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

TL;DR

This paper investigates the propagation of transition fronts in bi-stable chains, revealing that exact solutions are non-robust but simplified models can accurately predict front velocities and are robust to potential shape variations.

Contribution

It demonstrates that simplified single-degree-of-freedom models can reliably predict front dynamics despite variations in on-site potential details.

Findings

Exact solutions are non-robust and sensitive to potential details.

Simplified models accurately predict front velocity and tail parameters.

Front shape is weakly affected by the specific form of the on-site potential.

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. We demonstrate that these solutions are essentially non-robust. Various approximations for the on-site potential with the same basic parameters (height and coordinate of the potential barrier, energy effect and distance between the potential wells) lead to substantially different front velocities. Besides, inclusion of even weak nearest neighbor nonlinearity drastically modifies the front structure and parameters. The energy concentration in the front zone leads to a dominance of the nonlinear term. 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 accurate prediction of the front velocity and parameters of the oscillatory tail. Moreover, it turns out that the solution is robust - exact shape of the on-site potential weakly effects 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.