Propagation of transition fronts in nonlinear chains with non-degenerate on-site potentials

We address the problem of a front propagation in chains with a bi-stable nondegenerate on-site potential and a nonlinear gradient coupling. For a generic nonlinear coupling, one encounters a special regime of transitions, characterized by extremely narrow fronts, far supersonic velocities of propagation and long waves in the oscillatory tail. This regime can be qualitatively associated with a shock wave. The front propagation can be described with the help of a simple reduced-order model; the latter delivers a kinetic law, which is almost not sensitive to fine details of the on-site potential. Besides, it is possible to predict all main characteristics of the transition front, including its shape and frequency and amplitude of the oscillatory tail. The numerical results are in a good agreement with the analytical predictions. The suggested approach allows one to consider the effects of an external pre-load and on-site damping. When the damping is moderate, the analysis remains in the frame of the reduced-order model. It is possible to consider the solution for the front propagating in the damped chain as a perturbation of the undamped dynamics. This approach yield reasonable predictions. When the damping is high, the transition front enters a completely different asymptotic regime. The gradient nonlinearity generically turns negligible, and the propagating front converges to the exact solution obtained from a simple linear continuous model.