Coherent structures in nonlocal systems -- functional analytic tools

TL;DR

This paper develops functional analytic tools to analyze coherent structures like fronts and pulses in nonlocal systems, establishing Fredholm properties, center manifolds, and uniqueness results for small periodic waves.

Contribution

It introduces a framework for analyzing nonlocal spatial dynamics, including Fredholm theory, center manifold construction, and a Lyapunov center theorem for wave train uniqueness.

Findings

Fredholm properties of linear operators in nonlocal systems

Construction of center manifolds under optimal regularity

Uniqueness of small periodic wave trains using $C^1$-nonlinearity

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.