  Phase-matching of the third-harmonic generation process can be used to extend
the emission of radiation from Kerr microresonators into new spectral regions
far from the pump wavelength. Here, we present a theoretical mean-field model
for optical frequency combs in a dissipative and nonlinear $\chi^{(3)}$-based
cavity system with parametric coupling between fundamental and third-harmonic
waves. We investigate temporally dispersive dual-comb generation of
phase-matched combs with broad bandwidth, and report conditions for accessing a
multistable regime that simultaneously supports two types of coupled bright
cavity solitons. These bistable cavity solitons coexist for the same pump power
and frequency detuning, while featuring dissimilar amplitudes of their
individual field components. Third-harmonic generation frequency combs can
permit telecom pump laser sources to simultaneously directly access both the
near-infrared and the visible regions, which may be advantageous for the
development of optical clocks and sensing applications.
