Large Fronts in Nonlocally Coupled Systems Using Conley–Floer Homology

• Published in: Annales Henri Poincaré Vol. 24; no. 2; pp. 605 - 696
• Main Authors: Bakker, Bente Hilde, van den Berg, Jan Bouwe
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.02.2023; Springer Nature B.V
• Subjects: Classical and Quantum Gravitation; Continuity (mathematics); Dynamical systems; Dynamical Systems and Ergodic Theory; Elementary Particles; Fields (mathematics); Homology; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Original Paper; Physics; Physics and Astronomy; Quantum Field Theory; Quantum Physics; Relativity Theory; Theoretical; Secondary 53D40; Primary 37L60; 45J05
• ISSN: 1424-0637; 1424-0661
• DOI: 10.1007/s00023-022-01219-4

In this paper, we study travelling front solutions for nonlocal equations of the type ∂ t u = N ∗ S ( u ) + ∇ F ( u ) , u ( t , x ) ∈ R d . Here, N ∗ denotes a convolution-type operator in the spatial variable 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.