Coherent structures in nonlocal systems -- functional analytic tools

Abstract

We develop tools for the analysis of fronts, pulses, and wave trains in spatially extended systems with nonlocal coupling. We first determine Fredholm properties of linear operators, thereby identifying pointwise invertibility of the principal part together with invertibility at spatial infinity as necessary and sufficient conditions. We then build on the Fredholm theory to construct center manifolds for nonlocal spatial dynamics under optimal regularity assumptions, with reduced vector fields and phase space identified a posteriori through the shift on bounded solutions. As an application, we establish uniqueness of small periodic wave trains in a Lyapunov center theorem using only $C^1$-regularity of the nonlinearity.