Solitons in Bose-Einstein Condensates with Attractive Self-Interaction on a M\"obius Strip

TL;DR

This paper investigates matter-wave solitons in Bose-Einstein condensates confined on a M"obius strip, revealing stable and unstable vortex solitons, their properties, and dynamics through analytical and numerical methods.

Contribution

It introduces the first analysis of BEC solitons on a M"obius strip, including vortex states, stability criteria, and soliton dynamics in this unique topology.

Findings

Stable ground-state solitons below critical norm

Vortex solitons with specific quantum numbers are stable or unstable

Localized states are unstable

Abstract

We study the matter-wave solitons in Bose-Einstein condensate (BEC) trapped on a M\"{o}bius strip (MS), based on the respective Gross-Pitaevskii (GP) equation with the mean-field theory. In the linear regime, vortex states are characterized by quantum numbers, $n$ and $m$, corresponding to the transverse and circumferential directions, with the phase structure determined by the winding number (WN) $m$. Odd and even values of $n$ must associate, respectively, with integer and half-integer values of $m$, the latter ones requiring two cycles of motion around MS for returning to the initial phase. Using variational and numerical methods, we solve the GP equation with the attractive nonlinearity, producing a family of ground-state (GS) solitons for values of the norm below the critical one, above which the collapse sets in. Vortex solitons with $n=1,m=1$ and $% n=2,m=1/2$ are obtained in a numerical form. The vortex solitons with $% n=1,m=1$ are almost uniformly distributed in the azimuthal direction, while ones with $n=2,m=1/2$ form localized states. The Vakhitov-Kolokolov criterion and linear-stability analysis for the GS soliton solutions and vortices with $n=1,m=1$ demonstrates that they are completely stable, while the localized states with $n=2,m=1/2$ are completely unstable. Finally, the motion of solitons on the MS and the collision of two solitons are discussed.