Soliton-pursuing dynamics and pump-depletion locking in parametrically driven soliton crystals

We study parametrically driven soliton crystals in doubly resonant χ(2)-χ(3) cavities. Using numerical continuation, we map multi-soliton solution families versus driving strength and identify stability changes mediated by fold and Hopf bifurcations. Direct time-domain simulations confirm robust phase-locked soliton crystals at higher drive and reveal a Hopf-induced oscillatory regime that we term the soliton-pursuing state, where circulating solitons periodically exchange power and group velocity, producing slow oscillations of their separations. Increasing the drive shortens the pursuing period and quenches the oscillations, recovering stationary soliton crystals.