Fronts on a lattice

Abstract

Motivated by a model of a system of many particles at low densities, we consider a lattice differential equation with two uniform steady states and we investigate the existence of travelling fronts connecting them. This leads to a two-point boundary-value problem for a nonlinear delay-differential equation. We replace the original parabolic nonlinearity by a piecewise-linear function, where explicit computations are possible. We find monotone and nonmonotone fronts. Finally we also describe all the fronts such that the $\alpha$-limit is the unstable uniform state. For different values of the wave speed $c$ of the front we find bounded and unbounded as well as eventually periodic orbits, i.e., orbits $u_c (x)$ that are periodic for $x\geqslant x_{\text{per}}(c)$ for some $x_{\text{per}}(c)\in{{\mathbb {R}}}$.