  We study the mean-field dynamics and the reduced-dimension character of
two-mode Bose-Einstein condensates (BECs) in highly anisotropic traps. By means
of perturbative techniques, we show that the tightly confined (transverse)
degrees of freedom can be decoupled from the dynamical equations at the expense
of introducing additional effective three-body, attractive, intra- and
inter-mode interactions into the dynamics of the loosely confined
(longitudinal) degrees of freedom. These effective interactions are mediated by
changes in the transverse wave function. The perturbation theory is valid as
long as the nonlinear scattering energy is small compared to the transverse
energy scales. This approach leads to reduced-dimension mean-field equations
that optimally describe the evolution of a two-mode condensate in general
quasi-1D and quasi-2D geometries. We use this model to investigate the relative
phase and density dynamics of a two-mode, cigar-shaped $^{87}$Rb BEC. We study
the relative-phase dynamics in the context of a nonlinear Ramsey interferometry
scheme, which has recently been proposed as a novel platform for high-precision
interferometry. Numerical integration of the coupled, time-dependent,
three-dimensional, two-mode Gross-Pitaevskii equations for various atom numbers
shows that this model gives a considerably more refined analytical account of
the mean-field evolution than an idealized quasi-1D description.
