  The non-monotonicity of the local speed of sound in dense matter at baryon
number densities much higher than the nuclear saturation density ($n_0 \approx
0.16\,$fm$^{-3}$) suggests the possible existence of a non-convex
thermodynamics which will lead to a non-convex dynamics. Here, we explore the
rich and complex dynamics that an equation of state (EoS) with non-convex
regions in the pressure-density plane may develop as a result of genuinely
relativistic effects, without a classical counterpart. To this end, we have
introduced a phenomenological EoS, whose parameters can be restricted heeding
to causality and thermodynamic stability constraints. This EoS shall be
regarded as a toy-model with which we may mimic realistic (and far more
complex) EoS of practical use in the realm of Relativistic Hydrodynamics.
