Genuine nonlinearity and its connection to the modified Korteweg-de Vries equation in phase dynamics. (English) Zbl 1479.35748
Summary: The study of hyperbolic waves involves various notions which help characterise how these structures evolve. One important facet is the notion of 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.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35C20 | Asymptotic expansions of solutions to PDEs |
| 35L65 | Hyperbolic conservation laws |
| 78A60 | Lasers, masers, optical bistability, nonlinear optics |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 76B55 | Internal waves for incompressible inviscid fluids |
| 76B70 | Stratification effects in inviscid fluids |
| 35Q35 | PDEs in connection with fluid mechanics |
Keywords:
Lagrangian dynamics; modulation; internal waves; optics; nonlinear waves; hyperbolic systemsReferences:
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