Propagation of transition front in bi-stable nondegenerate chains: Model dependence and universality

Abstract

We consider a propagation of transition fronts in one-dimensional chains with bi-stable nondegenerate on-site potential. If one adopts linear coupling in the chain and piecewise linear on-site force, then it is possible to develop well-known exact solutions for the front and accompanying oscillatory tail. Our goal is to explore the sensitivity of these propagating-front solutions to fine details of the dynamical model. We numerically explore the linearly coupled chain with other shapes of the on-site potential with the same basic parameters (height of the potential barrier, energy effect and distance between the potential wells). Differences in the shapes of the on-site potential lead to a moderate modification of the front velocities. However, the front initiation may be substantially delayed due to possible localization of the initial excitation. Inclusion of a cubic nonlinearity in the nearest neighbor interaction drastically modifies the front structure and parameters. The energy concentration in the front zone leads to a dominance of the nonlinear term even if formally it is not too large. In this latter case, it turns out that the dynamics can be efficiently studied in terms of an equivalent model with a single degree of freedom. This estimation leads to an accurate prediction of the front velocity and the parameters of the oscillatory tail. Moreover, it turns out that the exact shape of the on-site potential almost does not affect the front parameters. This finding also conforms to the simplified model, since the latter invokes only the general shape characteristics of the on-site potential.