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

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.