A Bogomol'nyi-Prasad-Sommerfield bound with a first-order system in the $2D$ Gross-Pitaevskii equation

TL;DR

This paper introduces a new BPS bound for the 2D Gross-Pitaevskii equations, providing analytic solutions with fractional vorticity and linking the bound to a first-order PDE system.

Contribution

It presents the first analytic solutions with fractional vorticity in the Gross-Pitaevskii framework using a novel BPS bound and associated first-order system.

Findings

Derived a new BPS bound for 2D Gross-Pitaevskii equations.

Obtained analytic solutions with fractional vorticity in an annulus.

Connected the BPS system to hydrodynamical interpretations.

Abstract

A novel Bogomol'nyi-Prasad-Sommerfield (BPS) bound for the Gross-Pitaevskii equations in two spatial dimensions is presented. The energy can be bounded from below in terms of the combination of two boundary terms, one related to the vorticity (but ``dressed'' by the condensate profile) and the second to the ``skewness'' of the configurations. The bound is saturated by configurations that satisfy a system of two first-order partial differential equations. When such a BPS system is satisfied, the Gross-Pitaevskii equations are also satisfied. The analytic solutions of this BPS system in the present manuscript represent configurations with fractional vorticity living in an annulus. Using these techniques, we present the first analytic examples of this kind. The hydrodynamical interpretation of the BPS system is discussed, and the implications of these results are outlined.