Traveling Waves in the McKean-Vlasov Equation under Sakaguchi-Kuramoto Interaction with Phase Frustration

Abstract

We study the McKean-Vlasov equation for weakly coupled oscillators for the Sakaguchi-Kuramoto model. While the original Kuramoto model with purely sinusoidal coupling provides a good description for small densely connected networks, time delays in large networks generate symmetry-breaking phase offsets. Sakaguchi and Kuramoto proposed the simplest extension that captures this effect by incorporating a mean-field frustration parameter into a single-mode interaction. We establish a continuous global phase transition from incoherence to a unique non-equilibrium traveling-wave state that takes the form of a rotating exponentially modified circular normal distribution. This is a novel skew extension of the von Mises distribution family that is parametrized by location parameter, concentration parameter, and skewness parameter that arises through exponential filtering of its Fourier spectrum. The extension is natural in that it preserves the fragile Bessel moment hierarchy, which ensures that the family remains globally identifiable, a property not shared by existing skew extensions. The equation for traveling waves reduces to a mean-field self-consistency condition for the concentration parameter and the skewness parameter. The latter plays a dual role, statistically as a skewness parameter and dynamically as the effective frustration in that it is the wave speed. Existence and uniqueness are proven by showing that the normalized mean resultant map (the asymmetric deformation of the normalized Bessel ratio)is strictly monotone along the isogones, rendering it a globally invertible map between natural (bare) and mean (effective) parameters. The proof combines tools from geometric function theory and analytic combinatorics.