  About two decades ago it was discovered that systems of nonlocally coupled
oscillators can exhibit unusual symmetry-breaking patterns composed of coherent
and incoherent regions. Since then such patterns, called chimera states, have
been the subject of intensive study but mostly in the stationary case when the
coarse-grained system dynamics remains unchanged over time. Nonstationary
coherence-incoherence patterns, in particular periodically breathing chimera
states, were also reported, however not investigated systematically because of
their complexity. In this paper we suggest a semi-analytic solution to the
above problem providing a mathematical framework for the analysis of breathing
chimera states in a ring of nonlocally coupled phase oscillators. Our approach
relies on the consideration of an integro-differential equation describing the
long-term coarse-grained dynamics of the oscillator system. For this equation
we specify a class of solutions relevant to breathing chimera states. We derive
a self-consistency equation for these solutions and carry out their stability
analysis. We show that our approach correctly predicts macroscopic features of
breathing chimera states. Moreover, we point out its potential application to
other models which can be studied using the Ott-Antonsen reduction technique.
