# Bifurcation to coherent structures in nonlocally coupled systems

## TL;DR

This paper investigates how localized spike solutions bifurcate from uniform states in nonlocally coupled systems, extending classical results to higher dimensions and using a direct asymptotic approach.

## Contribution

It introduces a novel direct method for analyzing bifurcations in nonlocal systems, avoiding center manifold reductions, and generalizes known results to higher-dimensional and symmetric settings.

## Key findings

- Bifurcation of spike solutions from constant states is established.
- The approach applies to higher dimensions and symmetric kernels.
- Leading order asymptotics and Newton corrections are derived.

## Abstract

We show bifurcation of localized spike solutions from spatially constant states in systems of nonlocally coupled equations in the whole space. The main assumptions are a generic bifurcation of saddle-node or transcritical type for spatially constant profiles, and a symmetry and second moment condition on the convolution kernel. The results extend well known results for spots, spikes, and fronts, in locally coupled systems on the real line, and for radially symmetric profiles in higher space dimensions. Rather than relying on center manifolds, we pursue a more direct approach, deriving leading order asymptotics and Newton corrections for error terms. The key ingredient is smoothness of Fourier multipliers arising from discrepancies between nonlocal operators and their local long-wavelength approximations.