Formation of localized structures in bistable systems through nonlocalspatial coupling. I. General framework

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 that nonlocal terms can induce spatial oscil-
lations in the front tails, allowing for the creation of localized structures, emerging from pinning
between two fronts. In parameter space the region where fronts are oscillatory is limited by three
transitions: the modulational instability of the homogeneous state, the Belyakov-Devaney transition
in which monotonic fronts acquire spatial oscillations with infinite wavelength, and a crossover in
which monotonically decaying fronts develop oscillations with a finite wavelength. We show how
these transitions are organized by codimension 2 and 3 points and illustrate how by changing the
parameters of the nonlocal coupling it is possible to bring the system into the region where localized
structures can be formed.