Formation of localized structures in bistable systems through nonlocal spatial coupling

The present work studies the influence of nonlocal spatial coupling on the existence of localized structures in 1-dimensional extended systems. We consider systems described by a real field with a nonlocal coupling that has a linear dependence on the field. Leveraging spatial dynamics we provide a general framework to understand the effect of the nonlocality on the shape of the fronts connecting two stable states. In particular we show how three nonlocal kernels, mod-exponential, Gaussian and Mexican-hat kernel (with attractive and repelling regions)
can induce spatial oscillations in the front tails, allowing for the creation of localized structures, emerging from pinning between two fronts.