Systems that can be classified as oscillatory media consist of small oscillating elements that interact with each other via some form of coupling. Interest in these systems stems in part from their ability to generate beautiful structures, including target patterns and spiral waves.  When coupling between oscillators occurs over long spatial scales, and for certain parameter values, the set of unstable waven umbers that generate these patterns is no longer constrained to a narrow band. This then leads to interesting new structures called spiral chimeras, which are solutions that look like spiral waves in the far field but have an incoherent core where the 'oscillators' are no longer in synchrony with the rest of the pattern. As a first step in understanding this phenomenon, we rigorously study the existence of 'standard' spiral waves in an oscillating chemical reaction where the source of the nonlocal coupling is due to a fast-diffusing component. Our approach is based on the method of multiple-scales and Lyapunov-Schmidt reduction, which allow us to rigorously derive a nonlocal amplitude equation for these patterns. 

Examples include oscillating chemical reactions like the Belusov-Zhabotinsky reaction, colonies of yeast cells, and under certain assumptions heart and brain tissue.