  A two-time scale asymptotic method has been introduced to analyze the
multimodal mean-field Kuramoto-Sakaguchi model of oscillator synchronization in
the high-frequency limit. The method allows to uncouple the probability density
in different components corresponding to the different peaks of the oscillator
frequency distribution. Each component evolves toward a stationary state in a
comoving frame and the overall order parameter can be reconstructed by
combining them. Synchronized phases are a combination of traveling waves and
incoherent solutions depending on parameter values. Our results agree very well
with direct numerical simulations of the nonlinear Fokker-Planck equation for
the probability density. Numerical results have been obtained by finite
differences and a spectral method in the particular case of bimodal (symmetric
and asymmetric) frequency distribution with or without external field. We also
recover in a very easy and intuitive way the only other known analytical
results: those corresponding to reflection-symmetric bimodal frequency
distributions near bifurcation points.
