# Large fronts in nonlocally coupled systems using Conley-Floer homology

## TL;DR

This paper introduces a topological Conley--Floer homology framework to analyze and establish the existence of travelling front solutions in nonlocal equations with delay terms, overcoming the lack of a natural phase space.

## Contribution

It develops a novel Morse-type Conley--Floer homology theory for nonlocal delay equations without a phase space, enabling robust analysis of travelling fronts.

## Key findings

- Established existence of travelling front solutions.
- Provided multiplicity results for solutions.
- Developed a new transversality theory for non-phase space systems.

## Abstract

In this paper we study travelling front solutions for nonlocal equations of the type \begin{equation} \partial_t u = N * S(u) + \nabla F(u), \qquad u(t,x) \in \mathbf{R}^d. \end{equation} Here $N *$ denotes a convolution-type operator in the spatial variable $x \in \mathbf{R}$, either continuous or discrete. We develop a Morse-type theory, the Conley--Floer homology, which captures travelling front solutions in a topologically robust manner, by encoding fronts in the boundary operator of a chain complex. The equations describing the travelling fronts involve both forward and backward delay terms, possibly of infinite range. Consequently, these equations lack a natural phase space, so that classic dynamical systems tools are not at our disposal. We therefore develop, from scratch, a general transversality theory, and a classification of bounded solutions, in the absence of a phase space. In various cases the resulting Conley--Floer homology can be interpreted as a homological Conley index for multivalued vector fields. Using the Conley--Floer homology we derive existence and multiplicity results on travelling front solutions.