AbstractHyperbolic partial differential equations are central to many areas of science/engineering and the foundations of many modern technologies including semiconductors plasmas, fluids, environmental science. A general property of nonlinear hyperbolic wave equations is the eventual formation of discontinuities (shocks) in the evolving signal and a central issue in shock propagation is the identification of physically acceptable shock solutions. In this presentation we consider discontinuous solutions to hyperbolic systems of conservation laws that violate classical entropy criteria. The main focus is on hyperbolic systems which are not genuinely nonlinear and as a consequence have a much richer and complex structure. A key observation is that under-compressive shocks may arise in these systems where the number of characteristics entering the discontinuity is now smaller than that imposed by the classical Lax shock inequalities. Such shocks have been shown to be sensitive to the form of the dissipative-dispersive mechanisms present in the physical system.