# Vortex Pairs in the Discrete Nonlinear Schr\"odinger Equation

## TL;DR

This paper investigates how discreteness affects vortex interactions in the nonlinear Schrödinger equation, revealing that strong discreteness can stabilize vortex pairs into stationary states, with dynamics approaching continuum behavior.

## Contribution

It demonstrates that discreteness can halt vortex pairs into stable configurations and explores the transition to continuum-like behavior in vortex dynamics.

## Key findings

- Strong discreteness stabilizes vortex pairs into stationary states.
- Counterwinding vortices exhibit approaching motion beyond a critical coupling.
- Continuum limit restores rigid rotation of vortex pairs.

## Abstract

The aim of the present work is to examine the role of discreteness in the interaction of both co-winding and counter-winding vortices in the context of the nonlinear Schr{\"o}dinger equation. Contrary to the well-known rotation of same charge vortices, and translation of opposite charge vortices, we find that strong discreteness is able to halt both types of pairs into stationary, potentially stable configurations up to a critical inter-site coupling strength. Past the relevant critical point the behavior is also somewhat counter-intuitive as, for instance, counterwinding vortices start moving but also approach each other. This lateral motion becomes weaker as the continuum limit is approached and we conjecture that genuine traveling appears only at the continuum limit. Analogous features arise in the cowinding where the discrete coherent structure pair spirals outward, with rigid rotation being restored only in the continuum limit.