On the nonlinear Schr\"odinger equation with a toroidal trap in the strong confinement regime

TL;DR

This paper demonstrates that in the strong confinement regime, solutions to the 3D nonlinear Schrödinger equation with a toroidal trap can be effectively described by 1D models, and it establishes the stability of ring solitons relevant to Bose-Einstein condensates.

Contribution

It provides a rigorous dimension reduction from 3D to 1D for the nonlinear Schrödinger equation with a toroidal trap and proves the stability of the resulting ring solitons.

Findings

Global solutions approximate 1D periodic NLS solutions under strong confinement.

Construction of a steady state as a constrained energy minimizer reduces to 1D ground states.

Proof of local uniqueness and orbital stability of 3D ring solitons.

Abstract

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.