Existence of solutions to the infinite dimensional Kuramoto model

Abstract

The Kuramoto model is a kinetic model of phase-coupled oscillators. This model has a long legacy, starting from Kuramoto's text, \textit{Oscillations, Waves, and Turbulence}, which provided the foundation and initial analysis of the model. Kuramoto defined convenient measures of the average phase and variance to describe the population's macroscopic state. Since Kuramoto's original publication, researchers have worked to better characterize solution dynamics. In a much-cited paper, Ott and Antonsen showed that in some cases solutions adhere to a convenient ansatz, which greatly simplifies model analysis. Here we combine the method of characteristics with an iterative technique to prove existence and uniqueness of solutions to the infinite dimensional Kuramoto model. Our result differs from the result of Ott and Antonsen in that we require the initial data is twice continuously differentiable and show existence of continuously differentiable solutions, whereas the ansatz of Ott and Antonsen requires the solution and initial data are twice continuously differentiable and belong to a special class of Fourier series. We believe the alternative approach to model analysis developed here has the potential to open doors to future results on the extensively studied Kuramoto model.