Hasimoto Transformation of General Flows Expressed in the Frenet Frame

TL;DR

This paper generalizes the Hasimoto transformation to arbitrary flows on space curves, linking geometric evolution to nonlinear equations, and analyzes energy conservation properties of binormal flows.

Contribution

It extends the Hasimoto transformation to more general vector fields in the Frenet frame, providing a broader framework for analyzing curve dynamics.

Findings

Binormal flows generally preserve length.

Bending energy is not conserved in general flows.

The generalized transformation maps complex flows to nonlinear equations.

Abstract

A one-dimensional space curve in $\mathbb{R}^{3}$ is a useful nonlinear medium for modeling vortex filaments and biological soft-matter capable of supporting a variety of wave motions. The Hasimoto transformation defines a mapping between the kinematic evolution of a space curve and nonlinear scalar equations evolving its intrinsic curve geometry. This mapping is quite robust and able to transform general vector fields expressed in the Frenet frame, resulting in a fully nonlinear integro-differential evolution equation, whose coefficient structure is defined by the coordinates of the flow in the Frenet frame. In this paper, we generalize the Hasimoto map to arbitrary flows defined on space curves, which we test against several existing kinematic flows. After this, we consider the time dynamics of length and bending energy to see that binormal flows are generally length preserving, and bending energy is fragile and unlikely to be conserved in the general case.