# Space Curves from Nonlinear Schr\"odinger Solutions: A Direct Approach

## TL;DR

This paper introduces a direct method to construct space curves from any nonlinear Schrödinger (NLS) solution, enabling new ways to analyze vortex filament evolution with potential applications in physical systems.

## Contribution

It presents a novel direct construction approach linking NLS solutions to space curves using ordered integrals and Magnus expansion, simplifying the mapping process.

## Key findings

- Provides a new explicit mapping from NLS solutions to space curves.
- Utilizes ordered integrals and Magnus expansion for the construction.
- Highlights potential applications in physical systems analysis.

## Abstract

The connection between vortex filament evolution in the local induction approximation and non-linear Schr\"odinger (NLS) equation by Hasimoto [H. Hasimoto, J. Fluid Mechanics 51, (1972) 477] has led to space curves corresponding to NLS solitons in the past. Utilizing this map, we propose a direct construction of parametric curve evolution from any NLS solution. It includes ordered (or nested) integrals of products of local matrices akin to the causal evolution of quantum theory, necessitating the implementation of the Magnus expansion. Such a straightforward mapping may be a simple tool to study the evolution of various systems of physical concern, although the actual computation can be a challenge for most NLS solutions.