  We study synchronization in a system of phase-only oscillators residing on
the sites of a one-dimensional periodic lattice. The oscillators interact with
a strength that decays as a power law of the separation along the lattice
length and is normalized by a size-dependent constant. The exponent $\alpha$ of
the power law is taken in the range $0 \le \alpha <1$. The oscillator frequency
distribution is symmetric about its mean (taken to be zero), and is
non-increasing on $[0,\infty)$. In the continuum limit, the local density of
oscillators evolves in time following the continuity equation that expresses
the conservation of the number of oscillators of each frequency under the
dynamics. This equation admits as a stationary solution the unsynchronized
state uniform both in phase and over the space of the lattice. We perform a
linear stability analysis of this state to show that when it is unstable,
different spatial Fourier modes of fluctuations have different stability
thresholds beyond which they grow exponentially in time with rates that depend
on the Fourier modes. However, numerical simulations show that at long times,
all the non-zero Fourier modes decay in time, while only the zero Fourier mode
(i.e., the "mean-field" mode) grows in time, thereby dominating the instability
process and driving the system to a synchronized state. Our theoretical
analysis is supported by extensive numerical simulations.
