Dynamics of Vortices in Bose Fluids and Classical Continuous Spin Models

Abstract

Dynamics of two-dimensional (2d) vortices in Bose fluids governed by the Gross-Pitaevskii equation or the nonlinear Schrödinger equation and planar Heisenberg ferromagnets in an external field is studied. This is done by introducing a velocity potential corresponding to 2d point vortices and by assuming the time-dependence of field variables to occur only through the position of vortex cores. Equations of motion for these model systems are shown to be reducible to canonical equations of motion for 2d vortices which are identical in form to those for 2d point vortices in incompressible and inviscid fluids due to singularity natures of vortex cores. From the result for the integrability or the non-integrability of the motion of Euler vortices in 2d fluid dynamics, the non-integrability of the equations of motion for the systems are deduced. Discussions are also given on the dynamics of vortices in the 2d O(3)-nonlinear σ model and phonons in spin fluids associated with the classical XY model in an external field.