Nonlinear waves in granular crystals

We investigate excitations in one-dimensional granular crystals, i.e. chains of solid beads with nonlinear interaction forces. These chains have attracted attention because of their relative experimental simplicity and interesting dynamics. They can be modelled as chains of coupled oscillators in a model reminiscent of the celebrated Fermi-Pasta-Ulam (FPU) equations. The nonlinearity of the interaction leads to complex behaviour, including solitary waves and discrete breathers—periodic oscillations that are localised in space. We review the theory of granular chains starting from the contact law between two spheres and then consider some mathematical and numerical aspects of discrete breathers. We try to link some features of wave scattering by discrete breathers to Fano resonances, a resonant suppression of transmission observed in several other physical systems.