  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.
