  We generalize the Kuramoto model for the synchronization transition of
globally coupled phase oscillators to populations by incorporating an
additional heterogeneity with the coupling strength, where each oscillator pair
interacts with different coupling strength weighted by a genera; function of
their natural frequency. The expression for the critical coupling can be
straightforwardly extended to a generalized explicit formula analytically, and
s self-consistency approach is developed to predict the stationary states in
the thermodynamic limit. The landau damping effect is further revealed by means
of the linear stability analysis and resonance poles theory above the critical
threshold which turns to be far more generic. Furthermore, the dimensionality
reduction technique of the Ott-Antonsen is implemented to capture the
analytical description of relaxation dynamics of the steady states valid on a
globally attracting manifold. 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.
