Large Fronts in Nonlocally Coupled Systems Using Conley–Floer Homology

Abstract

AbstractIn this paper, we study travelling front solutions for nonlocal equations of the type $$\begin{aligned} \partial _t u = N * S(u) + \nabla F(u), \qquad u(t,x) \in {{\textbf {R}}}^d. \end{aligned}$$ ∂ t u = N ∗ S ( u ) + ∇ F ( u ) , u ( t , x ) ∈ R d . Here, $$N *$$ N ∗ denotes a convolution-type operator in the spatial variable $$x \in {{\textbf {R}}}$$ x ∈ 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.