Solutions to the Master Equations Governing Fractional Vortices

TL;DR

This paper establishes sharp existence and uniqueness results for solutions to the master equations governing fractional vortices, providing necessary and sufficient conditions in periodic domains and no restrictions in planar cases.

Contribution

It introduces new variational methods to determine existence and uniqueness of solutions for fractional vortex equations, with explicit bounds in periodic settings.

Findings

Necessary and sufficient conditions for existence in periodic domains.

No restrictions on vortex numbers in planar cases.

Solutions are uniquely determined by vortex locations and winding numbers.

Abstract

By means of variational methods, in this paper, we establish sharp existence results for solutions of the master equations governing `fractional multiple vortices.' In the doubly periodic situation, the conditions for existence are both necessary and sufficient and give the upper bounds for the vortex numbers in terms of the size of the periodic cell domain. In the planar situation, there is no restriction on the vortex numbers. In both situations, the solutions are uniquely determined by the prescribed locations and the local winding numbers of the vortices.