  We theoretically study the stationary-state vortex lattice configurations of
rotating spin-orbit- and coherently-coupled spin-1 Bose-Einstein condensates
trapped in quasi-two-dimensional harmonic potentials. The combined effects of
rotation, spin-orbit and coherent couplings are analyzed systematically from
the single-particle perspective. Through the single-particle Hamiltonian, which
is exactly solvable for one-dimensional coupling, under specific coupling and
rotation strengths, we illustrate that a boson in these rotating spin-orbit-
and coherently-coupled condensates are subjected to effective toroidal,
symmetric double-well, or asymmetric double-well potentials. In the presence of
mean-field interactions, using the coupled Gross-Pitaevskii formalism at
moderate to high rotation frequencies, the analytically obtained effective
potential minima and the numerically obtained coarse-grained density maxima
position are in excellent agreement. On rapid rotation, we further find that
the spin expectation per particle of an antiferromagnetic spin-1 Bose-Einstein
condensate approaches unity indicating a similarity in the response with
ferromagnetic spin-orbit-coupled condensates.
