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The study of hyperbolic waves involves various notions which helpcharacterise how these structures evolve. One important facet is the notion of\emph{genuine nonlinearity}, namely the ability for shocks and rarefactions toform instead of contact discontinuities. In the context of the WhithamModulation equations, this paper demonstrate that a loss of genuinenonlinearity leads to the appearance of a dispersive set of dynamics in theform of the modified Korteweg de-Vries equation governing the evolution of thewaves instead. Its form is universal in the sense that its coefficients can bewritten entirely using linear properties of the underlying waves such as theconservation laws and linear dispersion relation. This insight is applied totwo systems of physical interest, one an optical model and the other astratified hydrodynamics experiment, to demonstrate how it can be used toprovide insight into how waves in these systems evolve when genuinenonlinearity is lost.

Genuine Nonlinearity and its Connection to the Modified Korteweg - de Vries Equation in Phase Dynamics