Target patterns in a 2D array of oscillators with nonlocal coupling

We analyze the effect of adding a weak, localized, inhomogeneity to a two dimensional array of oscillators with nonlocal coupling. We propose and also justify a model for the phase dynamics in this system. Our model is a generalization of a viscous eikonal equation that is known to describe the phase modulation of traveling waves in reaction-diffusion systems. We show the existence of a branch of target pattern solutions that bifurcates from the spatially homogeneous state when epsilon, the strength of the inhomogeneity, is nonzero and we also show that these target patterns have an asymptotic wavenumber that is small beyond all orders in epsilon. The strategy of our proof is to pose a good ansatz for an approximate form of the solution and use the implicit function theorem to prove the existence of a solution in its vicinity. The analysis presents two challenges. First, the linearization about the homogeneous state is a convolution operator of diffusive type and hence not invertible on the usual Sobolev spaces. Second, a regular perturbation expansion in epsilon does not provide a good ansatz for applying the implicit function theorem since the nonlinearities play a major role in determining the relevant approximation, which also needs to be 'correct' to all orders in epsilon. We overcome these two points by proving Fredholm properties for the linearization in appropriate Kondratiev spaces and using a refined ansatz for the approximate solution which was obtained using matched asymptotics.