# Hasimoto variables, generalized vortex filament equations, Heisenberg models and Schrodinger maps arising from group-invariant NLS systems

## TL;DR

This paper explores the deep geometric connections among various integrable systems like NLS, vortex filament, Heisenberg models, and Schrödinger maps within Hermitian symmetric spaces, introducing a generalized Hasimoto variable.

## Contribution

It extends the geometric relationships to Hermitian symmetric spaces using a generalized Hasimoto variable derived from parallel moving frames.

## Key findings

- Established new geometric links among integrable systems in symmetric spaces.
- Applied the method to complex projective space CP^N to demonstrate the results.
- Provided a generalized framework for analyzing vortex filament and Schrödinger map equations.

## Abstract

The deep geometrical relationships holding among the NLS equation, the vortex filament equation,the Heisenberg spin model, and the Schrodinger map equation are extended to the general setting of Hermitian symmetric spaces. New results are obtained by utilizing a generalized Hasimoto variable which arises from applying the general theory of parallel moving frames. The example of complex projective space CP^N= SU(N+1)/U(N) is used to illustrate the method and results.