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

TL;DR

This paper links the loss of genuine nonlinearity in hyperbolic wave systems to the emergence of modified Korteweg-de Vries dynamics, providing a universal framework to understand wave evolution in physical models.

Contribution

It demonstrates that loss of genuine nonlinearity leads to dispersive dynamics described by the mKdV equation, with coefficients derived from linear wave properties.

Findings

Loss of genuine nonlinearity results in mKdV-type wave behavior.

The mKdV coefficients depend solely on linear properties of the waves.

Application to optical and hydrodynamic systems shows practical relevance.

Abstract

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