  We consider the 3D cubic nonlinear Schr\"odinger equation (NLS) with a strong
toroidal trap. In the first part, we show that as the confinement is
strengthened, a large class of global solutions to the time-dependent model can
be described by 1D flows solving the 1D periodic NLS (Theorem 1.4). In the
second part, we construct a steady state as a constrained energy minimizer, and
prove its dimension reduction to the well-known 1D periodic ground state
(Theorem 1.6 and 1.7). Then, employing the dimension reduction limit, we
establish the local uniqueness and the orbital stability of the 3D ring soliton
(Theorem 1.8). These results justify the emergence of stable quasi-1D periodic
dynamics for Bose-Einstein condensates on a ring in physics experiments.

Analysis of PDEs