Polar director structure of SmAP${}_{\text{F}}$ phase of bent-core liquid crystals in thin planar cells with bias electric field

We study the polar director structure in thin planar cells filled with bent-core liquid crystals in the ferroelectric smectic-A phase (SmAP${}_{\text{F}}$). We analyze a continuum phenomenological model proposed in the physics literature and present rigorous proofs of the existence and uniqueness of the equilibrium solutions. We further investigate the qualitative properties of nontrivial solutions and examine the effects of a bias electric field, surface anchoring, and cell thickness on the polar director configuration. Our results are consistent with previous experimental and numerical simulations reported in the physics literature. In addition, our analysis reveals new parameter-dependent behaviors supported by our numerical simulations and extends results reported from previous literature.