  We study the asymptotic phase concentration phenomena for the
Kuramoto-Sakaguchi(K-S) equation in a large coupling strength regime. For this,
we analyze the detailed dynamics of the order parameters such as the amplitude
and the average phase. For the infinite ensemble of oscillators with the
identical natural frequency, we show that the total mass distribution
concentrates on the average phase asymptotically, whereas the mass around the
antipodal point of the average phase decays to zero exponentially fast in any
positive coupling strength regime. Thus, generic initial kinetic densities
evolve toward the Dirac measure concentrated on the average phase. In contrast,
for the infinite ensemble with distributed natural frequencies, we find a
certain time-dependent interval whose length can be explicitly quantified in
terms of the coupling strength. Provided that the coupling strength is
sufficiently large, the mass on such an interval is eventually non-decreasing
over the time. We also show that the amplitude order parameter has a positive
lower bound that depends on the size of support of the distribution function
for the natural frequencies and the coupling strength. The proposed asymptotic
lower bound on the order parameter tends to unity, as the coupling strength
increases to infinity. This is reminiscent of practical synchronization for the
Kuramoto model, in which the diameter for the phase configuration is inversely
proportional to the coupling strength. Our results for the K-S equation
generalize the results in [19] on the emergence of phase-locked states for the
Kuramoto model in a large coupling strength regime.
