Closed curves in R^3: a characterization in terms of curvature and torsion, the Hasimoto map and periodic solutions of the Filament Equation
Abstract
If a curve in R^3 is closed, then the curvature and the torsion are periodic functions satisfying some additional constraints. We show that these constraints can be naturally formulated in terms of the spectral problem for a 2x2 matrix differential operator. This operator arose in the theory of the selffocusing Nonlinear Schrodinger Equation. A simple spectral characterization of Bloch varieties generating periodic solutions of the Filament Equation is obtained. We show that the method of isoperiodic deformations suggested earlier by the authors for constructing periodic solutions of soliton equations can be naturally applied to the Filament Equation.
 Publication:

eprint arXiv:dgga/9703020
 Pub Date:
 March 1997
 DOI:
 10.48550/arXiv.dgga/9703020
 arXiv:
 arXiv:dgga/9703020
 Bibcode:
 1997dg.ga.....3020G
 Keywords:

 Mathematics  Differential Geometry;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 LaTeX, 27 pages, macros "amssym.def" used