  Recently, the first-order synchronization transition has been studied in
systems of coupled phase oscillators. In this paper, we propose a framework to
investigate the synchronization in the frequency-weighted Kuramoto model with
all-to-all couplings. A rigorous mean-field analysis is implemented to predict
the possible steady states. Furthermore, a detailed linear stability analysis
proves that the incoherent state is only neutrally stable below the
synchronization threshold. Nevertheless, interestingly, the amplitude of the
order parameter decays exponentially (at least for short time) in this regime,
resembling the Landau damping effect in plasma physics. Moreover, the explicit
expression for the critical coupling strength is determined by both the
mean-field method and linear operator theory. The mechanism of bifurcation for
the incoherent state near the critical point is further revealed by the
amplitude expansion theory, which shows that the oscillating standing wave
state could also occur in this model for certain frequency distributions. Our
theoretical analysis and numerical results are consistent with each other,
which can help us understand the synchronization transition in general networks
with heterogenous couplings.
