Peristalsis by pulses of activity

Transition fronts and their universality classes

Steadily moving transition (switching) fronts, associated with local transformation, symmetry breaking, or collapse, are among the most important dynamic coherent structures. The nonlinear mechanical waves of this type play a major role in many modern applications involving the transmission of mechanical information in systems ranging from crystal lattices and metamaterials to macroscopic civil engineering structures. While many different classes of such dynamic fronts are known, the interrelation between them remains obscure. Here we consider a minimal prototypical mechanical system, the Fermi-Pasta-Ulam (FPU) chain with piecewise linear nonlinearity, and show that there are exactly three distinct classes of switching fronts, which differ fundamentally in how (and whether) they produce and transport oscillations. The fact that all three types of fronts could be obtained as explicit Wiener-Hopf solutions of the same discrete FPU problem allows one to identify the exact mathematical origin of the particular features of each class. To make the underlying Hamiltonian dynamics analytically transparent, we construct a minimal quasicontinuum approximation of the FPU model that captures all three classes of the fronts and reveals interrelation between them. This approximation is of higher order than conventional ones (KdV, Boussinesq) and involves mixed space-time derivatives. The proposed framework unifies previous attempts to classify the mechanical transition fronts as radiative, dispersive, topological, or compressive and categorizes them instead as irreducible types of dynamic lattice defects.

The Wiener-Hopf technique, its generalizations and applications: constructive and approximate methods

This paper reviews the modern state of the Wiener-Hopf factorization method and its generalizations. The main constructive results for matrix Wiener-Hopf problems are presented, approximate methods are outlined and the main areas of applications are mentioned. The aim of the paper is to offer an overview of the development of this method, and demonstrate the importance of bringing together pure and applied analysis to effectively employ the Wiener-Hopf technique.