# Inhomogeneous Heisenberg Spin Chain and Quantum Vortex Filament as Non-Holonomically Deformed NLS Systems

## TL;DR

This paper explores how inhomogeneous Heisenberg spin chains and quantum vortex filaments can be modeled as non-holonomically deformed NLS systems, revealing new integrable structures with semi-classical features.

## Contribution

It demonstrates that these physical systems are particular non-holonomic deformations of the NLS equation, with specific spectral restrictions and semi-classical signatures.

## Key findings

- Identifies non-holonomic deformations of NLS for spin chains and vortex filaments.
- Shows these deformations have specific spectral orders.
- Reveals semi-classical signatures in the generalized systems.

## Abstract

Through the Hasimoto map, various dynamical systems can be mapped to different integrodifferential generalizations of Nonlinear Schrodinger (NLS) family of equations some of which are known to be integrable. Two such continuum limits, corresponding to the inhomogeneous XXX Heisenberg spin chain [Balakrishnan, J. Phys. C 15, L1305 (1982)] and that of a thin vortex filament moving in a superfluid with drag [Shivamoggi, Eur. Phys. J. B 86, 275 (2013) 86; Van Gorder, Phys. Rev. E 91, 053201 (2015)], are shown to be particular non-holonomic deformations (NHDs) of the standard NLS system involving generalized parameterizations. Crucially, such NHDs of the NLS system are restricted to specific spectral orders that exactly complements NHDs of the original physical systems. The specific non-holonomic constraints associated with these integrodifferential generalizations additionally posses distinct semi-classical signature.