  We establish an exact partial differential equation to model Kerr comb
generation in whispering-gallery mode resonators. This equation is a variant of
the Lugiato-Lefever equation that includes higher-order dispersion and
nonlinearity. This spatio-temporal model, whose main variable is the total
intracavity field, is significantly more suitable than the modal expansion
approach for the theoretical understanding and the numerical simulation of
wide-span combs. It allows us to explore pulse formation in which a large
number of modes interact cooperatively. This versatile approach can be
straightforwardly extended to include higher-order dispersion, as well as other
phenomena like Raman, Brillouin and Rayleigh scattering. We demonstrate for the
first time that when the dispersion is anomalous, Kerr comb generation can
arise as the spectral signature of dissipative cavity solitons, leading to
wide-span combs with low pumping.
