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SIGMA 4 (2008), 044, 10 pages arXiv:0712.3105
https://doi.org/10.3842/SIGMA.2008.044
Generalized Hasimoto Transform of One-Dimensional Dispersive Flows into Compact Riemann Surfaces
Eiji Onodera
Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
Received December 18, 2007, in final form May 14, 2008; Published online May 20, 2008
Abstract
We study the structure of differential equations of
one-dimensional dispersive flows into compact Riemann surfaces.
These equations geometrically generalize two-sphere valued systems
modeling the motion of vortex filament. We define a
generalized Hasimoto transform by constructing a good moving
frame, and reduce the equation with values in the induced bundle
to a complex valued equation which is easy to handle. We also
discuss the relationship between our reduction and the theory of
linear dispersive partial differential equations.
Key words:
dispersive flow; Schrödinger map; geometric analysis; moving frame; Hasimoto transform; vortex filament.
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