Supersonic Fronts and Pulses in a Lattice with Hardening-Softening Interactions

This talk is based on recent joint work with Lev Truskinovsky (ESPCI ParisTech). We consider a version of the classical Hamiltonian Fermi-Pasta-Ulam problem with nonlinear force-strain relation in which a hardening response is taken over by a softening regime above a critical strain value. We show that in addition to pulses (solitary waves) this discrete system also supports non-topological and dissipation-free fronts (kinks). Moreover, we demonstrate that these two types of supersonic traveling wave solutions belong to the same family. Within this family, solitary waves exist for continuous ranges of velocity that extend up to a limiting speed corresponding to kinks. As the kink velocity limit is approached from above or below, the solitary waves become progressively more broad and acquire the structure of a kink-antikink bundle. We investigate stability of the obtained solutions via Floquet analysis and direct numerical simulations. To motivate and support our study of the discrete problem we also analyze a quasicontinuum approximation with temporal dispersion. We show that this model captures the main effects observed in the discrete problem.