Abstract

First-order hyperbolic PDEs are generally conceived as the vanishing-viscosity limit of a dissipative system; their solutions generically exhibit shock singularities and entropy decay.  In contrast, dispersive wave equations often exhibit solitary waves and dispersive shock waves.  Strikingly, first-order systems with periodically-varying coefficients can exhibit solitary wave formation as well as shock formation.  I will review what is known about such systems, including our understanding of the effective dispersive mechanisms and criteria for dispersive- or shock-like behavior of solutions.  I will show examples of solitary wave formation and shock suppression in applications including elasticity, non-dispersive water wave models, and compressible gas dynamics.  We will see that, in an appropriate regime, these systems seem to behave like near-integrable dispersive wave models.  Many interesting questions about them remain to be explored.