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We address the problem of a front propagation in chains with a bi-stablenondegenerate on-site potential and a nonlinear gradient coupling. For ageneric nonlinear coupling, one encounters a special regime of transitions,characterized by extremely narrow fronts, far supersonic velocities ofpropagation and long waves in the oscillatory tail. This regime can bequalitatively associated with a shock wave. The front propagation can bedescribed with the help of a simple reduced-order model; the latter delivers akinetic law, which is almost not sensitive to fine details of the on-sitepotential. Besides, it is possible to predict all main characteristics of thetransition front, including its shape and frequency and amplitude of theoscillatory tail. The numerical results are in a good agreement with theanalytical predictions. The suggested approach allows one to consider theeffects of an external pre-load and on-site damping. When the damping ismoderate, the analysis remains in the frame of the reduced-order model. It ispossible to consider the solution for the front propagating in the damped chainas a perturbation of the undamped dynamics. This approach yield reasonablepredictions. When the damping is high, the transition front enters a completelydifferent asymptotic regime. The gradient nonlinearity generically turnsnegligible, and the propagating front converges to the exact solution obtainedfrom a simple linear continuous model.

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