Synchronization behavior in a ternary phase model

Abstract

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.The damped, driven nonlinear Schrödinger equation (NLSE) with cubic nonlinearity is ubiquitous in nonlinear science. This equation models a broad range of physical systems, for instance charge-density waves, Josephson junctions, ferromagnets in microwave fields, quantum Hall ferromagnets, radio-frequency-driven plasmas, shear flows in liquid crystals, ocean and atmospheric waves, and spatial and temporal nonlinear waves in optical resonators with Kerr nonlinearity. These nonlinear wave phenomena can all exhibit sharply-peaked patterns of spatially and/or temporally localized pulses. In these systems, pulsation can be a signature of synchronization between many oscillating degrees of freedom. A previously derived model 1 explicitly relates the existence of such a pulse to the synchronization of its Fourier modes. Here, we analyze this model in the context of general coupled-oscillator systems and investigate its synchronization behavior.

Emerging material systems for integrated optical Kerr frequency combs

“…The advantage compared to mode locking through a saturable absorber is that the response time of a Kerr medium is very fast. Pulse formation based on microcombs, on the other hand, is said to occur through spontaneous selfsynchronization, i.e., mode locking here does not rely on a saturable absorber [249][250][251]. Microcombs are therefore efficient pulsed laser sources [216].…”

Algebraic theory of quantum synchronization and limit cycles under dissipation

“…It is also what many people conjure to mind when thinking about synchronization. As expected, it has been extensively studied in the past, both theoretically [8,[120][121][122][123][124], and for its applications [125][126][127][128]. Much of this progress, which has been primarily focused on controlling synchronization within chaotic systems, has built upon the seminal work of Pecora and Carroll [8] who formulated criteria based on the signs of Lyapunov exponents.…”

Algebraic Theory of Quantum Synchronization and Limit Cycles under Dissipation

“…It is also what many people conjure to mind when thinking about synchronization. As expected, it has been extensively studied in the past, both theoretically [8,[105][106][107][108][109], and for its applications [110][111][112][113]. Much of this progress, which has been primarily focused on controlling synchronization within chaotic systems, has built upon the seminal work of Pecora and Carroll [8] who formulated criterion based on the signs of Lyapunov exponents.…”