Generalization of Hasimoto's transformation

TL;DR

This paper generalizes Hasimoto's transformation, linking vortex filament dynamics in curved spaces to the nonlinear Schrödinger equation and providing a geometric interpretation of the transformation function.

Contribution

It extends Hasimoto's transformation to curved manifolds and offers a geometric interpretation of the associated complex function.

Findings

Vortex filament dynamics in curved spaces relate to the nonlinear Schrödinger equation.

Provides a geometric interpretation of Hasimoto's function .

Generalizes the transformation to manifolds of constant curvature.

Abstract

In this paper, we generalize the famous Hasimoto's transformation by showing that the dynamics of a closed unidimensional vortex filament embedded in a three-dimensional manifold of constant curvature gives rise under Hasimoto's transformation to the non-linear Schrodinger equation. We also give a natural interpretation of the function \psi introduced by Hasimoto in terms of moving frames associated to a natural complex bundle over the filament.