Multiple soliton regimes enabled by parametric down-conversion in $\chi^{(2)}+\chi^{(3)}$ microresonators
Half-harmonic generation in microresonators with mixed $\chi^{(2)}$ and $\chi^{(3)}$ nonlinearities provides access to a rich landscape of two-colour soliton states and frequency combs. We numerically demonstrate the formation of both bright and topological two-colour solitons at relatively modest pump powers and phase-mismatch parameters. At higher powers and larger phase mismatches, we identify a regime where parametric down-conversion assists the formation of dark solitons in the normally dispersive pump field. While the dark-soliton core is primarily shaped by the Kerr nonlinearity, nonlinear coupling between the pump and its half-harmonic modes enables soliton excitation using a standard laser-frequency scanning technique, a capability not normally associated with dark solitons. The simulations are performed using parameters representative of lithium-niobate microresonators pumped near 775 nm and generating a half-harmonic near 1550 nm. These results reveal how the interplay of $\chi^{(2)}$ and $\chi^{(3)}$ nonlinearities can substantially expand the range and types of experimentally accessible soliton states in a single integrated frequency-comb platform.