Abstract

Authors:  J. D’Ambroise, R. Carretero-Gonzalez, P. Schmelcher, P.G. Kevrekidis The phenomena of Bose-Einstein condensation (BECs) has been observed on various underlying geometries such as optical lattices with varying shapes and symmetries, rings, cylinders, cones and of particular interest here, as confined to the surface of a torus. From a mathematical perspective it is particularly interesting to explore how the various dark and bright nonlinear waves interact with the underlying geometric properties of the various background spaces in which they lie.  In this project we study the existence, stability, and dynamics of vortex structures for the nonlinear Schr\”{o}dinger (NLS) equation on the surface of a torus.   One can also derive the effective interaction of the vortices, viewed here as point particles, and the reduced  particle model is shown to be in excellent agreement with the full NLS evolution.  A few varieties of stationary  vortex dipoles and quadrapoles are identified and continued along the torus aspect ratio and along the frequency parameter of the solution.  In this work the various localization and stability properties of such solutions are detailed, and the windows of stability (and instability) for these solutions are identified.