  We study the behavior of nonlinear waves in a two-dimensional medium with
density and stress relation that vary periodically in space. Efficient
approximate Riemann solvers are developed for the corresponding
variable-coefficient first-order hyperbolic system. We present direct numerical
simulations of this multiscale problem, focused on the propagation of a single
localized perturbation in media with strongly varying impedance. For the
conditions studied, we find little evidence of shock formation. Instead,
solutions consist primarily of solitary waves. These solitary waves are
observed to be stable over long times and to interact in a manner approximately
like solitons. The system considered has no dispersive terms; these solitary
waves arise due to the material heterogeneity, which leads to strong
reflections and effective dispersion.
