Vortex Dynamics in Any Dimension

Loading...
Thumbnail Image
Date
Authors
Staley, Phillip Lee
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
We adapt the general method of moving frames and time-dependent submanifold geometry to investigate the <i>skew-mean curvature flow</i> of a 2-dimensional surface in R4; this is a natural generalization of the <i>vortex filament equation</i> (VFE) for the motion a vortex membrane in 4-dimensional space. (Haller and Vizman, Khesin, Shashinkanth.) We propose a geometric framework (the <i>Hasimoto bundle</i>) in which the connection between the VFE and the nonlinear Schr&ouml;dinger equation (through the well-known <i>Hasimoto transformation</i>) becomes transparent, and that easily generalizes to the general codimension-2 case. Such framework reveals that the main obstruction to the existence of a parallel frame and to defining a natural Hasimoto transformation is the non-zero curvature of the normal bundle.
Description
Keywords
Citation