On the Schr\"{o}dinger map for regular helical polygons in the hyperbolic space

TL;DR

This paper investigates the evolution of regular polygonal curves with nonzero torsion in hyperbolic space under the Schrödinger map, revealing new dynamic behaviors and instabilities similar to Euclidean cases through theoretical and numerical analysis.

Contribution

It extends the understanding of Schrödinger map dynamics for polygonal curves in hyperbolic space, highlighting differences from Euclidean space and providing new numerical insights.

Findings

Trajectory of points exhibits variants of Riemann's non-differentiable function.

Smooth solutions show instability similar to Euclidean counterparts.

Numerical results align with recent theoretical findings by Banica and Vega.

Abstract

The main purpose is to describe the evolution of $\Xt = \Xs \wedge_- \Xss,$ with $\X(s,0)$ a regular polygonal curve with a nonzero torsion in the 3-dimensional hyperbolic space. Unlike in the Euclidean space, a nonzero torsion implies two different helical curves. However, recent techniques developed by de la Hoz, Kumar, and Vega help us in describing the evolution at rational times both theoretically and numerically, and thus, the similarities and differences. Numerical experiments show that the trajectory of the point $\X(0,t)$ exhibits new variants of Riemann's non-differentiable function whose structure depends on the initial torsion in the problem. As a result, with these new solutions, it is shown that the smooth solutions (helices, straight line) in the hyperbolic space show the same instability as displayed by their Euclidean counterparts and curves with zero-torsion. These numerical observations are in agreement with some recent theoretical results obtained by Banica and Vega.