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

TL;DR

This paper analyzes traveling wave solutions in the McKean-Vlasov equation with Sakaguchi-Kuramoto interaction, revealing a phase transition from incoherence to coherence characterized by a propagating von Mises distribution.

Contribution

It establishes the existence of traveling wave solutions and a phase transition in the Sakaguchi-Kuramoto model with phase frustration, using an asymmetrical Bessel function extension.

Findings

Existence of a continuous phase transition from incoherence to coherence.

Traveling wave solutions characterized by an asymmetrically extended von Mises distribution.

Reduction of the problem to a system of two equations involving the order parameter and wave speed.

Abstract

We study the McKean-Vlasov equation for weakly coupled oscillators subject to the Sakaguchi-Kuramoto interaction. While the Kuramoto interaction provides a good approximation for small, densely connected networks, time delays in larger networks lead to symmetry-breaking phase offsets (frustrations). The Sakaguchi-Kuramoto interaction is the simplest such generalization, featuring a single frustration parameter. We establish the existence of a continuous global phase transition from incoherence to coherence, in the form of a propagating asymmetrically extended von Mises probability distribution function (AvMPDF). The corresponding traveling wave equation reduces to a system of two equations in two unknowns: the order parameter for the AvMPDF and the wave speed. The analysis relies on an appropriate asymmetrical extension of the modified Bessel function.