  Fully describing light propagation in a rotating, anisotropic medium with
thermal nonlinearity requires modeling the interplay between nonlinear
refraction, birefringence, and the nonlinear group index. Incorporating these
factors into a generalized nonlinear Schr\"odinger equation and fitting them to
recent experimental results reveals two key relationships: the photon drag
effect can have a nonlinear component that is dependent on the motion of the
medium, and the temporal dynamics of the moving birefringent nonlinear medium
create distorted figure-eight-like transverse trajectories at the output. The
beam trajectory can be accurately modelled with a full understanding of the
propagation effects. Efficiently modeling these effects and accurately
predicting the beam's output position has implications for optimizing
applications in velocimetry and beam-steering. Understanding the roles of
competitive nonlinearities gives insight into the creation or suppression of
nonlinear phenomena like self-action effects.
