Coupled Lugiato-Lefever equation analysis of bright soliton microcombs in photonic-crystal ring resonators.

We present a theoretical framework for bright soliton microcombs in photonic-crystal ring resonators based on coupled Lugiato-Lefever equations that resolve forward- and backward-propagating fields. By analyzing continuous-wave steady states, we show that mode splitting reshapes the resonance structure and fundamentally alters the origin of modulation instability (MI), allowing it to be dominated by the pumped mode rather than finite-frequency sidebands. This mechanism explains direct comb initiation and spontaneous formation of blue-detuned solitons. Using effective parameters linked to the underlying field coupling, we further estimate the soliton existence range and show that soliton states can persist even when they are not dynamically accessible through conventional pump scanning.