摘要： The ability to generate optical frequency combs in nonlinear resonators has attracted significant attention over the last decade. While the majority of studies have considered resonators dominated by the third-order Kerr nonlinearity, more recent investigations have shown that comb generation is also possible through second order nonlinear effects. Experimental works have in particular demonstrated that combs can arise in driven resonators via second-harmonic generation (SHG), both in the presence and absence of a phase-mismatch. Theoretical works have so far focussed almost exclusively on the case of perfect phase-matching. This is somewhat surprising, for it is well-known that phase-mismatched SHG can give rise to a strong effective Kerr nonlinearity coefficient whose sign can be controlled with the phase-mismatch. Here, we theoretically and numerically investigate the generation of Kerr-like frequency combs in resonators under conditions of phase-mismatched SHG. We show that the equations describing intracavity SHG tend towards the Lugiato-Lefever equation of Kerr nonlinear resonators when the phase-mismatch is very large. The resulting effective Kerr nonlinearity coefficient can still exceed that of typical fused silica microcavities. Importantly, the Kerr coefficient is negative if the phase-mismatch is negative, alluding to the possibility of Kerr-like combs in the regime of normal dispersion. In conclusion, we have shown that a cavity subject to phase-mismatched SHG can display pure Kerr cavity dynamics in the regime of very large phase-mismatch. Moreover, we find that localised soliton states can manifest themselves even for intermediate values of phase-mismatch.