  The dynamics of dissipative topological defects in a system of coupled phase
oscillators, arranged in one and two-dimensional arrays, is numerically
investigated using the Kuramoto model. After an initial rapid decay of the
number of topological defects, due to vortex-anti-vortex annihilation, we
identify a long-time (quasi) steady state where the number of defects is nearly
constant. We find that the number of topological defects at long times is
significantly smaller when the coupling between the oscillators is increased at
a finite rate rather than suddenly turned on. Moreover, the number of
topological defects scales with the coupling rate, analogous to the cooling
rate in KibbleZurek mechanism (KZM). Similar to the KZM, the dynamics of
topological defects is governed by two competing time scales: the dissipation
rate and the coupling rate. Reducing the number of topological defects improves
the long time coherence and order parameter of the system and enhances its
probability to reach a global minimal loss state that can be mapped to the
ground state of a classical XY spin Hamiltonian.
