Landau damping in the Kuramoto model

TL;DR

This paper analyzes the Landau damping phenomenon in the Kuramoto model, demonstrating asymptotic decay of the order parameter under certain stability conditions using PDE techniques, extending Landau damping concepts to coupled oscillators.

Contribution

It establishes a linear stability criterion and proves nonlinear Landau damping for the Kuramoto model with smooth frequency distributions, a novel application of PDE methods.

Findings

Asymptotic decay of the order parameter for small perturbations

Stability criterion aligns with standard coupling conditions

Polynomial rate of decay under the nonlinear dynamics

Abstract

We consider the Kuramoto model of globally coupled phase oscillators in its continuum limit, with individual frequencies drawn from a distribution with density of class $C^n$ ($n\geq 4$). A criterion for linear stability of the uniform stationary state is established which, for basic examples of frequency distributions, is equivalent to the standard condition on the coupling strength in the literature. We prove that, under this criterion, the Kuramoto order parameter, when evolved under the full nonlinear dynamics, asymptotically vanishes (with polynomial rate $n$) for every trajectory issued from sufficiently small $C^n$ perturbation. The proof uses techniques from the Analysis of PDEs and closely follows recent proofs of the nonlinear Landau damping in the Vlasov equation and Vlasov-HMF model.