Parametrically driven Kerr temporal soliton crystals
We theoretically investigate the dynamics of parametrically driven soliton crystals (PDSC) and their associated frequency combs in doubly resonant cavities with quadratic and cubic nonlinearities. We show that, in a regime with strong pump--signal walk-off where the homogeneous state is unstable, noise-seeded dynamics relaxes toward stable, equally spaced multi-soliton states. At fixed detuning, the driving strength acts as the primary control parameter for the soliton number. Pump signal walkoff extends pump depletion from a local perturbation to a global constraint, enabling long-range soliton interactions; in combination with the pump phase, this mechanism stabilizes the crystal's equal spacing and preserves comb coherence. Additionally, we identify a novel nonlinear state in parametric soliton crystals in which circulating solitons periodically alternate their intensities and group velocities, a phenomenon we term the {\it soliton-pursuing} state. Furthermore, due to the phase-selective nature of the optical parametric process, we show how different configurations of soliton phases determine the optical frequency combs, enabling odd-harmonic and subharmonic-like combs.