Conformal maps and superfluid vortex dynamics on curved and bounded surfaces: The case of an elliptical boundary

• Published in: SciPost physics Vol. 17; no. 2; p. 039
• Main Authors: Caldara, Matteo, Richaud, Andrea, Massignan, Pietro, Fetter, Alexander L.
• Format: Journal Article
• Language: English
• Published: SciPost 01.08.2024
• ISSN: 2542-4653; 2542-4653
• DOI: 10.21468/SciPostPhys.17.2.039

Recent advances in cold-atom platforms have made real-time dynamics accessible, renewing interest in the motion of superfluid vortices in two-dimensional domains. Here we show that the energy and the trajectories of arbitrary vortex configurations may be computed on a complicated (curved or bounded) surface, provided that one knows a conformal map that links the latter to a simpler domain (like the full plane, or a circular boundary). We also prove that Hamilton’s equations based on the vortex energy agree with the complex dynamical equations for the vortex dynamics, demonstrating that the vortex trajectories are constant-energy curves. We use these ideas to study the dynamics of vortices in a two-dimensional incompressible superfluid with an elliptical boundary, and we derive an analytical expression for the complex potential describing the hydrodynamic flow throughout the fluid. For a vortex inside an elliptical boundary, the orbits are nearly self-similar ellipses.