  Recent atom interferometry (AI) experiments involving Bose--Einstein
condensates (BECs) have been conducted under extreme conditions of volume and
interrogation time. Numerical solution of the standard mean-field theory
applied to these experiments presents a nearly intractable challenge. We
present an approximate variational model that provides rapid approximate
solutions of the rotating-frame Gross--Pitaevskii equation for a power-law
potential. This model is well-suited to the design and analysis of AI
experiments involving BECs that are split and later recombined to form an
interference pattern. We derive the equations of motion of the variational
parameters for this model and illustrate how the model can be applied to the
sequence of steps in a recent AI experiment where BECs were used to implement a
dual-Sagnac atom interferometer rotation sensor. We use this model to
investigate the impact of finite-size and interaction effects on the
single-Sagnac-interferometer phase shift.
