  We solve a longstanding stability problem for the Kuramoto model of coupled
oscillators. This system has attracted mathematical attention, in part because
of its applications in fields ranging from neuroscience to condensed-matter
physics, and also because it provides a beautiful connection between nonlinear
dynamics and statistical mechanics. The model consists of a large population of
phase oscillators with all-to-all sinusoidal coupling. The oscillators'
intrinsic frequencies are randomly distributed across the population according
to a prescribed probability density, here taken to be unimodal and symmetric
about its mean. As the coupling between the oscillators is increased, the
system spontaneously synchronizes: the oscillators near the center of the
frequency distribution lock their phases together and run at the same
frequency, while those in the tails remain unlocked and drift at different
frequencies. Although this ``partially locked'' state has been observed in
simulations for decades, its stability has never been analyzed mathematically.
Part of the difficulty is in formulating a reasonable infinite-N limit of the
model. Here we describe such a continuum limit, and prove that the
corresponding partially locked state is, in fact, neutrally stable, contrary to
what one might have expected. The possible implications of this result are
discussed.
