Traveling Speed of Clusters in the Kuramoto-Sakaguchi Model

We study a variant of the Kuramoto-Sakaguchi model in which oscillators are divided into two groups, each characterized by its coupling constant and phase lag. Specifically, we consider the case that one coupling constant is positive and the other negative, and calculate numerically the traveling speed of two clusters emerging in the system and the average separation between them, as well as the order parameters for positive and negative oscillators, as the two coupling constants, phase lags, and the fraction of positive oscillators are varied. An expression explaining the dependence of the traveling speed on these parameters is obtained and is observed to fit the numerical data well. With the help of this, we describe the conditions for the traveling state to appear in the system.