Synchronization behavior in a ternary phase model

Localized traveling-wave solutions to a nonlinear Schrödinger 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.