First-order continuum models for nonlinear dispersive waves in the granular crystal lattice

We derive and analyze, analytically and numerically, two first-order continuum models to approximate the nonlinear dynamics of granular crystal lattices, focusing specifically on solitary waves, periodic waves, and dispersive shock waves. The dispersive shock waves predicted by the two continuum models are studied using modulation theory, DSW fitting techniques, and direct numerical simulations. The PDE-based predictions show good agreement with the DSWs generated by the discrete model simulation of the granular lattice itself, even in cases where no precompression is present and the lattice is purely nonlinear. Such an effective description could prove useful for future, more analytically amenable approximations of the original lattice system.