Integrable curve dynamics.

Abstract

The Heisenberg Model of the integrable evolution of a continuous spin chain can be used to describe an integrable dynamics of curves in R ³. The role of orthonormal frames of the curve is explored. In this framework a second Poisson structure for the Heisenberg Model is derived and the relation between the Heisenberg Model and the cubic Non-Linear Schrodinger Equation is explained. The Frenet frame of a curve is shown to be a Legendrian curve in the space of orthonormal frames with respect to a natural contact structure. As a consequence, generic singularities of the solution of the Heisenberg Model and topological invariants of the curve are computed. The family of multi-phase solutions of the Heisenberg Model and the corresponding curves are constructed with techniques of algebraic geometry. The relation with the Non-Linear Schrodinger Equation is explained also in this context. A formula for the Backlund transformation for the Heisenberg Model is derived and applied to construct orbits homoclinic to planar circles. As a result singular knots are obtained.