# Synchronization Behavior in a Ternary Phase Model

## TL;DR

This paper investigates a ternary phase model with quenched disorder, revealing first-order synchronization transitions with hysteresis, contrasting with Kuramoto model behavior, supported by theoretical analysis of the infinite-oscillator limit.

## Contribution

It introduces and analyzes a novel ternary coupling phase model with quenched disorder, providing insights into its synchronization transitions and theoretical predictions in the infinite-oscillator limit.

## Key findings

- First-order transitions with hysteresis observed for Gaussian and uniform disorder.
- Theoretical predictions derived from the infinite-oscillator limit.
- Comparison shows stark differences from Kuramoto model behavior.

## Abstract

Localized traveling-wave solutions to a nonlinear Schrodinger equation were recently shown to be a consequence of Fourier mode synchronization. The reduced dynamics describing mode interaction take the form of a phase model with novel ternary coupling. We analyze this model in the presence of quenched disorder and explore transitions to partial and complete synchronization. For both Gaussian and uniform disorder, first-order transitions with hysteresis are observed. These results are compared with the phenomenology of the Kuramoto model which exhibits starkly different behavior. An infinite-oscillator limit of the model is derived and solved to provide theoretical predictions for the observed transitions. Treatment of the nonlocal ternary coupling in this limit sheds some light on the model's novel structure.