Motion of Space Curves in Three-dimensional Minkowski Space $R_1^{3}$, SO(2,1) Spin Equation and Defocusing Nonlinear Schr\"odinger Equation

TL;DR

This paper explores the dynamics of space curves in Minkowski space, linking their evolution to the defocusing nonlinear Schrödinger equation through a geometric interpretation of the SO(2,1) spin chain.

Contribution

It establishes a novel connection between space curve dynamics in Minkowski space and the defocusing NLSE via the SO(2,1) spin chain, providing a geometric perspective.

Findings

Derived evolution equations for curvature and torsion in Minkowski space.

Mapped the SO(2,1) spin chain to the space curve, linking it to the defocusing NLSE.

Provided a geometric derivation of the associated linear eigenvalue problem.

Abstract

We consider the dynamics of moving curves in three-dimensional Minkowski space $R_1^{3}$ and deduce the evolution equations for the curvature and torsion of the curve. Next by mapping a continuous SO(2,1) Heisenberg spin chain on the space curve in $R_1^{3}$, we show that the defocusing nonlinear Schr\"odinger equation(NLSE) can be identified with the spin chain, thereby giving a geometrical interpretation of it. The associated linear eigenvalue problem is also obtained in a geometrical way.