity is predicted
by comparing the RHIC, SPS and cosmic ray data. This is important for LHC
experiments. Cherenkov gluons may be responsible for the asymmetry of dilepton
mass spectra near rho-meson, observed in the SPS experiment with excess in the
low-mass wing of the resonance. This feature is predicted to be common for all
resonances. The "color rainbow" quantum effect might appear according to higher
order terms of in-medium QCD if the nuclear permittivity depends on color.
