Learning the Position of Image Vortices from Data

The point vortex model is an idealized model for describing the dynamics of many vortices with numerical efficiency, and has been shown to be powerful in modeling the dynamics of vortices in a superfluid. The model can be extended to describe vortices in fluids with a well defined boundary, as an image vortex can be added to the equations of motion to impose the correct velocity profile at the boundary. The mathematical formulation of the image vortex depends on the boundary in question, and is well known for a wide variety of problems, although the formulation of an image vortex in a fluid with a soft boundary remains under debate, as the boundary condition is ill-posed. Such a boundary is common-place in the dynamics of a vortex in an ultra-cold atomic Bose-Einstein condensate, for example, which is typically trapped in a harmonic potential. In order to address this problem, the Sparse Identification of Nonlinear Dynamics framework is applied to data from mean-field simulations to extract an approximate point vortex model for a vortex in a circular power law trapping potential. A formulation for the position of an image vortex in such a trap is presented, and the accuracy of this model is evaluated.