Geometric Quantum Hydrodynamics and the Evolution of Vortex Lines
he simplest manifestation of a circulatory field occurs when a vortex line breaks the simple connectivity of an otherwise irrotational fluid. Arnold's program tells us that the evolution of such a vortex is Hamiltonian with respect to the arclength. The simplest models of the geometric dynamics prescribed to the vortex line by the induced ambient flow are provided by a cubic focusing nonlinear Schrodinger evolution of the line's curvature and torsion. Past the first approximation, the geometry is evolved by a nonlinear integro-differential Schrodinger equation. While the line-length Hamiltonian is still preserved under this evolution of curvature and torsion, the rich structure provided by the integrable cubic Schrodinger equation is not. However, perturbing off of the lowest-order approximation one finds that the vortex medium supports L2 gain/loss mechanisms, with enhanced dispersion, allowing for curvature abnormalities along the vortex to decompose into helical Kelvin modes. In the field of ultracold/ultraquantum turbulence, these helical excitations can allow for a tangled web of vortex lines to relax by insonification of the ultracold atomic gas. In this talk, we outline the mathematical structures needed to generalize first-order approximations of vortex line motion and show how these generalizations allow for dynamics consistent with the qualities of quantum hydrodynamics. Specifically, we use the Biot-Savart integral to arrive at an exact representation of the binormal component of the velocity field induced by a vortex line which is then mapped to a Schrodinger type evolution capable of supporting the generation of helical waves.