Kinks in chains with on-site bistable nondegenerate potential: Beyond traveling waves.

Abstract

This paper revisits the well-known transition fronts (kinks) in chains of coupled oscillators with nondegenerate on-site potentials. Usually, such transition fronts are considered in terms of traveling-wave solutions. We explore the loss of stability of such traveling waves. Generically, it corresponds to one of the common scenarios for fixed points of discrete maps. For example, one can encounter the quasiperiodic kink propagation (due to Hopf bifurcation), or the Feigenbaum cascade of period doublings, leading to a chaoticlike propagation pattern. The aforementioned scenarios show up, for instance, for triparabolic and φ^{4} on-site potentials. Numeric evidence suggests that the loss of stability occurs due to resonances between the frequency associated with the kink propagation, and the linear band gaps of the chain. Particular resonance mechanisms are model dependent. For the classical Atkinson-Cabrera model with a biparabolic on-site potential, the stability threshold is estimated by the simple means of linear algebra. The loss of stability in this model occurs through Hopf bifurcation. The results are in good agreement with numerical simulations.