Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 044, 10 pages      arXiv:0712.3105      http://dx.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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