  The nuclear rainbow observed in the elastic $\alpha$-nucleus and light
heavy-ion scattering is proven to be due to the refraction of the scattering
wave by a deep, attractive real optical potential. The nuclear rainbow pattern,
established as a broad oscillation of the Airy minima in the elastic cross
section, originates from an interference of the refracted far-side scattering
amplitudes. It is natural to expect a similar rainbow pattern also in the
inelastic scattering of a nucleus-nucleus system that exhibits a pronounced
rainbow pattern in the elastic channel. Although some feature of the nuclear
rainbow in the inelastic nucleus-nucleus scattering was observed in experiment,
the measured inelastic cross sections exhibit much weaker rainbow pattern,
where the Airy oscillation is suppressed and smeared out. To investigate this
effect, a novel method of the near-far decomposition of the inelastic
scattering amplitude is proposed to explicitly reveal the coupled partial-wave
contributions to the inelastic cross section. Using the new decomposition
method, our coupled channel analysis of the elastic and inelastic
$^{12}$C+$^{12}$C and $^{16}$O+$^{12}$C scattering at the refractive energies
shows unambiguously that the suppression of the nuclear rainbow pattern in the
inelastic scattering cross section is caused by a destructive interference of
the partial waves of different multipoles. However, the inelastic scattering
remains strongly refractive in these cases, where the far-side scattering is
dominant at medium and large angles like that observed in the elastic
scattering.
