Choreographies in the Discrete Nonlinear Schr\"odinger Equations

TL;DR

This paper investigates periodic solutions, including choreographies, in the discrete nonlinear Schr"odinger equation, revealing many stable orbits through numerical continuation and stability analysis.

Contribution

It introduces specialized numerical continuation methods to locate and analyze choreographic solutions in the DNLSE, including stability properties.

Findings

Existence of infinitely many choreographies in DNLSE

Many choreographies are stable according to Floquet analysis

Numerical methods effectively locate and verify periodic solutions

Abstract

We study periodic solutions of the discrete nonlinear Schr\"{o}dinger equation (DNLSE) that bifurcate from a symmetric polygonal relative equilibrium containing $n$ sites. With specialized numerical continuation techniques and a varying physically relevant parameter we can locate interesting orbits, including infinitely many choreographies. Many of the orbits that correspond to choreographies are stable, as indicated by Floquet multipliers that are extracted as part of the numerical continuation scheme, and as verified \textit{a posteriori} by simple numerical integration. We discuss the physical relevance and the implications of our results.