Large Fronts in Nonlocally Coupled Systems Using Conley–Floer Homology

Abstract

In this paper, we study travelling front solutions for nonlocal equations of the type ∂tu=N∗S(u)+∇F(u),u(t,x)∈Rd.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\partial _t u = N * S(u) + \ abla F(u), \\qquad u(t,x) \\in {{\ extbf {R}}}^d. \\end{aligned}$$\\end{document}Here, N * denotes a convolution-type operator in the spatial variable x in {{textbf {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.